993 results
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2. A Note on a Paper by Wong and Heyde
- Author
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Mikhail Urusov and Aleksandar Mijatović
- Subjects
Statistics and Probability ,Statistics::Theory ,Pure mathematics ,60G44, 60G48, 60H10, 60J60 ,General Mathematics ,Applied probability ,01 natural sciences ,FOS: Economics and business ,010104 statistics & probability ,60G48 ,FOS: Mathematics ,60G44 ,0101 mathematics ,60J60 ,Mathematics ,Local martingales versus true martingales ,010102 general mathematics ,Probability (math.PR) ,stochastic exponential ,Exponential function ,Mathematik ,60H10 ,Statistics, Probability and Uncertainty ,Martingale (probability theory) ,Quantitative Finance - General Finance ,General Finance (q-fin.GN) ,Mathematics - Probability ,Counterexample - Abstract
In this note we re-examine the analysis of the paper "On the martingale property of stochastic exponentials" by B. Wong and C.C. Heyde, Journal of Applied Probability, 41(3):654-664, 2004. Some counterexamples are presented and alternative formulations are discussed., Comment: To appear in Journal of Applied Probability, 11 pages
- Published
- 2011
3. Epidemics with carriers: A note on a paper of Dietz
- Author
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F. Downton
- Subjects
Statistics and Probability ,Entire population ,education.field_of_study ,General Mathematics ,010102 general mathematics ,Population ,01 natural sciences ,Short interval ,010104 statistics & probability ,0101 mathematics ,Statistics, Probability and Uncertainty ,education ,Demography ,Mathematics - Abstract
In a recent paper Weiss (1965) has suggested a simple model for a carrier-borne epidemic such as typhoid. He considers a population (of size m) of susceptibles into which a number (k) of carriers is introduced. These carriers exhibit no overt symptoms and are only detectable by the discovery of infected persons. He supposed that after the initial introduction of the carriers, the population remains entirely closed and no new carriers arise. The epidemic then progresses until either all the carriers have been traced and isolated or until the entire population has succumbed to the disease.
- Published
- 1967
4. 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
5. 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
6. Martingale decomposition of an L2 space with nonlinear stochastic integrals
- Author
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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
7. 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
8. 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
9. 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
10. 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
11. 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
12. 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
13. On exponential limit laws for hitting times of rare sets for Harris chains and processes
- Author
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Peter W. Glynn
- Subjects
Harris recurrent Markov process ,Statistics and Probability ,Exponential distribution ,60G70 ,General Mathematics ,01 natural sciences ,Time reversibility ,Combinatorics ,Hitting time ,010104 statistics & probability ,60J05 ,60K05 ,60J25 ,60F05 ,Phase-type distribution ,Limit (mathematics) ,0101 mathematics ,Mathematics ,Markov chain ,010102 general mathematics ,regenerative process ,Harris chain ,Harris recurrent Markov chain ,Markov property ,Statistics, Probability and Uncertainty - Abstract
This paper provides a simple proof for the fact that the hitting time to an infrequently visited subset for a one-dependent regenerative process converges weakly to an exponential distribution. Special cases are positive recurrent Harris chains and Harris processes. The paper further extends this class of limit theorems to ‘rewards’ that are cumulated to the hitting time of such a rare set.
- Published
- 2011
14. Generalized Increasing Convex and Directionally Convex Orders
- Author
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Mhamed Mesfioui and Michel Denuit
- Subjects
Statistics and Probability ,Convex analysis ,Convex hull ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Proper convex function ,Convex set ,Subderivative ,01 natural sciences ,Combinatorics ,010104 statistics & probability ,Convex optimization ,Convex polytope ,Convex combination ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In this paper, the componentwise increasing convex order, the upper orthant order, the upper orthant convex order, and the increasing directionally convex order for random vectors are generalized to hierarchical classes of integral stochastic order relations. The elements of the generating classes of functions possess nonnegative partial derivatives up to some given degrees. Some properties of these new stochastic order relations are studied. Particular attention is paid to the comparison of weighted sums of the respective components of ordered random vectors. By providing a unified derivation of standard multivariate stochastic orderings, the present paper shows how some well-known results derive from a common principle.
- Published
- 2010
15. The Early Stage Behaviour of a Stochastic SIR Epidemic with Term-Time Forcing
- Author
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Mathias Lindholm and Tom Britton
- Subjects
Statistics and Probability ,education.field_of_study ,General Mathematics ,010102 general mathematics ,Population ,Context (language use) ,Intersection graph ,01 natural sciences ,Giant component ,010104 statistics & probability ,Probability theory ,Statistics ,Probability distribution ,0101 mathematics ,Statistics, Probability and Uncertainty ,Epidemic model ,education ,Branching process ,Mathematics - Abstract
This thesis is concerned with the study of stochastic epidemic models for infectious diseases in heterogeneous populations. All diseases treated are of SIR type, i.e. individuals are either Susceptible, Infectious or Recovered (and immune). The transitions between these states are according to S to I to R. The thesis consists of five papers. Papers I and II treat approximations for the distribution of the time to extinction. In Paper I, a sub-community version of the SIR model with demography is considered. The interest is in how the distribution of the time to extinction is affected by varying the degree of interaction between the sub-communities. Paper II is concerned with a two-type version of Bartlett's model. The distribution of the time to extinction is studied when the difference in susceptibility/infectivity between the types of individuals is varied. Papers III and IV treat random intersection graphs with tunable clustering. In Paper III a Reed-Frost epidemic is run on such a random intersection graph. The critical parameter R_0 and the probability of a large outbreak are derived and it is investigated how these quantities are affected by the clustering in the graph. In Paper IV the interest is in the component structure of such a graph, i.e. the size and the emergence of a giant component is studied. The last paper, Paper V, treats the situation when a simple epidemic is running in a varying environment. A varying environment is in this context any external factor that affects the contact rate in the population, but is itself unaffected by the population. The model treated is a term-time forced version of the stochastic general epidemic where the contact rate is modelled by an alternating renewal process. A threshold parameter R_* and the probability of a large outbreak are derived and studied.
