126 results
Search Results
2. On a new stochastic model for cascading failures
- Author
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Hyunju Lee
- Subjects
Statistics and Probability ,Stochastic modelling ,General Mathematics ,010102 general mathematics ,Residual ,01 natural sciences ,Stochastic ordering ,Cascading failure ,010104 statistics & probability ,Control theory ,Component (UML) ,Life test ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In this paper, to model cascading failures, a new stochastic failure model is proposed. In a system subject to cascading failures, after each failure of the component, the remaining component suffers from increased load or stress. This results in shortened residual lifetimes of the remaining components. In this paper, to model this effect, the concept of the usual stochastic order is employed along with the accelerated life test model, and a new general class of stochastic failure models is generated.
- Published
- 2020
3. On moderate deviations in Poisson approximation
- Author
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Qingwei Liu and Aihua Xia
- Subjects
Statistics and Probability ,Random graph ,Matching (graph theory) ,Distribution (number theory) ,General Mathematics ,Probability (math.PR) ,010102 general mathematics ,Poisson distribution ,01 natural sciences ,Birthday problem ,Normal distribution ,010104 statistics & probability ,symbols.namesake ,FOS: Mathematics ,Rare events ,symbols ,Applied mathematics ,Moderate deviations ,0101 mathematics ,Statistics, Probability and Uncertainty ,Primary 60F05, secondary 60E15 ,Mathematics - Probability ,Mathematics - Abstract
In this paper, we first use the distribution of the number of records to demonstrate that the right tail probabilities of counts of rare events are generally better approximated by the right tail probabilities of Poisson distribution than {those} of normal distribution. We then show the moderate deviations in Poisson approximation generally require an adjustment and, with suitable adjustment, we establish better error estimates of the moderate deviations in Poisson approximation than those in \cite{CFS}. Our estimates contain no unspecified constants and are easy to apply. We illustrate the use of the theorems in six applications: Poisson-binomial distribution, matching problem, occupancy problem, birthday problem, random graphs and 2-runs. The paper complements the works of \cite{CC92,BCC95,CFS}., 29 pages and 5 figures
- Published
- 2020
4. Martingale decomposition of an L2 space with nonlinear stochastic integrals
- Author
-
Clarence Simard
- Subjects
Statistics and Probability ,Optimization problem ,General Mathematics ,010102 general mathematics ,Stochastic calculus ,01 natural sciences ,010104 statistics & probability ,Nonlinear system ,Integrator ,Bounded function ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Lp space ,Martingale (probability theory) ,Brownian motion ,Mathematics - Abstract
This paper generalizes the Kunita–Watanabe decomposition of an $L^2$ space. The generalization comes from using nonlinear stochastic integrals where the integrator is a family of continuous martingales bounded in $L^2$ . This result is also the solution of an optimization problem in $L^2$ . First, martingales are assumed to be stochastic integrals. Then, to get the general result, it is shown that the regularity of the family of martingales with respect to its spatial parameter is inherited by the integrands in the integral representation of the martingales. Finally, an example showing how the results of this paper, with the Clark–Ocone formula, can be applied to polynomial functions of Brownian integrals.
- Published
- 2019
5. Approximate lumpability for Markovian agent-based models using local symmetries
- Author
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Wasiur R. KhudaBukhsh, Arnab Auddy, Heinz Koeppl, and Yann Disser
- Subjects
Statistics and Probability ,Random graph ,Markov chain ,General Mathematics ,Probability (math.PR) ,Lumpability ,Neighbourhood (graph theory) ,Markov process ,020206 networking & telecommunications ,0102 computer and information sciences ,02 engineering and technology ,01 natural sciences ,symbols.namesake ,60J28 ,010201 computation theory & mathematics ,Approximation error ,Local symmetry ,FOS: Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,State space ,Applied mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Probability ,Mathematics - Abstract
We study a Markovian agent-based model (MABM) in this paper. Each agent is endowed with a local state that changes over time as the agent interacts with its neighbours. The neighbourhood structure is given by a graph. In a recent paper [Simon et al. 2011], the authors used the automorphisms of the underlying graph to generate a lumpable partition of the joint state space ensuring Markovianness of the lumped process for binary dynamics. However, many large random graphs tend to become asymmetric rendering the automorphism-based lumping approach ineffective as a tool of model reduction. In order to mitigate this problem, we propose a lumping method based on a notion of local symmetry, which compares only local neighbourhoods of vertices. Since local symmetry only ensures approximate lumpability, we quantify the approximation error by means of Kullback-Leibler divergence rate between the original Markov chain and a lifted Markov chain. We prove the approximation error decreases monotonically. The connections to fibrations of graphs are also discussed., Comment: 28 pages, 4 figures
- Published
- 2019
6. Comparison results for M/G/1 queues with waiting and sojourn time deadlines
- Author
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Yoshiaki Inoue
- Subjects
Statistics and Probability ,Waiting time ,Discrete mathematics ,021103 operations research ,Service time ,General Mathematics ,0211 other engineering and technologies ,Comparison results ,02 engineering and technology ,01 natural sciences ,010104 statistics & probability ,M/G/1 queue ,0101 mathematics ,Statistics, Probability and Uncertainty ,Queue ,Mathematics - Abstract
This paper considers two variants of M/G/1 queues with impatient customers, which are denoted by M/G/1+Gw and M/G/1+Gs. In the M/G/1+Gw queue customers have deadlines for their waiting times, and they leave the system immediately if their services do not start before the expiration of their deadlines. On the other hand, in the M/G/1+Gs queue customers have deadlines for their sojourn times, where customers in service also immediately leave the system when their deadlines expire. In this paper we derive comparison results for performance measures of these models. In particular, we show that if the service time distribution is new better than used in expectation, then the loss probability in the M/G/1+Gs queue is greater than that in the M/G/1+Gw queue.
- Published
- 2019
7. A note on the simulation of the Ginibre point process
- Author
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Laurent Decreusefond, Anaïs Vergne, Ian Flint, Data, Intelligence and Graphs (DIG), Laboratoire Traitement et Communication de l'Information (LTCI), Institut Mines-Télécom [Paris] (IMT)-Télécom Paris-Institut Mines-Télécom [Paris] (IMT)-Télécom Paris, Département Informatique et Réseaux (INFRES), Télécom ParisTech, Mathématiques discrètes, Codage et Cryptographie (MC2), and Réseaux, Mobilité et Services (RMS)
- Subjects
Statistics and Probability ,Property (philosophy) ,Distribution (number theory) ,General Mathematics ,02 engineering and technology ,point process simulation ,01 natural sciences ,Point process ,Computer Science::Hardware Architecture ,010104 statistics & probability ,Determinantal point process ,0202 electrical engineering, electronic engineering, information engineering ,0101 mathematics ,ComputingMilieux_MISCELLANEOUS ,Mathematics ,60G60 ,Ginibre point process ,Plane (geometry) ,010102 general mathematics ,15A52 ,020206 networking & telecommunications ,Algebra ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,60K35 ,60G55 ,Statistics, Probability and Uncertainty ,Complex plane ,Random matrix - Abstract
The Ginibre point process (GPP) is one of the main examples of determinantal point processes on the complex plane. It is a recurring distribution of random matrix theory as well as a useful model in applied mathematics. In this paper we briefly overview the usual methods for the simulation of the GPP. Then we introduce a modified version of the GPP which constitutes a determinantal point process more suited for certain applications, and we detail its simulation. This modified GPP has the property of having a fixed number of points and having its support on a compact subset of the plane. See Decreusefond et al. (2013) for an extended version of this paper.