- Published
- 2009
16. An analysis of transient Markov decision processes
- Author
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E. J. Collins and Huw W. James
- Subjects
Statistics and Probability ,Bounded set ,General Mathematics ,010102 general mathematics ,Markov process ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Probability theory ,Bellman equation ,Bounded function ,symbols ,Calculus ,Countable set ,Applied mathematics ,Markov decision process ,Uniqueness ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
This paper is concerned with the analysis of Markov decision processes in which a natural form of termination ensures that the expected future costs are bounded, at least under some policies. Whereas most previous analyses have restricted attention to the case where the set of states is finite, this paper analyses the case where the set of states is not necessarily finite or even countable. It is shown that all the existence, uniqueness, and convergence results of the finite-state case hold when the set of states is a general Borel space, provided we make the additional assumption that the optimal value function is bounded below. We give a sufficient condition for the optimal value function to be bounded below which holds, in particular, if the set of states is countable.
- Published
- 2006
17. Hazard rate ordering of order statistics and systems
- Author
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Moshe Shaked and Jorge Navarro
- Subjects
Statistics and Probability ,Discrete mathematics ,Reliability theory ,Series (mathematics) ,Multivariate random variable ,General Mathematics ,010102 general mathematics ,Order statistic ,Function (mathematics) ,01 natural sciences ,Stochastic ordering ,010104 statistics & probability ,Probability theory ,Statistics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Rate function ,Mathematics - Abstract
Let X = (X1, X2, …, Xn) be an exchangeable random vector, and write X(1:i) = min{X1, X2, …, Xi}, 1 ≤ i ≤ n. In this paper we obtain conditions under which X(1:i) decreases in i in the hazard rate order. A result involving more general (that is, not necessarily exchangeable) random vectors is also derived. These results are applied to obtain the limiting behaviour of the hazard rate function of the lifetimes of various coherent systems in reliability theory. The notions of the Samaniego signatures and the minimal signatures of such systems are extensively used in the paper. An interesting relationship between these two signatures is obtained. The results are illustrated in a series of examples.
- Published
- 2006
18. Range of Asymptotic Behaviour of the Optimality Probability of the Expert and Majority Rules
- Author
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Luba Sapir and Daniel Berend
- Subjects
Statistics and Probability ,Majority rule ,Chain rule (probability) ,General Mathematics ,05 social sciences ,Decision rule ,01 natural sciences ,0506 political science ,010104 statistics & probability ,Probability theory ,Ranking ,050602 political science & public administration ,Econometrics ,Probability distribution ,0101 mathematics ,Statistics, Probability and Uncertainty ,Random variable ,Mathematics ,Optimal decision - Abstract
We study the uncertain dichotomous choice model. In this model, a group of expert decision makers is required to select one of two alternatives. The applications of this model are relevant to a wide variety of areas. A decision rule translates the individual opinions of the members into a group decision, and is optimal if it maximizes the probability of the group making a correct choice. In this paper, we assume the correctness probabilities of the experts to be independent random variables selected from some given distribution. Moreover, the ranking of the members in the group is (at least partly) known. Thus, one can follow rules based on this ranking. The extremes are the expert rule and the majority rule. The probabilities of the two extreme rules being optimal were compared in a series of early papers, for a variety of distributions. In most cases, the asymptotic behaviours of the probabilities of the two extreme rules followed the same patterns. Do these patterns hold in general? If not, what are the ranges of possible asymptotic behaviours of the probabilities of the two extreme rules being optimal? In this paper, we provide satisfactory answers to these questions.
- Published
- 2006
19. A renewal-process-type expression for the moments of inverse subordinators
- Author
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Andreas Nordvall Lagerås
- Subjects
Statistics and Probability ,Pure mathematics ,education.field_of_study ,Markov chain ,Subordinator ,General Mathematics ,010102 general mathematics ,Population ,01 natural sciences ,Combinatorics ,Cox process ,010104 statistics & probability ,Iterated function system ,Markov renewal process ,Renewal theory ,0101 mathematics ,Statistics, Probability and Uncertainty ,education ,Mathematics ,Central limit theorem - Abstract
This thesis consists of four papers.In paper 1, we prove central limit theorems for Markov chains under (local) contraction conditions. As a corollary we obtain a central limit theorem for Markov chains associated with iterated function systems with contractive maps and place-dependent Dini-continuous probabilities.In paper 2, properties of inverse subordinators are investigated, in particular similarities with renewal processes. The main tool is a theorem on processes that are both renewal and Cox processes.In paper 3, distributional properties of supercritical and especially immortal branching processes are derived. The marginal distributions of immortal branching processes are found to be compound geometric.In paper 4, a description of a dynamic population model is presented, such that samples from the population have genealogies as given by a Lambda-coalescent with mutations. Depending on whether the sample is grouped according to litters or families, the sampling distribution is either regenerative or non-regenerative.
- Published
- 2005
20. The moment problem for some Wiener functionals: corrections to previous proofs (with an appendix by H. L. Pedersen)
- Author
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Per Hörfelt
- Subjects
Statistics and Probability ,Geometric Brownian motion ,Pure mathematics ,Class (set theory) ,Generalization ,General Mathematics ,Mathematical analysis ,Gauss ,010102 general mathematics ,Space (mathematics) ,01 natural sciences ,Moment problem ,010104 statistics & probability ,Probability theory ,0101 mathematics ,Statistics, Probability and Uncertainty ,Brownian motion ,Mathematics - Abstract
In this paper, we describe a class of Wiener functionals that are ‘indeterminate by their moments’, that is, whose distributions are not uniquely determined by their moments. In particular, it is proved that the integral of a geometric Brownian motion is indeterminate by its moments and, moreover, shown that previous proofs of this result are incorrect. The main result of this paper is based on geometric inequalities in Gauss space and on a generalization of the Krein criterion due to H. L. Pedersen.
- Published
- 2005
21. Decomposition property in a discrete-time queue with multiple input streams and service interruptions
- Author
-
Fumio Ishizaki
- Subjects
Statistics and Probability ,Independent and identically distributed random variables ,Queueing theory ,Mathematical optimization ,Markov chain ,Queue management system ,General Mathematics ,010102 general mathematics ,Fork–join queue ,01 natural sciences ,Computer Science::Performance ,010104 statistics & probability ,Multilevel queue ,Decomposition method (queueing theory) ,0101 mathematics ,Statistics, Probability and Uncertainty ,Computer Science::Operating Systems ,Queue ,Mathematics - Abstract
This paper studies a discrete-time single-server queue with two independent inputs and service interruptions. One of the inputs to the queue is an independent and identically distributed process. The other is a much more general process and it is not required to be Markov nor is it required to be stationary. The service interruption process is also general and it is not required to be Markov or to be stationary. This paper shows that a stochastic decomposition property for the virtual waiting-time process holds in the discrete-time single-server queue with service interruptions. To the best of the author's knowledge, no stochastic decomposition results for virtual waiting-time processes in non-work-conserving queues, such as queues with service interruptions, have been obtained before and only work-conserving queues have been studied in the literature.
- Published
- 2004
22. On-line parameter estimation for a failure-prone system subject to condition monitoring
- Author
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Daming Lin and Viliam Makis
- Subjects
Statistics and Probability ,Mathematical optimization ,Discretization ,Estimation theory ,General Mathematics ,Condition-based maintenance ,010102 general mathematics ,Condition monitoring ,Markov process ,Observable ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,symbols ,Range (statistics) ,0101 mathematics ,Statistics, Probability and Uncertainty ,Projection (set theory) ,Algorithm ,Mathematics - Abstract
In this paper, we study the on-line parameter estimation problem for a partially observable system subject to deterioration and random failure. The state of the system evolves according to a continuous-time homogeneous Markov process with a finite state space. The state of the system is hidden except for the failure state. When the system is operating, only the information obtained by condition monitoring, which is related to the working state of the system, is available. The condition monitoring observations are assumed to be in continuous range, so that no discretization is required. A recursive maximum likelihood (RML) algorithm is proposed for the on-line parameter estimation of the model. The new RML algorithm proposed in the paper is superior to other RML algorithms in the literature in that no projection is needed and no calculation of the gradient on the surface of the constraint manifolds is required. A numerical example is provided to illustrate the algorithm.
- Published
- 2004
23. Perpetual American put options in a level-dependent volatility model
- Author
-
Erik Ekström
- Subjects
Statistics and Probability ,General Mathematics ,010102 general mathematics ,Trinomial tree ,Implied volatility ,Put–call parity ,01 natural sciences ,Binary option ,010104 statistics & probability ,Asian option ,Finite difference methods for option pricing ,0101 mathematics ,Statistics, Probability and Uncertainty ,Put option ,Moneyness ,Mathematical economics ,Mathematics - Abstract
This thesis, consisting of six papers and a summary, studies the area of continuous time financial mathematics. A unifying theme for many of the problems studied is the implications of possible mis-specifications of models. Intimately connected with this question is, perhaps surprisingly, convexity properties of option prices. We also study qualitative behavior of different optimal stopping boundaries appearing in option pricing.In Paper I a new condition on the contract function of an American option is provided under which the option price increases monotonically in the volatility. It is also shown that American option prices are continuous in the volatility.In Paper II an explicit pricing formula for the perpetual American put option in the Constant Elasticity of Variance model is derived. Moreover, different properties of this price are studied.Paper III deals with the Russian option with a finite time horizon. It is shown that the value of the Russian option solves a certain free boundary problem. This information is used to analyze the optimal stopping boundary.A study of perpetual game options is performed in Paper IV. One of the main results provides a condition under which the value of the option is increasing in the volatility.In Paper V options written on several underlying assets are considered. It is shown that, within a large class of models, the only model for the stock prices that assigns convex option prices to all convex contract functions is geometric Brownian motion.Finally, in Paper VI it is shown that the optimal stopping boundary for the American put option is convex in the standard Black-Scholes model.