- Published
- 2015
8. Partially informed investors: hedging in an incomplete market with default
- Author
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Paola Tardelli
- Subjects
Statistics and Probability ,exponential utility ,General Mathematics ,backward stochastic differential equation ,93E11 ,01 natural sciences ,default time ,Unobservable ,010104 statistics & probability ,Stochastic differential equation ,Order (exchange) ,Bellman equation ,Incomplete markets ,Econometrics ,49L20 ,Asset (economics) ,0101 mathematics ,Mathematics ,dynamic programming ,Stochastic control ,Actuarial science ,Optimal investment ,010102 general mathematics ,filtering ,93E03 ,Exponential utility ,Statistics, Probability and Uncertainty - Abstract
In a defaultable market, an investor trades having only partial information about the behavior of the market. Taking into account the intraday stock movements, the risky asset prices are modelled by marked point processes. Their dynamics depend on an unobservable process, representing the amount of news reaching the market. This is a marked point process, which may have common jump times with the risky asset price processes. The problem of hedging a defaultable claim is studied. In order to discuss all these topics, in this paper we examine stochastic control problems using backward stochastic differential equations (BSDEs) and filtering techniques. The goal of this paper is to construct a sequence of functions converging to the value function, each of these is the unique solution of a suitable BSDE.
- Published
- 2015
9. The limiting failure rate for a convolution of life distributions
- Author
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Henry W. Block, Thomas H. Savits, and Naftali A. Langberg
- Subjects
failure rate function ,Statistics and Probability ,education.field_of_study ,decreasing failure rate ,Component (thermodynamics) ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Population ,Block (permutation group theory) ,Monotonic function ,Failure rate ,Limiting ,Reliability ,01 natural sciences ,increasing failure rate ,Convolution ,62N05 ,010104 statistics & probability ,convolution ,60K10 ,0101 mathematics ,Statistics, Probability and Uncertainty ,education ,Mathematics - Abstract
In this paper we investigate the limiting behavior of the failure rate for the convolution of two or more life distributions. In a previous paper on mixtures, Block, Mi and Savits (1993) showed that the failure rate behaves like the limiting behavior of the strongest component. We show a similar result here for convolutions. We also show by example that unlike a mixture population, the ultimate direction of monotonicity does not necessarily follow that of the strongest component.
- Published
- 2015
10. Quasistochastic matrices and Markov renewal theory
- Author
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Gerold Alsmeyer
- Subjects
Statistics and Probability ,Markov kernel ,General Mathematics ,perpetuity ,01 natural sciences ,age-dependent multitype branching process ,010104 statistics & probability ,Matrix (mathematics) ,random difference equation ,60K05 ,Markov renewal process ,Quasistochastic matrix ,60J45 ,Nonnegative matrix ,Renewal theory ,Markov renewal equation ,0101 mathematics ,Eigenvalues and eigenvectors ,Mathematics ,Discrete mathematics ,Markov chain ,010102 general mathematics ,Stochastic matrix ,Stone-type decomposition ,60K15 ,Markov renewal theorem ,spread out ,60J10 ,Statistics, Probability and Uncertainty ,Markov random walk - Abstract
Let 𝓈 be a finite or countable set. Given a matrix F = (F ij ) i,j∈𝓈 of distribution functions on R and a quasistochastic matrix Q = (q ij ) i,j∈𝓈 , i.e. an irreducible nonnegative matrix with maximal eigenvalue 1 and associated unique (modulo scaling) positive left and right eigenvectors u and v, the matrix renewal measure ∑ n≥0 Q n ⊗ F *n associated with Q ⊗ F := (q ij F ij ) i,j∈𝓈 (see below for precise definitions) and a related Markov renewal equation are studied. This was done earlier by de Saporta (2003) and Sgibnev (2006, 2010) by drawing on potential theory, matrix-analytic methods, and Wiener-Hopf techniques. In this paper we describe a probabilistic approach which is quite different and starts from the observation that Q ⊗ F becomes an ordinary semi-Markov matrix after a harmonic transform. This allows us to relate Q ⊗ F to a Markov random walk {(M n , S n )} n≥0 with discrete recurrent driving chain {M n } n≥0. It is then shown that renewal theorems including a Choquet-Deny-type lemma may be easily established by resorting to standard renewal theory for ordinary random walks. The paper concludes with two typical examples.
- Published
- 2014
11. Asymptotic Bounds for the Distribution of the Sum of Dependent Random Variables
- Author
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Ruodu Wang
- Subjects
Statistics and Probability ,General Mathematics ,Structure (category theory) ,Value (computer science) ,91E30 ,01 natural sciences ,value at risk ,Combinatorics ,010104 statistics & probability ,0502 economics and business ,60E05 ,Limit (mathematics) ,0101 mathematics ,Mathematics ,Discrete mathematics ,050208 finance ,05 social sciences ,Expected shortfall ,Distribution (mathematics) ,Dependence bound ,complete mixability ,modeling uncertainty ,60E15 ,Marginal distribution ,Statistics, Probability and Uncertainty ,Random variable ,Value at risk - Abstract
Suppose that X 1, …, X n are random variables with the same known marginal distribution F but unknown dependence structure. In this paper we study the smallest possible value of P(X 1 + · · · + X n < s) over all possible dependence structures, denoted by m n,F (s). We show that m n,F (ns) → 0 for s no more than the mean of F under weak assumptions. We also derive a limit of m n,F (ns) for any s ∈ R with an error of at most n -1/6 for general continuous distributions. An application of our result to risk management confirms that the worst-case value at risk is asymptotically equivalent to the worst-case expected shortfall for risk aggregation with dependence uncertainty. In the last part of this paper we present a dual presentation of the theory of complete mixability and give dual proofs of theorems in the literature on this concept.
- Published
- 2014
12. Optimal allocation of relevations in coherent systems
- Author
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Jiandong Zhang, Yiying Zhang, and Rongfang Yan
- Subjects
Statistics and Probability ,Mathematical optimization ,Order (business) ,General Mathematics ,Optimal allocation ,Statistics, Probability and Uncertainty ,Stochastic ordering ,Mathematics - Abstract
In this paper we study the allocation problem of relevations in coherent systems. The optimal allocation strategies are obtained by implementing stochastic comparisons of different policies according to the usual stochastic order and the hazard rate order. As special cases of relevations, the load-sharing and minimal repair policies are further investigated. Sufficient (and necessary) conditions are established for various stochastic orderings. Numerical examples are also presented as illustrations.
- Published
- 2021
13. General drawdown of general tax model in a time-homogeneous Markov framework
- Author
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Shu Li, Florin Avram, and Bin Li
- Subjects
Statistics and Probability ,Markov chain ,General Mathematics ,Mathematical finance ,Probability (math.PR) ,Markov process ,Regret ,CUSUM ,Lévy process ,symbols.namesake ,FOS: Mathematics ,symbols ,Drawdown (economics) ,Optimal stopping ,Statistics, Probability and Uncertainty ,Mathematical economics ,Mathematics - Probability ,Mathematics - Abstract
Drawdown/regret times feature prominently in optimal stopping problems, in statistics (CUSUM procedure), and in mathematical finance (Russian options). Recently it was discovered that a first passage theory with more general drawdown times, which generalize classic ruin times, may be explicitly developed for spectrally negative Lévy processes [9, 20]. In this paper we further examine the general drawdown-related quantities in the (upward skip-free) time-homogeneous Markov process, and then in its (general) tax process by noticing the pathwise connection between general drawdown and the tax process.