- Published
- 2003
24. Geometric bounds on certain sublinear functionals of geometric Brownian motion
- Author
-
Per Hörfelt
- Subjects
Statistics and Probability ,Sublinear function ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Upper and lower bounds ,010305 fluids & plasmas ,Combinatorics ,Moment problem ,010104 statistics & probability ,Probability theory ,Bounded function ,0103 physical sciences ,Log-normal distribution ,0101 mathematics ,Statistics, Probability and Uncertainty ,Borel measure ,Random variable ,Mathematics - Abstract
Suppose that {X s , 0 ≤ s ≤ T} is an m-dimensional geometric Brownian motion with drift, μ is a bounded positive Borel measure on [0,T], and ϕ : ℝ m → [0,∞) is a (weighted) l q (ℝ m )-norm, 1 ≤ q ≤ ∞. The purpose of this paper is to study the distribution and the moments of the random variable Y given by the L p (μ)-norm, 1 ≤ p ≤ ∞, of the function s ↦ ϕ(X s ), 0 ≤ s ≤ T. By using various geometric inequalities in Wiener space, this paper gives upper and lower bounds for the distribution function of Y and proves that the distribution function is log-concave and absolutely continuous on every open subset of the distribution's support. Moreover, the paper derives tail probabilities, presents sharp moment inequalities, and shows that Y is indetermined by its moments. The paper will also discuss the so-called moment-matching method for the pricing of Asian-styled basket options.
- Published
- 2003
25. The classification of matrix GI/M/1-type Markov chains with a tree structure and its applications to queueing
- Author
-
Qi-Ming He
- Subjects
Statistics and Probability ,Discrete mathematics ,Queueing theory ,Markov chain ,General Mathematics ,Variable-order Markov model ,010102 general mathematics ,01 natural sciences ,Continuous-time Markov chain ,010104 statistics & probability ,symbols.namesake ,Tree structure ,Matrix analytic method ,Jacobian matrix and determinant ,symbols ,Examples of Markov chains ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In this paper, we study the classification of matrix GI/M/1-type Markov chains with a tree structure. We show that the Perron–Frobenius eigenvalue of a Jacobian matrix provides information for classifying these Markov chains. A fixed-point approach is utilized. A queueing application is presented to show the usefulness of the classification method developed in this paper.
- Published
- 2003
26. Denumerable-state continuous-time Markov decision processes with unbounded transition and reward rates under the discounted criterion
- Author
-
Weiping Zhu and Xianping Guo
- Subjects
Statistics and Probability ,Markov kernel ,Markov chain ,General Mathematics ,010102 general mathematics ,Markov process ,State (functional analysis) ,Transition rate matrix ,01 natural sciences ,Birth–death process ,010104 statistics & probability ,symbols.namesake ,symbols ,Countable set ,Markov decision process ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematical economics ,Mathematics - Abstract
In this paper, we consider denumerable-state continuous-time Markov decision processes with (possibly unbounded) transition and reward rates and general action space under the discounted criterion. We provide a set of conditions weaker than those previously known and then prove the existence of optimal stationary policies within the class of all possibly randomized Markov policies. Moreover, the results in this paper are illustrated by considering the birth-and-death processes with controlled immigration in which the conditions in this paper are satisfied, whereas the earlier conditions fail to hold.
- Published
- 2002
27. The delay distribution of a type k customer in a first-come-first-served MMAP[K]/PH[K]/1 queue
- Author
-
Chris Blondia and B. Van Houdt
- Subjects
Discrete mathematics ,Statistics and Probability ,Queueing theory ,mmap ,M/G/k queue ,General Mathematics ,010102 general mathematics ,01 natural sciences ,010104 statistics & probability ,First-come, first-served ,Probability distribution ,Phase-type distribution ,Markovian arrival process ,0101 mathematics ,Statistics, Probability and Uncertainty ,Queue ,Algorithm ,Mathematics - Abstract
This paper presents an algorithmic procedure to calculate the delay distribution of a type k customer in a first-come-first-served (FCFS) discrete-time queueing system with multiple types of customers, where each type has different service requirements (the MMAP[K]/PH[K]/1 queue). First, we develop a procedure, using matrix analytical methods, to handle arrival processes that do not allow batch arrivals to occur. Next, we show that this technique can be generalized to arrival processes that do allow batch arrivals to occur. We end the paper by presenting some numerical examples.
- Published
- 2002
28. Finite-size corrections to Poisson approximations of rare events in renewal processes
- Author
-
John L. Spouge
- Subjects
Statistics and Probability ,General Mathematics ,010102 general mathematics ,Generating function ,Poisson distribution ,01 natural sciences ,Point process ,symbols.namesake ,010104 statistics & probability ,Poisson point process ,Rare events ,symbols ,Calculus ,Applied mathematics ,Renewal theory ,0101 mathematics ,Statistics, Probability and Uncertainty ,Asymptotic expansion ,Residual time ,Mathematics - Abstract
Consider a renewal process. The renewal events partition the process into i.i.d. renewal cycles. Assume that on each cycle, a rare event called 'success’ can occur. Such successes lend themselves naturally to approximation by Poisson point processes. If each success occurs after a random delay, however, Poisson convergence may be relatively slow, because each success corresponds to a time interval, not a point. In 1996, Altschul and Gish proposed a finite-size correction to a particular approximation by a Poisson point process. Their correction is now used routinely (about once a second) when computers compare biological sequences, although it lacks a mathematical foundation. This paper generalizes their correction. For a single renewal process or several renewal processes operating in parallel, this paper gives an asymptotic expansion that contains in successive terms a Poisson point approximation, a generalization of the Altschul-Gish correction, and a correction term beyond that.
- Published
- 2001
29. Point process convergence of stochastic volatility processes with application to sample autocorrelation
- Author
-
Richard A. Davis and Thomas Mikosch
- Subjects
Statistics and Probability ,Stationary process ,Stochastic volatility ,Autoregressive conditional heteroskedasticity ,General Mathematics ,010102 general mathematics ,Sample (statistics) ,01 natural sciences ,Point process ,010104 statistics & probability ,Heavy-tailed distribution ,Econometrics ,Statistical physics ,Marginal distribution ,0101 mathematics ,Statistics, Probability and Uncertainty ,Random variable ,Mathematics - Abstract
The paper considers one of the standard processes for modeling returns in finance, the stochastic volatility process with regularly varying innovations. The aim of the paper is to show how point process techniques can be used to derive the asymptotic behavior of the sample autocorrelation function of this process with heavy-tailed marginal distributions. Unlike other non-linear models used in finance, such as GARCH and bilinear models, sample autocorrelations of a stochastic volatility process have attractive asymptotic properties. Specifically, in the infinite variance case, the sample autocorrelation function converges to zero in probability at a rate that is faster the heavier the tails of the marginal distribution. This behavior is analogous to the asymptotic behavior of the sample autocorrelations of independent identically distributed random variables.
- Published
- 2001
30. On the dependence structure and bounds of correlated parallel queues and their applications to synchronized stochastic systems
- Author
-
Susan H. Xu and Haijun Li
- Subjects
Statistics and Probability ,Queueing theory ,Mathematical optimization ,Queue management system ,General Mathematics ,010102 general mathematics ,Fork–join queue ,01 natural sciences ,Measure (mathematics) ,Upper and lower bounds ,Orthant ,010104 statistics & probability ,First-come, first-served ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Spatial dependence ,Mathematics - Abstract
This paper studies the dependence structure and bounds of several basic prototypical parallel queueing systems with correlated arrival processes to different queues. The marked feature of our systems is that each queue viewed alone is a standard single-server queuing system extensively studied in the literature, but those queues are statistically dependent due to correlated arrival streams. The major difficulty in analysing those systems is that the presence of correlation makes the explicit computation of a joint performance measure either intractable or computationally intensive. In addition, it is not well understood how and in what sense arrival correlation will improve or deteriorate a system performance measure. The objective of this paper is to provide a better understanding of the dependence structure of correlated queueing systems and to derive computable bounds for the statistics of a joint performance measure. In this paper, we obtain conditions on arrival processes under which a performance measure in two systems can be compared, in the sense of orthant and supermodular orders, among different queues and over different arrival times. Such strong comparison results enable us to study both spatial dependence (dependence among different queues) and temporal dependence (dependence over different time instances) for a joint performance measure. Further, we derive a variety of upper and lower bounds for the statistics of a stationary joint performance measure. Finally, we apply our results to synchronized queueing systems, using the ideas combined from the theory of orthant and supermodular dependence orders and majorization with respect to weighted trees (Xu and Li (2000)). Our results reveal how a performance measure can be affected, favourably or adversely, by different types of dependencies.
- Published
- 2000
31. Approximate entropy for testing randomness
- Author
-
Andrew L. Rukhin
- Subjects
Discrete mathematics ,Statistics and Probability ,Random number generation ,General Mathematics ,010102 general mathematics ,Approximate entropy ,Joint entropy ,01 natural sciences ,Rényi entropy ,010104 statistics & probability ,Maximum entropy probability distribution ,Randomness tests ,Statistical physics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Randomness ,Joint quantum entropy ,Mathematics - Abstract
This paper arose from interest in assessing the quality of random number generators. The problem of testing randomness of a string of binary bits produced by such a generator gained importance with the wide use of public key cryptography and the need for secure encryption algorithms. All such algorithms are based on a generator of (pseudo) random numbers; the testing of such generators for randomness became crucial for the communications industry where digital signatures and key management are vital for information processing. The concept of approximate entropy has been introduced in a series of papers by S. Pincus and co-authors. The corresponding statistic is designed to measure the degree of randomness of observed sequences. It is based on incremental contrasts of empirical entropies based on the frequencies of different patterns in the sequence. Sequences with large approximate entropy must have substantial fluctuation or irregularity. Alternatively, small values of this characteristic imply strong regularity, or lack of randomness, in a sequence. Pincus and Kalman (1997) evaluated approximate entropies for binary and decimal expansions of e, π, √2 and √3 with the surprising conclusion that the expansion of √3 demonstrated much less irregularity than that of π. Tractable small sample distributions are hardly available, and testing randomness is based, as a rule, on fairly long strings. Therefore, to have rigorous statistical tests of randomness based on this approximate entropy statistic, one needs the limiting distribution of this characteristic under the randomness assumption. Until now this distribution remained unknown and was thought to be difficult to obtain. To derive the limiting distribution of approximate entropy we modify its definition. It is shown that the approximate entropy as well as its modified version converges in distribution to a χ2-random variable. The P-values of approximate entropy test statistics for binary expansions of e, π and √3 are plotted. Although some of these values for √3 digits are small, they do not provide enough statistical significance against the randomness hypothesis.