- Published
- 2021
14. On transform orders for largest claim amounts
- Author
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Yiying Zhang
- Subjects
Statistics and Probability ,Set (abstract data type) ,Hazard (logic) ,Skewness ,General Mathematics ,Regular polygon ,Applied mathematics ,Statistics, Probability and Uncertainty ,Star (graph theory) ,Majorization ,Upper and lower bounds ,Mathematics ,Exponential function - Abstract
This paper investigates the ordering properties of largest claim amounts in heterogeneous insurance portfolios in the sense of some transform orders, including the convex transform order and the star order. It is shown that the largest claim amount from a set of independent and heterogeneous exponential claims is more skewed than that from a set of independent and homogeneous exponential claims in the sense of the convex transform order. As a result, a lower bound for the coefficient of variation of the largest claim amount is established without any restrictions on the parameters of the distributions of claim severities. Furthermore, sufficient conditions are presented to compare the skewness of the largest claim amounts from two sets of independent multiple-outlier scaled claims according to the star order. Some comparison results are also developed for the multiple-outlier proportional hazard rates claims. Numerical examples are presented to illustrate these theoretical results.
- Published
- 2021
15. Computing minimal signature of coherent systems through matrix-geometric distributions
- Author
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Fatih Tank and Serkan Eryilmaz
- Subjects
Statistics and Probability ,Sequence ,Matrix (mathematics) ,General Mathematics ,Computation ,Binary number ,Statistics, Probability and Uncertainty ,Representation (mathematics) ,Random variable ,Algorithm ,Signature (logic) ,Expression (mathematics) ,Mathematics - Abstract
Signatures are useful in analyzing and evaluating coherent systems. However, their computation is a challenging problem, especially for complex coherent structures. In most cases the reliability of a binary coherent system can be linked to a tail probability associated with a properly defined waiting time random variable in a sequence of binary trials. In this paper we present a method for computing the minimal signature of a binary coherent system. Our method is based on matrix-geometric distributions. First, a proper matrix-geometric random variable corresponding to the system structure is found. Second, its probability generating function is obtained. Finally, the companion representation for the distribution of matrix-geometric distribution is used to obtain a matrix-based expression for the minimal signature of the coherent system. The results are also extended to a system with two types of components.
- Published
- 2021
16. Stochastic properties of generalized finite mixture models with dependent components
- Author
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Narayanaswamy Balakrishnan and Ebrahim Amini-Seresht
- Subjects
Statistics and Probability ,Independent and identically distributed random variables ,Finite mixture ,General Mathematics ,Applied mathematics ,Statistics, Probability and Uncertainty ,Mixture model ,Stochastic ordering ,Mathematics - Abstract
In this paper we consider a new generalized finite mixture model formed by dependent and identically distributed (d.i.d.) components. We then establish results for the comparisons of lifetimes of two such generalized finite mixture models in two different cases: (i) when the two mixture models are formed from two random vectors $\textbf{X}$ and $\textbf{Y}$ but with the same weights, and (ii) when the two mixture models are formed with the same random vectors but with different weights. Because the lifetimes of k-out-of-n systems and coherent systems are special cases of the mixture model considered, we used the established results to compare the lifetimes of k-out-of-n systems and coherent systems with respect to the reversed hazard rate and hazard rate orderings.
- Published
- 2021
17. Risk-sensitive average continuous-time Markov decision processes with unbounded transition and cost rates
- Author
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Yonghui Huang and Xin Guo
- Subjects
Statistics and Probability ,0209 industrial biotechnology ,Sequence ,General Mathematics ,Multiplicative function ,02 engineering and technology ,State (functional analysis) ,01 natural sciences ,Dynamic programming ,010104 statistics & probability ,020901 industrial engineering & automation ,Applied mathematics ,Countable set ,Markov decision process ,Uniqueness ,0101 mathematics ,Statistics, Probability and Uncertainty ,Finite set ,Mathematics - Abstract
This paper considers risk-sensitive average optimization for denumerable continuous-time Markov decision processes (CTMDPs), in which the transition and cost rates are allowed to be unbounded, and the policies can be randomized history dependent. We first derive the multiplicative dynamic programming principle and some new facts for risk-sensitive finite-horizon CTMDPs. Then, we establish the existence and uniqueness of a solution to the risk-sensitive average optimality equation (RS-AOE) through the results for risk-sensitive finite-horizon CTMDPs developed here, and also prove the existence of an optimal stationary policy via the RS-AOE. Furthermore, for the case of finite actions available at each state, we construct a sequence of models of finite-state CTMDPs with optimal stationary policies which can be obtained by a policy iteration algorithm in a finite number of iterations, and prove that an average optimal policy for the case of infinitely countable states can be approximated by those of the finite-state models. Finally, we illustrate the conditions and the iteration algorithm with an example.
- Published
- 2021
18. Sensitivity of mean-field fluctuations in Erlang loss models with randomized routing
- Author
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Ravi R. Mazumdar and Thirupathaiah Vasantam
- Subjects
FOS: Computer and information sciences ,Statistics and Probability ,General Mathematics ,02 engineering and technology ,Blocking (statistics) ,01 natural sciences ,010104 statistics & probability ,60F17, 60M20, 68M20 ,Server ,FOS: Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,Applied mathematics ,Sensitivity (control systems) ,Limit (mathematics) ,0101 mathematics ,Computer Science::Operating Systems ,Queue ,Mathematics ,Central limit theorem ,Computer Science - Performance ,Probability (math.PR) ,020206 networking & telecommunications ,Erlang (unit) ,Exponential function ,Computer Science::Performance ,Performance (cs.PF) ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Statistics, Probability and Uncertainty ,Mathematics - Probability - Abstract
In this paper, we study a large system of $N$ servers each with capacity to process at most $C$ simultaneous jobs and an incoming job is routed to a server if it has the lowest occupancy amongst $d$ (out of N) randomly selected servers. A job that is routed to a server with no vacancy is assumed to be blocked and lost. Such randomized policies are referred to JSQ(d) (Join the Shortest Queue out of $d$) policies. Under the assumption that jobs arrive according to a Poisson process with rate $N\lambda^{(N)}$ where $\lambda^{(N)}=\sigma-\frac{\beta}{\sqrt{N}}$, $\sigma\in\mb{R}_+$ and $\beta\in\mb{R}$, we establish functional central limit theorems (FCLTs) for the fluctuation process in both the transient and stationary regimes when service time distributions are exponential. In particular, we show that the limit is an Ornstein-Uhlenbeck process whose mean and variance depend on the mean-field of the considered model. Using this, we obtain approximations to the blocking probabilities for large $N$, where we can precisely estimate the accuracy of first-order approximations., Comment: 29 pages
- Published
- 2021
19. The martingale comparison method for Markov processes
- Author
-
Benedikt Köpfer and Ludger Rüschendorf
- Subjects
Statistics and Probability ,Property (philosophy) ,Process (engineering) ,General Mathematics ,010102 general mathematics ,Banach space ,Markov process ,Characterization (mathematics) ,01 natural sciences ,Set (abstract data type) ,010104 statistics & probability ,symbols.namesake ,Transfer (group theory) ,Mathematics::Probability ,symbols ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Martingale (probability theory) ,Mathematics - Abstract
Comparison results for Markov processes with respect to function-class-induced (integral) stochastic orders have a long history. The most general results so far for this problem have been obtained based on the theory of evolution systems on Banach spaces. In this paper we transfer the martingale comparison method, known for the comparison of semimartingales to Markovian semimartingales, to general Markov processes. The basic step of this martingale approach is the derivation of the supermartingale property of the linking process, giving a link between the processes to be compared. This property is achieved using the characterization of Markov processes by the associated martingale problem in an essential way. As a result, the martingale comparison method gives a comparison result for Markov processes under a general alternative but related set of regularity conditions compared to the evolution system approach.