- Published
- 2000
32. Bounded normal approximation in simulations of highly reliable Markovian systems
- Author
-
Bruno Tuffin
- Subjects
Statistics and Probability ,Mathematical optimization ,Stochastic process ,General Mathematics ,010102 general mathematics ,Markov process ,Bounded deformation ,01 natural sciences ,symbols.namesake ,010104 statistics & probability ,Approximation error ,Bounded function ,symbols ,Applied mathematics ,State space ,0101 mathematics ,Statistics, Probability and Uncertainty ,Bounded inverse theorem ,Mathematics ,Central limit theorem - Abstract
In this paper, we give necessary and sufficient conditions to ensure the validity of confidence intervals, based on the central limit theorem, in simulations of highly reliable Markovian systems. We resort to simulations because of the frequently huge state space in practical systems. So far the literature has focused on the property of bounded relative error. In this paper we focus on ‘bounded normal approximation’ which asserts that the approximation of the normal law, suggested by the central limit theorem, does not deteriorate as the reliability of the system increases. Here we see that the set of systems with bounded normal approximation is (strictly) included in the set of systems with bounded relative error.
- Published
- 1999
33. Excursions of birth and death processes, orthogonal polynomials, and continued fractions
- Author
-
Fabrice Guillemin and Didier Pinchon
- Subjects
Statistics and Probability ,Laplace transform ,General Mathematics ,Mathematical analysis ,010102 general mathematics ,Riemann–Stieltjes integral ,Stieltjes transformation ,01 natural sciences ,010104 statistics & probability ,Orthogonal polynomials ,Applied mathematics ,Ergodic theory ,Fraction (mathematics) ,0101 mathematics ,Statistics, Probability and Uncertainty ,Continued fraction ,Random variable ,Mathematics - Abstract
On the basis of the Karlin and McGregor result, which states that the transition probability functions of a birth and death process can be expressed via the introduction of an orthogonal polynomial system and a spectral measure, we investigate in this paper how the Laplace transforms and the distributions of different transient characteristics related to excursions of a birth and death process can be expressed by means of the basic orthogonal polynomial system and the spectral measure. This allows us in particular to give a probabilistic interpretation of the series introduced by Stieltjes to study the convergence of the fundamental continued fraction associated with the system. Throughout the paper, we pay special attention to the case when the birth and death process is ergodic. Under the assumption that the spectrum of the spectral measure is discrete, we show how the distributions of different random variables associated with excursions depend on the fundamental continued fraction, the orthogonal polynomial system and the spectral measure.
- Published
- 1999
34. Analysis of a two-queue model with Bernoulli schedules
- Author
-
Duan-Shin Lee
- Subjects
Statistics and Probability ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Fredholm integral equation ,01 natural sciences ,Riemann boundary value problem ,010104 statistics & probability ,symbols.namesake ,Bernoulli's principle ,Unit circle ,symbols ,Applied mathematics ,Boundary value problem ,Bernoulli scheme ,0101 mathematics ,Statistics, Probability and Uncertainty ,Bernoulli process ,Queue ,Mathematics - Abstract
In this paper we analyze a single server two-queue model with Bernoulli schedules. This discipline is very flexible and contains the exhaustive and 1-limited disciplines as special cases. We formulate the queueing system as a Riemann boundary value problem with shift. The boundary value problem is solved by exploring a Fredholm integral equation around the unit circle. Some numerical examples are presented at the end of the paper.
- Published
- 1997
35. A correspondence between product-form batch-movement queueing networks and single-movement networks
- Author
-
W. Henderson, J. L. Coleman, Peter G. Taylor, and Charles E. M. Pearce
- Subjects
Statistics and Probability ,Discrete mathematics ,Queueing theory ,Mathematical optimization ,General Mathematics ,010102 general mathematics ,BCMP network ,Loss network ,01 natural sciences ,010104 statistics & probability ,Evolving networks ,Product (mathematics) ,Jackson network ,Layered queueing network ,State space ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
A number of recent papers have exhibited classes of queueing networks, with batches of customers served and routed through the network, which have generalised product-form equilibrium distributions. In this paper we look at these from a new viewpoint. In particular we show that, under standard assumptions, for a network to possess an equilibrium distribution that factorises into a product form over the nodes of the network for all possible transition rates, it is necessary and sufficient that it be equivalent to a suitably-defined single-movement network. We consider also the form of the state space for such networks.
- Published
- 1997
36. Simple random walk statistics. Part I: Discrete time results
- Author
-
Wolfgang Panny and Walter Katzenbeisser
- Subjects
Statistics and Probability ,Stochastic process ,General Mathematics ,Order statistic ,010102 general mathematics ,Process (computing) ,Markov process ,Simple random sample ,Random walk ,01 natural sciences ,symbols.namesake ,010104 statistics & probability ,Discrete time and continuous time ,Statistics ,symbols ,Point (geometry) ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In a famous paper, Dwass (1967) proposed a method to deal with rank order statistics, which constitutes a unifying framework to derive various distributional results. In the present paper an alternative method is presented, which allows us to extend Dwass's results in several ways, namely arbitrary endpoints, horizontal steps and arbitrary probabilities for the three step types. Regarding these extensions the pertaining rank order statistics are extended as well to simple random walk statistics. This method has proved appropriate to generalize all results given by Dwass. Moreover, these discrete time results can be taken as a starting point to derive the corresponding results for randomized random walks by means of a limiting process.
- Published
- 1996
37. Peaks and Eulerian numbers in a random sequence
- Author
-
Di Warren and E. Seneta
- Subjects
Statistics and Probability ,Stochastic process ,General Mathematics ,010102 general mathematics ,Eulerian path ,Random permutation ,01 natural sciences ,Combinatorics ,symbols.namesake ,Permutation ,010104 statistics & probability ,Distribution (mathematics) ,Rate of convergence ,symbols ,0101 mathematics ,Statistics, Probability and Uncertainty ,Cumulant ,Mathematics ,Central limit theorem - Abstract
We consider the exact distribution of the number of peaks in a random permutation of the integers 1, 2, ···, n. This arises from a test of whether n successive observations from a continuous distribution are i.i.d. The Eulerian numbers, which figure in the p.g.f., are then shown to provide a link between the simpler problem of ascents (which has been thoroughly analysed) and both our problem of peaks and similar problems on the circle. This link then permits easy deduction of certain general properties, such as linearity in n of the cumulants, in the more complex settings. Since the focus of the paper is on exact distributional results, a uniform bound on the deviation from the limiting normal is included. A secondary purpose of the paper is synthesis, beginning with the more familiar setting of peaks and troughs.
- Published
- 1996
38. Risk-sensitive average continuous-time Markov decision processes with unbounded transition and cost rates
- Author
-
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
39. Sensitivity of mean-field fluctuations in Erlang loss models with randomized routing
- Author
-
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
40. Limit distributions for the Bernoulli meander
- Author
-
Lajos Takács
- Subjects
Statistics and Probability ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Mathematical analysis ,Brownian meander ,Geometry ,Random walk ,Infinity ,01 natural sciences ,010104 statistics & probability ,Bernoulli's principle ,Meander (mathematics) ,Local time ,Limit (mathematics) ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics ,media_common - Abstract
This paper is concerned with the distibutions and the moments of the area and the local time of a random walk, called the Bernoulli meander. The limit behavior of the distributions and the moments is determined in the case where the number of steps in the random walk tends to infinity. The results of this paper yield explicit formulas for the distributions and the moments of the area and the local time for the Brownian meander.
- Published
- 1995
41. On the asymptotic distribution of the maximum number of infectives in epidemic models by immigration
- Author
-
V. M. Abramov
- Subjects
Birth and death process ,Statistics and Probability ,Limit distribution ,General Mathematics ,010102 general mathematics ,Asymptotic distribution ,Limiting ,01 natural sciences ,Birth–death process ,010104 statistics & probability ,Statistics ,Applied mathematics ,Almost surely ,0101 mathematics ,Statistics, Probability and Uncertainty ,Martingale (probability theory) ,Epidemic model ,Mathematics - Abstract
This paper considers the asymptotic distribution of the maximum number of infectives in an epidemic model by showing that, as the initial number of susceptibles converges to infinity, the process of infectives converges almost surely to a birth and death process. The model studied here is more general than usual (see e.g. Bailey (1975), Bharucha-Reid (1960), Keilson (1979)) in that it incorporates immigration and the limiting birth and death process is non-linear. The main novelty of the present paper is the martingale approach used to prove the above-mentioned convergence. MAXIMUM NUMBER OF INFECTIVES; GAMBLER'S RUIN PROBLEM; MARTINGALE; BIRTH AND DEATH PROCESS AMS 1991 SUBJECT CLASSIFICATION: PRIMARY 92D30 SECONDARY 60J20; 60G42; 60G40; 60J80
- Published
- 1994
42. Applications of the hazard rate ordering in reliability and order statistics
- Author
-
Emad El-Neweihi, Philip J. Boland, and Frank Proschan
- Subjects
Reliability theory ,Independent and identically distributed random variables ,Statistics and Probability ,Exponential distribution ,General Mathematics ,Order statistic ,Hazard ratio ,010102 general mathematics ,Stochastic ordering ,01 natural sciences ,Combinatorics ,010104 statistics & probability ,Statistics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Majorization ,Random variable ,Mathematics - Abstract
The hazard rate ordering is an ordering for random variables which compares lifetimes with respect to their hazard rate functions. It is stronger than the usual stochastic order for random variables, yet is weaker than the likelihood ratio ordering. The hazard rate ordering is particularly useful in reliability theory and survival analysis, owing to the importance of the hazard rate function in these areas. In this paper earlier work on the hazard rate ordering is reviewed, and extensive new results related to coherent systems are derived. Initially we fix the form of a coherent structure and investigate the effect on the hazard rate function of the system when we switch the lifetimes of its components from the vector (T 1, · ··, Tn ) to the vector (T′ 1, · ··, T′n ), where the hazard rate functions of the two vectors are assumed to be comparable in some sense. Although the hazard rate ordering is closed under the formation of series systems, we see that this is not the case for parallel systems even when the system is a two-component parallel system with exponentially distributed lifetimes. A positive result shows that for two-component parallel systems with proportional hazards (λ 1 r(t), λ 2 r(t))), the more diverse (λ 1, λ2 ) is in the sense of majorization the stronger is the system in the hazard rate ordering. Unfortunately even this result does not extend to parallel systems with more than two components, demonstrating again the delicate nature of the hazard rate ordering. The principal result of the paper concerns the hazard rate ordering for the lifetime of a k-out-of-n system. It is shown that if τ k|n is the lifetime of a k-out-of-n system, then τ k|n is greater than τ k+ 1|n in the hazard rate ordering for any k. This has an interesting interpretation in the language of order statistics. For independent (not necessarily identically distributed) lifetimes T 1, · ··, Tn , we let Tk:n represent the kth order statistic (in increasing order). Then it is shown that Tk + 1:n is greater than Tk:n in the hazard rate ordering for all k = 1, ···, n − 1. The result does not, however, extend to the stronger likelihood ratio order.