- Published
- 2021
20. Diffusion approximations for randomly arriving expert opinions in a financial market with Gaussian drift
- Author
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Jörn Sass, Ralf Wunderlich, and Dorothee Westphal
- Subjects
Statistics and Probability ,050208 finance ,General Mathematics ,Gaussian ,05 social sciences ,Ornstein–Uhlenbeck process ,Filter (signal processing) ,Kalman filter ,01 natural sciences ,Unobservable ,010104 statistics & probability ,symbols.namesake ,0502 economics and business ,symbols ,Applied mathematics ,Limit (mathematics) ,0101 mathematics ,Statistics, Probability and Uncertainty ,Portfolio optimization ,Diffusion (business) ,Mathematics - Abstract
This paper investigates a financial market where stock returns depend on an unobservable Gaussian mean reverting drift process. Information on the drift is obtained from returns and randomly arriving discrete-time expert opinions. Drift estimates are based on Kalman filter techniques. We study the asymptotic behavior of the filter for high-frequency experts with variances that grow linearly with the arrival intensity. The derived limit theorems state that the information provided by discrete-time expert opinions is asymptotically the same as that from observing a certain diffusion process. These diffusion approximations are extremely helpful for deriving simplified approximate solutions of utility maximization problems.
- Published
- 2021
21. On geometric and algebraic transience for block-structured Markov chains
- Author
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Xiuqin Li, Wendi Li, and Yuanyuan Liu
- Subjects
Statistics and Probability ,Pure mathematics ,Queueing theory ,Markov chain ,General Mathematics ,010102 general mathematics ,Type (model theory) ,01 natural sciences ,Stability (probability) ,010104 statistics & probability ,System parameters ,0101 mathematics ,Statistics, Probability and Uncertainty ,Variety (universal algebra) ,Algebraic number ,Mathematics ,Block (data storage) - Abstract
Block-structured Markov chains model a large variety of queueing problems and have many important applications in various areas. Stability properties have been well investigated for these Markov chains. In this paper we will present transient properties for two specific types of block-structured Markov chains, including M/G/1 type and GI/M/1 type. Necessary and sufficient conditions in terms of system parameters are obtained for geometric transience and algebraic transience. Possible extensions of the results to continuous-time Markov chains are also included.
- Published
- 2020
22. Maximizing the pth moment of the exit time of planar brownian motion from a given domain
- Author
-
Maher Boudabra and Greg Markowsky
- Subjects
Statistics and Probability ,Coupling ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,Domain (mathematical analysis) ,Moment (mathematics) ,010104 statistics & probability ,Planar ,Conformal symmetry ,Point (geometry) ,0101 mathematics ,Statistics, Probability and Uncertainty ,Brownian motion ,Mathematics - Abstract
In this paper we address the question of finding the point which maximizes the pth moment of the exit time of planar Brownian motion from a given domain. We present a geometrical method for excluding parts of the domain from consideration which makes use of a coupling argument and the conformal invariance of Brownian motion. In many cases the maximizing point can be localized to a relatively small region. Several illustrative examples are presented.
- Published
- 2020
23. Perron–Frobenius theory for kernels and Crump–Mode–Jagers processes with macro-individuals
- Author
-
Serik Sagitov
- Subjects
Statistics and Probability ,Pure mathematics ,Markov kernel ,General Mathematics ,media_common.quotation_subject ,Probabilistic logic ,Structure (category theory) ,Atom (order theory) ,Infinity ,Interpretation (model theory) ,Limit (mathematics) ,Statistics, Probability and Uncertainty ,Kernel (category theory) ,Mathematics ,media_common - Abstract
Perron–Frobenius theory developed for irreducible non-negative kernels deals with so-called R-positive recurrent kernels. If the kernel M is R-positive recurrent, then the main result determines the limit of the scaled kernel iterations $R^nM^n$ as $n\to\infty$ . In Nummelin (1984) this important result is proven using a regeneration method whose major focus is on M having an atom. In the special case when $M=P$ is a stochastic kernel with an atom, the regeneration method has an elegant explanation in terms of an associated split chain. In this paper we give a new probabilistic interpretation of the general regeneration method in terms of multi-type Galton–Watson processes producing clusters of particles. Treating clusters as macro-individuals, we arrive at a single-type Crump–Mode–Jagers process with a naturally embedded renewal structure.
- Published
- 2020
24. Flooding and diameter in general weighted random graphs
- Author
-
Thomas Mountford and Jacques Saliba
- Subjects
Statistics and Probability ,General Mathematics ,media_common.quotation_subject ,flooding time ,01 natural sciences ,Combinatorics ,010104 statistics & probability ,FOS: Mathematics ,0101 mathematics ,diameter ,Mathematics ,media_common ,60C05, 05C80, 90B15 ,Random graph ,first passage percolation ,Probability (math.PR) ,010102 general mathematics ,1st passage percolation ,First passage percolation ,Infinity ,continuous branching process ,configuration model ,Flooding (computer networking) ,Exponential function ,Vertex (geometry) ,Weight distribution ,Graph (abstract data type) ,Statistics, Probability and Uncertainty ,Mathematics - Probability ,MathematicsofComputing_DISCRETEMATHEMATICS - Abstract
In this paper we study first passage percolation on a random graph model, the configuration model. We first introduce the notions of weighted diameter, which is the maximum of the weighted lengths of all optimal paths between any two vertices in the graph, and the flooding time, which represents the time (weighted length) needed to reach all the vertices in the graph starting from a uniformly chosen vertex. Our result consists in describing the asymptotic behavior of the diameter and the flooding time, as the number of vertices n tends to infinity, in the case where the weight distribution G has an exponential tail behavior, and proving that this category of distributions is the largest possible for which the asymptotic behavior holds.
- Published
- 2020
25. On the behavior of the failure rate and reversed failure rate in engineering systems
- Author
-
Mahdi Tavangar
- Subjects
Statistics and Probability ,021103 operations research ,General Mathematics ,Cumulative distribution function ,Order statistic ,0211 other engineering and technologies ,Failure rate ,02 engineering and technology ,01 natural sciences ,Stochastic ordering ,010104 statistics & probability ,System failure ,Statistics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Random variable ,Reliability (statistics) ,Mathematics - Abstract
In this paper the behaviour of the failure rate and reversed failure rate of an n-component coherent system is studied, where it is assumed that the lifetimes of the components are independent and have a common cumulative distribution function F. Sufficient conditions are provided under which the system failure rate is increasing and the corresponding reversed failure rate is decreasing. We also study the stochastic and ageing properties of doubly truncated random variables for coherent systems.
- Published
- 2020
26. On the extension of signature-based representations for coherent systems with dependent non-exchangeable components
- Author
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Jorge Navarro and Juan Fernández-Sánchez
- Subjects
Statistics and Probability ,Discrete mathematics ,Independent and identically distributed random variables ,Class (set theory) ,General Mathematics ,Reliability (computer networking) ,010102 general mathematics ,Copula (linguistics) ,Extension (predicate logic) ,01 natural sciences ,Signature (logic) ,010104 statistics & probability ,0101 mathematics ,Statistics, Probability and Uncertainty ,Representation (mathematics) ,Mathematics - Abstract
The signature representation shows that the reliability of the system is a mixture of the reliability functions of the k-out-of-n systems. The first representation was obtained for systems with independent and identically distributed (IID) components and after it was extended to exchangeable (EXC) components. The purpose of the present paper is to extend it to the class of systems with identically distributed (ID) components which have a diagonal-dependent copula. We prove that this class is much larger than the class with EXC components. This extension is used to compare systems with non-EXC components.