- Published
- 1994
43. Queueing networks by negative customers and negative queue lengths
- Author
-
W. Henderson
- Subjects
Service (business) ,Statistics and Probability ,Queueing theory ,Mathematical optimization ,Distribution (number theory) ,Node (networking) ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Product (business) ,010104 statistics & probability ,State dependence ,Routing (electronic design automation) ,0101 mathematics ,Statistics, Probability and Uncertainty ,Queue ,Mathematics - Abstract
A number of papers have recently appeared in the literature in which customers, in moving from node to node in the network arrive as either a positive customer or as a batch of negative customers. A positive customer joining its queue increases the number of customers at the queue by 1 and each negative customer decreases the queue length by 1, if possible. It has been shown that the equilibrium distribution for these networks assumes a geometric product form, that certain partial balance equations prevail and that the parameters of the geometric distributions are, as in Jackson networks, the service facility throughputs of customers. In this paper the previous work is generalised by allowing state dependence into both the service and routing intensities and by allowing the possibility, although not the necessity, for negative customers to build up at the nodes.
- Published
- 1993
44. On strict stationarity and ergodicity of a non-linear ARMA model
- Author
-
Edward Susko and Jian Liu
- Subjects
Statistics and Probability ,Markov chain ,General Mathematics ,010102 general mathematics ,Ergodicity ,Boundary (topology) ,Context (language use) ,01 natural sciences ,010104 statistics & probability ,Moving average ,Tweedie distribution ,Calculus ,Irreducibility ,Applied mathematics ,Autoregressive–moving-average model ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
Two recent papers by Petruccelli and Woolford (1984) and Chan et al. (1985) showed that the key element governing ergodicity of a threshold AR(1) model is the joint behavior of the two linear AR(1) pieces falling in the two boundary threshold regimes. They used essentially the necessary and sufficient conditions for ergodicity of a general Markov chain of Tweedie (1974), (1975) in a rather clever manner. However, it is difficult to extend the results to the more general threshold ARMA models. Besides, irreducibility is also required to apply Tweedie's results. In this paper, instead of pursuing the ideas in Tweedie's results, we shall develop a criterion similar in spirit to the technique used by Beneš (1967) in the context of continuous-time Markov chains. Consequently, we derive a necessary and sufficient condition for existence of a strictly stationary solution of a general non-linear ARMA model to be introduced in Section 2 of this paper. This condition is then applied to the threshold ARMA(1, q) model to yield a sufficient condition for strict stationarity which is identical to the condition given by Petruccelli and Woolford (1984) for the threshold AR(1). Hence, the conjecture that the moving average component does not affect stationarity is partially verified. Furthermore, under an additional irreducibility assumption, ergodicity of a non-linear ARMA model is established. The paper then concludes with a necessary condition for stationarity of the threshold ARMA(1, q) model.
- Published
- 1992
45. 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
46. Diffusion approximations for randomly arriving expert opinions in a financial market with Gaussian drift
- Author
-
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
47. Dynamic multivariate mean residual life functions
- Author
-
Moshe Shaked and J. George Shanthikumar
- Subjects
Statistics and Probability ,Reliability theory ,Multivariate statistics ,Multivariate random variable ,General Mathematics ,010102 general mathematics ,Context (language use) ,Residual ,Conditional expectation ,01 natural sciences ,Stochastic ordering ,010104 statistics & probability ,Statistics ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Partially ordered set ,Mathematics - Abstract
In this paper we introduce and study a dynamic notion of mean residual life (mrl) functions in the context of multivariate reliability theory. Basic properties of these functions are derived and their relationship to the multivariate conditional hazard rate functions is studied. A partial ordering, called the mrl ordering, of non-negative random vectors is introduced and its basic properties are presented. Its relationship to stochastic ordering and to other related orderings (such as hazard rate ordering) is pointed out. Using this ordering it is possible to introduce a weak notion of positive dependence of random lifetimes. Some properties of this positive dependence notion are given. Finally, using the mrl ordering, a dynamic notion of multivariate DMRL (decreasing mean residual life) is introduced and studied. The relationship of this multivariate DMRL notion to other notions of dynamic multivariate aging is highlighted in this paper.
- Published
- 1991
48. The Galton-Watson predator-prey process
- Author
-
Wolfgang J. Bühler and John Coffey
- Subjects
Statistics and Probability ,education.field_of_study ,Offspring ,General Mathematics ,010102 general mathematics ,Population ,Probabilistic logic ,Particle (ecology) ,01 natural sciences ,Galton–Watson process ,Predation ,010104 statistics & probability ,Statistics ,0101 mathematics ,Statistics, Probability and Uncertainty ,education ,Predator ,Mathematics ,Branching process - Abstract
A probabilistic predator-prey model is constructed using linked discrete-time branching-type processes. A necessary and sufficient condition for positive pro- bability of survival of both populations is given. In this paper, we introduce a new probabilistic model for the relationship between the sizes of a population of predators and a population of prey. This process is formulated in terms of a modified two-type discrete-time branching process. The population of predators is described by an ordinary Galton-Watson process. In the population of prey, each surviving particle of the nth generation produces offspring independently of all other particles. However, not all of these prey offspring are allowed to reproduce. Instead, each member of the (n + 1)th generation of predators consumes a random number of these offspring prey particles. (If there are not enough prey offspring to go around, then they are all eaten and the prey population is extinct.) The main result of this paper is a necessary and sufficient condition for there to be positive probability of indefinitely long survival for both populations.
- Published
- 1991
49. On geometric and algebraic transience for block-structured Markov chains
- Author
-
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
50. 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
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