- Published
- 2020
27. Optimal stopping for measure-valued piecewise deterministic Markov processes
- Author
-
Maud Joubaud, Bertrand Cloez, Benoîte de Saporta, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Mathématiques, Informatique et STatistique pour l'Environnement et l'Agronomie (MISTEA), Institut National de la Recherche Agronomique (INRA)-Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro), Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro), Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), Region Occitanie, Region Ile-de-France, French National Research Agency (ANR), Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro)-Institut National de la Recherche Agronomique (INRA), and de Saporta, Benoîte
- Subjects
Statistics and Probability ,[MATH.MATH-PR] Mathematics [math]/Probability [math.PR] ,Mathematical optimization ,General Mathematics ,Population ,Markov process ,01 natural sciences ,Measure (mathematics) ,measure space ,010104 statistics & probability ,03 medical and health sciences ,symbols.namesake ,Bellman equation ,FOS: Mathematics ,population dynamics ,Optimal stopping ,0101 mathematics ,education ,030304 developmental biology ,Mathematics ,dynamic programming ,0303 health sciences ,Sequence ,education.field_of_study ,Markov processes ,Probability (math.PR) ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Dynamic programming ,optimal stopping ,symbols ,Piecewise ,Statistics, Probability and Uncertainty ,Mathematics - Probability - Abstract
This paper investigates the random horizon optimal stopping problem for measure-valued piecewise deterministic Markov processes (PDMPs). This is motivated by population dynamics applications, when one wants to monitor some characteristics of the individuals in a small population. The population and its individual characteristics can be represented by a point measure. We first define a PDMP on a space of locally finite measures. Then we define a sequence of random horizon optimal stopping problems for such processes. We prove that the value function of the problems can be obtained by iterating some dynamic programming operator. Finally we prove via a simple counter-example that controlling the whole population is not equivalent to controlling a random lineage.
- Published
- 2020
28. Generalized stacked contact process with variable host fitness
- Author
-
Eric Foxall and Nicolas Lanchier
- Subjects
Statistics and Probability ,Contact process ,education.field_of_study ,Host (biology) ,General Mathematics ,Probability (math.PR) ,Population ,Integer lattice ,Forest-fire model ,Spin system ,01 natural sciences ,010305 fluids & plasmas ,Variable (computer science) ,60K35 ,Spatial model ,0103 physical sciences ,FOS: Mathematics ,Statistics, Probability and Uncertainty ,010306 general physics ,Biological system ,education ,Mathematics - Probability ,Mathematics - Abstract
The stacked contact process is a three-state spin system that describes the co-evolution of a population of hosts together with their symbionts. In a nutshell, the hosts evolve according to a contact process while the symbionts evolve according to a contact process on the dynamic subset of the lattice occupied by the host population, indicating that the symbiont can only live within a host. This paper is concerned with a generalization of this system in which the symbionts may affect the fitness of the hosts by either decreasing (pathogen) or increasing (mutualist) their birth rate. Standard coupling arguments are first used to compare the process with other interacting particle systems and deduce the long-term behavior of the host-symbiont system in several parameter regions. The mean-field approximation of the process is also studied in detail and compared to the spatial model. Our main result focuses on the case where unassociated hosts have a supercritical birth rate whereas hosts associated to a pathogen have a subcritical birth rate. In this case, the mean-field model predicts coexistence of the hosts and their pathogens provided the infection rate is large enough. For the spatial model, however, only the hosts survive on the one-dimensional integer lattice., 23 pages, 4 figures
- Published
- 2020
29. The alpha-mixture of survival functions
- Author
-
Nader Ebrahimi, Ehsan S. Soofi, and Majid Asadi
- Subjects
Statistics and Probability ,Property (philosophy) ,General Mathematics ,020206 networking & telecommunications ,Failure rate ,02 engineering and technology ,01 natural sciences ,Stochastic ordering ,Interpretation (model theory) ,010104 statistics & probability ,Alpha (programming language) ,Monotone polygon ,Constant elasticity of substitution ,0202 electrical engineering, electronic engineering, information engineering ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
This paper presents a flexible family which we call the $\alpha$ -mixture of survival functions. This family includes the survival mixture, failure rate mixture, models that are stochastically closer to each of these conventional mixtures, and many other models. The $\alpha$ -mixture is endowed by the stochastic order and uniquely possesses a mathematical property known in economics as the constant elasticity of substitution, which provides an interpretation for $\alpha$ . We study failure rate properties of this family and establish closures under monotone failure rates of the mixture’s components. Examples include potential applications for comparing systems.
- Published
- 2019
30. Persistence probability of a random polynomial arising from evolutionary game theory
- Author
-
Viet Viet Hung Pham, Van Hao Can, and Manh Hong Duong
- Subjects
Statistics and Probability ,Computer Science::Computer Science and Game Theory ,General Mathematics ,Evolutionary game theory ,FOS: Physical sciences ,Dynamical Systems (math.DS) ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,Applied mathematics ,Asymptotic formula ,Mathematics - Dynamical Systems ,0101 mathematics ,Quantitative Biology - Populations and Evolution ,Real line ,Gaussian process ,Mathematical Physics ,Mathematics ,Equilibrium point ,Sequence ,Probability (math.PR) ,010102 general mathematics ,Populations and Evolution (q-bio.PE) ,Mathematical Physics (math-ph) ,FOS: Biological sciences ,symbols ,Key (cryptography) ,Statistics, Probability and Uncertainty ,Persistence (discontinuity) ,Mathematics - Probability ,Analysis of PDEs (math.AP) - Abstract
In this paper, we obtain an asymptotic formula for the persistence probability in the positive real line of a random polynomial arising from evolutionary game theory. It corresponds to the probability that a multi-player two-strategy random evolutionary game has no internal equilibria. The key ingredient is to approximate the sequence of random polynomials indexed by their degrees by an appropriate centered stationary Gaussian process., revised version
- Published
- 2019
31. Sums of standard uniform random variables
- Author
-
Bin Wang, Tiantian Mao, and Ruodu Wang
- Subjects
Statistics and Probability ,050208 finance ,Uniform distribution (continuous) ,General Mathematics ,Probability (math.PR) ,05 social sciences ,Aggregate (data warehouse) ,Characterization (mathematics) ,Type (model theory) ,01 natural sciences ,Set (abstract data type) ,010104 statistics & probability ,Dimension (vector space) ,0502 economics and business ,FOS: Mathematics ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Marginal distribution ,Random variable ,Mathematics - Probability ,Mathematics - Abstract
In this paper, we analyze the set of all possible aggregate distributions of the sum of standard uniform random variables, a simply stated yet challenging problem in the literature of distributions with given margins. Our main results are obtained for two distinct cases. In the case of dimension two, we obtain four partial characterization results. An analytical result with full generality is not available in this case, and it seems to be out of reach with existing techniques. For dimension greater or equal to three, we obtain a full characterization of the set of aggregate distributions, which is the first complete characterization result of this type in the literature for any choice of continuous marginal distributions.
- Published
- 2019
32. The De Vylder–Goovaerts conjecture holds within the diffusion limit
- Author
-
Nabil Kazi-Tani, Stefan Ankirchner, Christophette Blanchet-Scalliet, Institut für Mathematik, Friedrich-Schiller-Universität = Friedrich Schiller University Jena [Jena, Germany], Probabilités, statistique, physique mathématique (PSPM), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Sciences Actuarielle et Financière (SAF), Université Claude Bernard Lyon 1 (UCBL), and Université de Lyon-Université de Lyon
- Subjects
Statistics and Probability ,Approximations of π ,Ruin probability ,General Mathematics ,Open problem ,[QFIN.RM]Quantitative Finance [q-fin]/Risk Management [q-fin.RM] ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,0502 economics and business ,Risk theory ,Applied mathematics ,0101 mathematics ,Diffusion (business) ,Gaussian process ,Mathematics ,050208 finance ,Conjecture ,05 social sciences ,Heavy traffic approximation ,Diffusion approximations ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Equalized claims ,Distribution (mathematics) ,Jump ,symbols ,Statistics, Probability and Uncertainty - Abstract
The De Vylder and Goovaerts conjecture is an open problem in risk theory, stating that the finite-time ruin probability in a standard risk model is greater than or equal to the corresponding ruin probability evaluated in an associated model with equalized claim amounts. Equalized means here that the jump sizes of the associated model are equal to the average jump in the initial model between 0 and a terminal time T.In this paper, we consider the diffusion approximations of both the standard risk model and its associated risk model. We prove that the associated model, when conveniently renormalized, converges in distribution to a Gaussian process satisfying a simple SDE. We then compute the probability that this diffusion hits the level 0 before time T and compare it with the same probability for the diffusion approximation for the standard risk model. We conclude that the De Vylder and Goovaerts conjecture holds for the diffusion limits.
- Published
- 2019
33. The N-star network evolution model
- Author
-
István Fazekas, Attila Perecsényi, and Csaba Noszály
- Subjects
Statistics and Probability ,Star network ,Random graph ,Discrete mathematics ,General Mathematics ,010102 general mathematics ,Joins ,Preferential attachment ,01 natural sciences ,Power law ,Doob–Meyer decomposition theorem ,010104 statistics & probability ,Asymptotic power ,0101 mathematics ,Statistics, Probability and Uncertainty ,Unit (ring theory) ,Mathematics - Abstract
A new network evolution model is introduced in this paper. The model is based on cooperations of N units. The units are the nodes of the network and the cooperations are indicated by directed links. At each evolution step N units cooperate, which formally means that they form a directed N-star subgraph. At each step either a new unit joins the network and it cooperates with N − 1 old units, or N old units cooperate. During the evolution both preferential attachment and uniform choice are applied. Asymptotic power law distributions are obtained both for in-degrees and for out-degrees.
- Published
- 2019
34. A martingale view of Blackwell’s renewal theorem and its extensions to a general counting process
- Author
-
Masakiyo Miyazawa and Daryl J. Daley
- Subjects
Statistics and Probability ,Semimartingale ,Counting process ,Stochastic process ,General Mathematics ,Renewal theorem ,Renewal theory ,Statistics, Probability and Uncertainty ,Intensity function ,Martingale (probability theory) ,Mathematical economics ,Mathematics - Abstract
Martingales constitute a basic tool in stochastic analysis; this paper considers their application to counting processes. We use this tool to revisit a renewal theorem and give extensions for various counting processes. We first consider a renewal process as a pilot example, deriving a new semimartingale representation that differs from the standard decomposition via the stochastic intensity function. We then revisit Blackwell’s renewal theorem, its refinements and extensions. Based on these observations, we extend the semimartingale representation to a general counting process, and give conditions under which asymptotic behaviour similar to Blackwell’s renewal theorem holds.
- Published
- 2019
35. Some explicit results on one kind of sticky diffusion
- Author
-
Song Shiyu, Yongjin Wang, and Yiming Jiang
- Subjects
Nonlinear Sciences::Chaotic Dynamics ,Statistics and Probability ,Financial application ,Diffusion process ,Valuation of options ,General Mathematics ,Piecewise ,Applied mathematics ,Call option ,Statistics, Probability and Uncertainty ,Diffusion (business) ,Value (mathematics) ,Mathematics - Abstract
In this paper we derive several explicit results on one special sticky diffusion process which is constructed as a time-changed version of a diffusion with no sticky points. A theorem concerning the process-related Green operators defined on some nonnegative piecewise continuous functions is provided. Then, based on this theorem, we explore the distributional properties of the sticky diffusion. A financial application is presented where we compute the value of the European vanilla call option written on the underlying with sticky price dynamics.
- Published
- 2019
36. Preservation of the mean residual life order for coherent and mixed systems
- Author
-
Bo Henry Lindqvist, Nana Wang, and Francisco J. Samaniego
- Subjects
Statistics and Probability ,021103 operations research ,Component (thermodynamics) ,General Mathematics ,Order statistic ,0211 other engineering and technologies ,Closure (topology) ,02 engineering and technology ,Residual ,01 natural sciences ,Stochastic ordering ,Signature (logic) ,010104 statistics & probability ,Statistical physics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Variety (universal algebra) ,Mathematics ,Counterexample - Abstract
The signature of a coherent system has been studied extensively in the recent literature. Signatures are particularly useful in the comparison of coherent or mixed systems under a variety of stochastic orderings. Also, certain signature-based closure and preservation theorems have been established. For example, it is now well known that certain stochastic orderings are preserved from signatures to system lifetimes when components have independent and identical distributions. This applies to the likelihood ratio order, the hazard rate order, and the stochastic order. The point of departure of the present paper is the question of whether or not a similar preservation result will hold for the mean residual life order. A counterexample is provided which shows that the answer is negative. Classes of distributions for the component lifetimes for which the latter implication holds are then derived. Connections to the theory of order statistics are also considered.
- Published
- 2019
37. Asymptotic expansions and saddlepoint approximations using the analytic continuation of moment generating functions
- Author
-
Ronald W. Butler
- Subjects
Statistics and Probability ,General Mathematics ,Analytic continuation ,Likelihood-ratio test ,Path (graph theory) ,Applied mathematics ,Statistics, Probability and Uncertainty ,Moment-generating function ,Asymptotic expansion ,Gradient descent ,Methods of contour integration ,Term (time) ,Mathematics - Abstract
Transform inversions, in which density and survival functions are computed from their associated moment generating function $\mathcal{M}$, have largely been based on methods which use values of $\mathcal{M}$ in its convergence region. Prominent among such methods are saddlepoint approximations and Fourier-series inversion methods, including the fast Fourier transform. In this paper we propose inversion methods which make use of values for $\mathcal{M}$ which lie outside of its convergence region and in its analytic continuation. We focus on the simplest and perhaps richest setting for applications in which $\mathcal{M}$ is either a meromorphic function in its analytic continuation, so that all of its singularities are poles, or else the singularities are isolated essential. Asymptotic expansions of finite- and infinite-orders are developed for density and survival functions using the poles of $\mathcal{M}$ in its analytic continuation. For finite-order expansions, the expansion error is a contour integral in the analytic continuation, which we approximate using the saddlepoint method based on following the path of steepest descent. Such saddlepoint error approximations accurately determine expansion errors and, thus, provide the means for determining the order of the expansion needed to achieve some preset accuracy. They also provide an additive correction term which increases accuracy of the expansion. Further accuracy is achieved by computing the expansion errors numerically using a contour path which ultimately tracks the steepest descent direction. Important applications include Wilks’ likelihood ratio test in MANOVA, compound distributions, and the Sparre Andersen and Cramér–Lundberg ruin models.
- Published
- 2019
38. A unified stability theory for classical and monotone Markov chains
- Author
-
Takashi Kamihigashi and John Stachurski
- Subjects
Statistics and Probability ,Markov chain ,General Mathematics ,05 social sciences ,Stochastic dominance ,01 natural sciences ,Stability (probability) ,010104 statistics & probability ,Total variation ,Monotone polygon ,Stability theory ,0502 economics and business ,Metric (mathematics) ,Applied mathematics ,050207 economics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics ,Probability measure - Abstract
In this paper we integrate two strands of the literature on stability of general state Markov chains: conventional, total-variation-based results and more recent order-theoretic results. First we introduce a complete metric over Borel probability measures based on ‘partial’ stochastic dominance. We then show that many conventional results framed in the setting of total variation distance have natural generalizations to the partially ordered setting when this metric is adopted.
- Published
- 2019
39. Variance estimates for random disc-polygons in smooth convex discs
- Author
-
Ferenc Fodor and Viktor Vígh
- Subjects
Statistics and Probability ,Mathematics::Complex Variables ,General Mathematics ,010102 general mathematics ,Regular polygon ,Boundary (topology) ,Metric Geometry (math.MG) ,Variance (accounting) ,Computer Science::Computational Geometry ,01 natural sciences ,Probability model ,Combinatorics ,010104 statistics & probability ,Mathematics - Metric Geometry ,Intersection ,FOS: Mathematics ,Mathematics::Metric Geometry ,Astrophysics::Earth and Planetary Astrophysics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Astrophysics::Galaxy Astrophysics ,Inscribed figure ,52A22, 60D05 ,Mathematics - Abstract
In this paper we prove asymptotic upper bounds on the variance of the number of vertices and the missed area of inscribed random disc-polygons in smooth convex discs whose boundary isC+2. We also consider a circumscribed variant of this probability model in which the convex disc is approximated by the intersection of random circles.
- Published
- 2018
40. Reliability assessment of system under a generalized run shock model
- Author
-
Yaning Yang, Min Xie, and Ming Gong
- Subjects
Statistics and Probability ,Sequence ,021103 operations research ,Markov chain ,General Mathematics ,0211 other engineering and technologies ,Markov process ,02 engineering and technology ,Measure (mathematics) ,Shock (mechanics) ,symbols.namesake ,Distribution (mathematics) ,Control theory ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,020201 artificial intelligence & image processing ,Phase-type distribution ,Statistics, Probability and Uncertainty ,Reliability (statistics) ,Mathematics - Abstract
In this paper we are concerned with modelling the reliability of a system subject to external shocks. In a run shock model, the system fails when a sequence of shocks above a threshold arrive in succession. Nevertheless, using a single threshold to measure the severity of a shock is too critical in real practice. To this end, we develop a generalized run shock model with two thresholds. We employ a phase-type distribution to model the damage size and the inter-arrival time of shocks, which is highly versatile and may be used to model many quantitative features of random phenomenon. Furthermore, we use the Markovian property to construct a multi-state system which degrades with the arrival of shocks. We also provide a numerical example to illustrate our results.
- Published
- 2018
41. Convergence rates for estimators of geodesic distances and Frèchet expectations
- Author
-
Olivier Bodart, Catherine Aaron, Aaron, Catherine, Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS), Université Clermont Auvergne [2017-2020] (UCA [2017-2020]), Modélisation mathématique, calcul scientifique (MMCS), Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Institut Camille Jordan [Villeurbanne] (ICJ), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Statistics and Probability ,Geodesic distance ,Pure mathematics ,Geodesic ,Geometric inference ,General Mathematics ,010102 general mathematics ,Dimension (graph theory) ,Estimator ,Boundary (topology) ,01 natural sciences ,Combinatorics ,Set (abstract data type) ,010104 statistics & probability ,Rate of convergence ,62-07, 62G05, 62G20, 62H99 ,[MATH.MATH-ST]Mathematics [math]/Statistics [math.ST] ,Convergence (routing) ,Probability distribution ,0101 mathematics ,Statistics, Probability and Uncertainty ,Fréchet expectations ,Statistics on manifolds ,[MATH.MATH-ST] Mathematics [math]/Statistics [math.ST] ,Mathematics - Abstract
Consider a sample 𝒳n={X1,…,Xn} of independent and identically distributed variables drawn with a probability distribution ℙX supported on a compact set M⊂ℝd. In this paper we mainly deal with the study of a natural estimator for the geodesic distance on M. Under rather general geometric assumptions on M, we prove a general convergence result. Assuming M to be a compact manifold of known dimension d′≤d, and under regularity assumptions on ℙX, we give an explicit convergence rate. In the case when M has no boundary, knowledge of the dimension d′ is not needed to obtain this convergence rate. The second part of the work consists in building an estimator for the Fréchet expectations on M, and proving its convergence under regularity conditions, applying the previous results.
- Published
- 2018
42. On optimal periodic dividend and capital injection strategies for spectrally negative Lévy models
- Author
-
Kazutoshi Yamazaki, José-Luis Pérez, Kouji Yano, and Kei Noba
- Subjects
Statistics and Probability ,050208 finance ,General Mathematics ,05 social sciences ,Poisson distribution ,01 natural sciences ,Arrival time ,Dividend payment ,Scale function ,010104 statistics & probability ,symbols.namesake ,Capital injection ,0502 economics and business ,Econometrics ,Reflection (physics) ,symbols ,Dividend ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
De Finetti’s optimal dividend problem has recently been extended to the case when dividend payments can be made only at Poisson arrival times. In this paper we consider the version with bail-outs where the surplus must be nonnegative uniformly in time. For a general spectrally negative Lévy model, we show the optimality of a Parisian-classical reflection strategy that pays the excess above a given barrier at each Poisson arrival time and also reflects from below at 0 in the classical sense.
- Published
- 2018
43. Uniform decomposition of probability measures: quantization, clustering and rate of convergence
- Author
-
Julien Chevallier, Analyse, Géométrie et Modélisation (AGM - UMR 8088), Centre National de la Recherche Scientifique (CNRS)-CY Cergy Paris Université (CY), Statistique pour le Vivant et l’Homme (SVH), Laboratoire Jean Kuntzmann (LJK ), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), and ANR-11-LABX-0023,MME-DII,Modèles Mathématiques et Economiques de la Dynamique, de l'Incertitude et des Interactions(2011)
- Subjects
Statistics and Probability ,Approximations of π ,Computer Science::Information Retrieval ,General Mathematics ,Quantization (signal processing) ,010103 numerical & computational mathematics ,Rate of convergence ,Mathematical Subject Classification: 60E15, 62E17, 60B10, 60F99 ,01 natural sciences ,Minimax approximation algorithm ,Clustering ,Uniform approximation ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,010104 statistics & probability ,Quantization ,Applied mathematics ,Wasserstein distance ,0101 mathematics ,Statistics, Probability and Uncertainty ,Cluster analysis ,Probability measure ,Mathematics - Abstract
The study of finite approximations of probability measures has a long history. In Xu and Berger (2017), the authors focused on constrained finite approximations and, in particular, uniform ones in dimensiond=1. In the present paper we give an elementary construction of a uniform decomposition of probability measures in dimensiond≥1. We then use this decomposition to obtain upper bounds on the rate of convergence of the optimal uniform approximation error. These bounds appear to be the generalization of the ones obtained by Xu and Berger (2017) and to be sharp for generic probability measures.
- Published
- 2018
44. The degree-wise effect of a second step for a random walk on a graph
- Author
-
Kenneth S. Berenhaut, Elizabeth J. Krizay, Hongyi Jiang, and Katelyn M. McNab
- Subjects
Statistics and Probability ,Combinatorics ,010104 statistics & probability ,Degree (graph theory) ,General Mathematics ,010102 general mathematics ,Friendship paradox ,Graph (abstract data type) ,0101 mathematics ,Statistics, Probability and Uncertainty ,Random walk ,01 natural sciences ,Mathematics - Abstract
In this paper we consider the degree-wise effect of a second step for a random walk on a graph. We prove that under the configuration model, for any fixed degree sequence the probability of exceeding a given degree threshold is smaller after two steps than after one. This builds on recent work of Krameret al.(2016) regarding the friendship paradox under random walks.
- Published
- 2018
45. A conditional limit theorem for high-dimensional ℓᵖ-spheres
- Author
-
Kavita Ramanan and Steven Soojin Kim
- Subjects
Statistics and Probability ,Pure mathematics ,Kullback–Leibler divergence ,Convex geometry ,Euclidean space ,General Mathematics ,Convex set ,01 natural sciences ,010101 applied mathematics ,010104 statistics & probability ,Probability theory ,Limit (mathematics) ,0101 mathematics ,Statistics, Probability and Uncertainty ,Rate function ,Event (probability theory) ,Mathematics - Abstract
The study of high-dimensional distributions is of interest in probability theory, statistics, and asymptotic convex geometry, where the object of interest is the uniform distribution on a convex set in high dimensions. The ℓp-spaces and norms are of particular interest in this setting. In this paper we establish a limit theorem for distributions on ℓp-spheres, conditioned on a rare event, in a high-dimensional geometric setting. As part of our proof, we establish a certain large deviation principle that is also relevant to the study of the tail behavior of random projections of ℓp-balls in a high-dimensional Euclidean space.
- Published
- 2018
46. Multi-point correlations for two-dimensional coalescing or annihilating random walks
- Author
-
Oleg Zaboronski, Roger Tribe, and James Lukins
- Subjects
Statistics and Probability ,Particle system ,Logarithm ,Differential equation ,General Mathematics ,010102 general mathematics ,Field (mathematics) ,Simple random sample ,Random walk ,01 natural sciences ,010104 statistics & probability ,Correlation function ,Initial value problem ,Statistical physics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In this paper we consider an infinite system of instantaneously coalescing rate 1 simple symmetric random walks on ℤ2, started from the initial condition with all sites in ℤ2 occupied. Two-dimensional coalescing random walks are a `critical' model of interacting particle systems: unlike coalescence models in dimension three or higher, the fluctuation effects are important for the description of large-time statistics in two dimensions, manifesting themselves through the logarithmic corrections to the `mean field' answers. Yet the fluctuation effects are not as strong as for the one-dimensional coalescence, in which case the fluctuation effects modify the large time statistics at the leading order. Unfortunately, unlike its one-dimensional counterpart, the two-dimensional model is not exactly solvable, which explains a relative scarcity of rigorous analytic answers for the statistics of fluctuations at large times. Our contribution is to find, for any N≥2, the leading asymptotics for the correlation functions ρN(x1,…,xN) as t→∞. This generalises the results for N=1 due to Bramson and Griffeath (1980) and confirms a prediction in the physics literature for N>1. An analogous statement holds for instantaneously annihilating random walks. The key tools are the known asymptotic ρ1(t)∼logt∕πt due to Bramson and Griffeath (1980), and the noncollision probability 𝒑NC(t), that no pair of a finite collection of N two-dimensional simple random walks meets by time t, whose asymptotic 𝒑NC(t)∼c0(logt)-(N2) was found by Cox et al. (2010). We re-derive the asymptotics, and establish new error bounds, both for ρ1(t) and 𝒑NC(t) by proving that these quantities satisfy effective rate equations; that is, approximate differential equations at large times. This approach can be regarded as a generalisation of the Smoluchowski theory of renormalised rate equations to multi-point statistics.
- Published
- 2018
47. New nonparametric classes of distributions in terms of mean time to failure in age replacement
- Author
-
Baha-Eldin Khaledi, Muhyiddin Izadi, and Maryam Sharafi
- Subjects
Statistics and Probability ,Mean time between failures ,021103 operations research ,General Mathematics ,0211 other engineering and technologies ,Nonparametric statistics ,02 engineering and technology ,Function (mathematics) ,01 natural sciences ,010104 statistics & probability ,Statistics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
The mean time to failure (MTTF) function in age replacement is used to evaluate the performance and effectiveness of the age replacement policy. In this paper, based on the MTTF function, we introduce two new nonparametric classes of lifetime distributions with nonmonotonic mean time to failure in age replacement; increasing then decreasing MTTF (IDMTTF) and decreasing then increasing MTTF (DIMTTF). The implications between these classes of distributions and some existing classes of nonmonotonic ageing classes are studied. The characterizations of IDMTTF and DIMTTF in terms of the scaled total time on test transform are also obtained.
- Published
- 2018
48. Reliability modeling of coherent systems with shared components based on sequential order statistics
- Author
-
Somayeh Ashrafi, Somayeh Zarezadeh, and Majid Asadi
- Subjects
Statistics and Probability ,021103 operations research ,Dependency (UML) ,Signature matrix ,General Mathematics ,Order statistic ,0211 other engineering and technologies ,02 engineering and technology ,Function (mathematics) ,01 natural sciences ,Stochastic ordering ,Set (abstract data type) ,010104 statistics & probability ,Mixing (mathematics) ,0101 mathematics ,Statistics, Probability and Uncertainty ,Algorithm ,Reliability (statistics) ,Mathematics - Abstract
In this paper we are concerned with the reliability properties of two coherent systems having shared components. We assume that the components of the systems are two overlapping subsets of a set of n components with lifetimes X1,...,Xn. Further, we assume that the components of the systems fail according to the model of sequential order statistics (which is equivalent, under some mild conditions, to the failure model corresponding to a nonhomogeneous pure-birth process). The joint reliability function of the system lifetimes is expressed as a mixture of the joint reliability functions of the sequential order statistics, where the mixing probabilities are the bivariate signature matrix associated to the structures of systems. We investigate some stochastic orderings and dependency properties of the system lifetimes. We also study conditions under which the joint reliability function of systems with shared components of order m can be equivalently written as the joint reliability function of systems of order n (n>m). In order to illustrate the results, we provide several examples.
- Published
- 2018
49. Univariate and multivariate stochastic orderings of residual lifetimes of live components in sequential (𝑛-𝑟+ 1)-out-of-𝑛 systems
- Author
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Narayanaswamy Balakrishnan, Abedin Haidari, and Ghobad Barmalzan
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Statistics and Probability ,Multivariate statistics ,Exponential distribution ,Component (thermodynamics) ,General Mathematics ,Reliability (computer networking) ,Order statistic ,Univariate ,Statistical physics ,Statistics, Probability and Uncertainty ,Residual ,Uncorrelated ,Mathematics - Abstract
Sequential order statistics can be used to describe the ordered lifetimes of components of a system when the failure of a component may affect the reliability of the remaining components. After a reliability system consisting of n components fails, some of its components may still be alive. In this paper we first establish some univariate stochastic orderings and ageing properties of the residual lifetimes of the live components in a sequential (n-r+1)-out-of-n system. We also obtain a characterizing result for the exponential distribution based on uncorrelated residual lifetimes of live components. Finally, we provide some sufficient conditions for comparing vectors of residual lifetimes of the live components from two sequential (n-r+1)-out-of-n systems. The results established here extend some well-known results in the literature.
- Published
- 2018
50. Reach of repulsion for determinantal point processes in high dimensions
- Author
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François Baccelli and Eliza O'Reilly
- Subjects
Statistics and Probability ,Boolean model ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Radius ,Palm calculus ,01 natural sciences ,Measure (mathematics) ,Point process ,010104 statistics & probability ,Distribution (mathematics) ,Point (geometry) ,Determinantal point process ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
Goldman (2010) proved that the distribution of a stationary determinantal point process (DPP) Φ can be coupled with its reduced Palm version Φ0,! such that there exists a point process η where Φ=Φ0,!∪η in distribution and Φ0,!∩η=∅. The points of η characterize the repulsive nature of a typical point of Φ. In this paper we use the first-moment measure of η to study the repulsive behavior of DPPs in high dimensions. We show that many families of DPPs have the property that the total number of points in η converges in probability to 0 as the space dimension n→∞. We also prove that for some DPPs, there exists an R∗ such that the decay of the first-moment measure of η is slowest in a small annulus around the sphere of radius √nR∗. This R∗ can be interpreted as the asymptotic reach of repulsion of the DPP. Examples of classes of DPP models exhibiting this behavior are presented and an application to high-dimensional Boolean models is given.
- Published
- 2018
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