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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.

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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.

In this paper we prove a quenched functional central limit theorem for a biased random walk on a supercritical Galton–Watson tree with leaves. This extends a result of Peres and Zeitouni (2008) where the case without leaves was considered. A conjecture of Ben Arous and Fribergh (2016) suggests an upper bound on the bias which we observe to be sharp.

A classical P��lya urn scheme is a Markov process whose evolution is encoded by a replacement matrix $(R_{i,j})_{1\leq i,j\leq d}$. At every discrete time-step, we draw a ball uniformly at random, denote its colour $c$, and replace it in the urn together with $R_{c,j}$ balls of colour $j$ (for all $1\leq j\leq d$). We are interested in multi-drawing P��lya urns, where the replacement rule depends on the random drawing of a set of $m$ balls from the urn (with or without replacement). This generalisation has already been studied in the literature, in particular by Kuba & Mahmoud (ArXiv:1503.09069 and 1509.09053), where second order asymptotic results are proved for $2$-colour urns under the balanced and the affinity assumptions. The main idea of this work is to apply stochastic approximation methods to this problem, which enables us to remove the affinity hypothesis of Kuba & Mahmoud and generalise the result to more-than-two-colour urns. We also give some partial results in the two-colour non-balanced case., This new arxiv version (v6) corrects a mistake that we discovered in the previous versions of this paper (v1-5). The mistake was in Theorem 1$(a)$ and in the last sentence of Theorem 4. In this new version, Theorem 1$(a)$ has been corrected, and Theorem 4 has been deleted

In this paper we consider the problem of simultaneously estimating rare-event probabilities for a class of Gaussian random fields. A conventional rare-event simulation method is usually tailored to a specific rare event and consequently would lose estimation efficiency for different events of interest, which often results in additional computational cost in such simultaneous estimation problems. To overcome this issue, we propose a uniformly efficient estimator for a general family of Hölder continuous Gaussian random fields. We establish the asymptotic and uniform efficiency of the proposed method and also conduct simulation studies to illustrate its effectiveness.

In this paper we present a set of results relating to the occupation time α(t) of a processX(·). The first set of results concerns exact characterizations of α(t), e.g. in terms of its transform up to an exponentially distributed epoch. In addition, we establish a central limit theorem (entailing that a centered and normalized version of α(t)∕tconverges to a zero-mean normal random variable ast→∞) and the tail asymptotics of ℙ(α(t)∕t≥q). We apply our findings to spectrally positive Lévy processes reflected at the infimum and establish various new occupation time results for the corresponding model.

In this paper, by using the exact tail asymptotics derived by Debicki, Hashorva and Ji (Ann. Probab. 2014), we proved the Gumbel limit theorem for the maximum of a class of non-homogeneous Gaussian random fields. By using the obtained results, we also derived the Gumbel laws for Shepp statistics of fractional Brownian motion and Gaussian integrated process as well as the Gumbel law for Storage process with fractional Brownian motion as

In this paper we provide a sufficient condition for mean residual life ordering of parallel systems with n ≥ 3 heterogeneous exponential components.

In this paper we study the existence of Lebesgue densities of one-dimensional Lévy processes. Equivalently, we show the absolute continuity of the distributions of one-dimensional Lévy processes. Compared with the previous literature, we consider Lévy processes with Lévy symbols of a logarithmic behavior at ∞.

In this paper we consider a diffusive stochastic predator–prey model with a nonlinear functional response and the randomness is assumed to be of Gaussian nature. A large deviation principle is established for solution processes of the considered model by implementing the weak convergence technique.

In this paper we deal with an optimal stopping problem whose objective is to maximize the probability of selecting k out of the last ℓ successes, given a sequence of independent Bernoulli trials of length N, where k and ℓ are predetermined integers satisfying 1≤k≤ℓ

In the paper we consider the following modification of a discrete-time branching process with stationary immigration. In each generation a binomially distributed subset of the population will be observed. The number of observed individuals constitute a partially observed branching process. After inspection both observed and unobserved individuals may change their offspring distributions. In the subcritical case we investigate the possibility of using the known estimators for the offspring mean and for the mean of the stationary-limiting distribution of the process when the observation of the population sizes is restricted. We prove that, if both the population and the number of immigrants are partially observed, the estimators are still strongly consistent. We also prove that the `skipped' version of the estimator for the offspring mean is asymptotically normal and the estimator of the stationary distribution's mean is asymptotically normal under additional assumptions.

In this paper we study the expected rank problem under full information. Our approach uses the planar Poisson approach from Gnedin (2007) to derive the expected rank of a stopping rule that is one of the simplest nontrivial examples combining rank dependent rules with threshold rules. This rule attains an expected rank lower than the best upper bounds obtained in the literature so far, in particular, we obtain an expected rank of 2.326 14.

Consider a d-type (dk≥2 individuals in the nth generation, pick k individuals at random from the nth generation and trace their lines of descent backward in time till they meet. In this paper, the limit behaviors of the distributions of the generation number of the most recent common ancestor of any k chosen individuals and of the whole population are studied for both critical and subcritical cases. Also, we investigate the limit distribution of the joint distribution of the generation number and their types.

In this paper we study the asymptotic behaviour near extinction of (sub-)critical continuous-state branching processes. In particular, we establish an analogue of Khintchine's law of the iterated logarithm near extinction time for a continuous-state branching process whose branching mechanism satisfies a given condition.

In this paper we study a special class of size dependent branching processes. We assume that for some positive integer K as long as the population size does not exceed level K, the process evolves as a discrete-time supercritical branching process, and when the population size exceeds level K, it evolves as a subcritical or critical branching process. It is shown that this process does die out in finite time T. The question of when the mean value E(T) is finite or infinite is also addressed.

In this paper we connect Archimedean survival processes (ASPs) with the theory of Markov copulas. ASPs were introduced by Hoyle and Mengütürk (2013) to model the realized variance of two assets. We present some new properties of ASPs related to their dependency structure. We study weak and strong Markovian consistency properties of ASPs. An ASP is weak Markovian consistent, but generally not strong Markovian consistent. Our results contain necessary and sufficient conditions for an ASP to be strong Markovian consistent. These properties are closely related to the concept of Markov copulas, which is very useful in modelling different dependence phenomena. At the end we present possible applications.

In this paper, we investigate the functional central limit theorem for stochastic processes associated to partial sums of additive functionals of reversible Markov chains with general spate space, under the normalization standard deviation of partial sums. For this case, we show that the functional central limit theorem is equivalent to the fact that the variance of partial sums is regularly varying with exponent 1 and the partial sums satisfy the CLT. It is also equivalent to the conditional CLT., 10 pages

In this paper we derive sufficient conditions for the permanence and ergodicity of a stochastic predator–prey model with a Beddington–DeAngelis functional response. The conditions obtained are in fact very close to the necessary conditions. Both nondegenerate and degenerate diffusions are considered. One of the distinctive features of our results is that they enable the characterization of the support of a unique invariant probability measure. It proves the convergence in total variation norm of the transition probability to the invariant measure. Comparisons to the existing literature and matters related to other stochastic predator–prey models are also given.

In this paper we consider an M/M/c queue modified to allow both mass arrivals when the system is empty and the workload to be removed. Properties of queues which terminate when the server becomes idle are firstly developed. Recurrence properties, equilibrium distribution, and equilibrium queue-size structure are studied for the case of resurrection and no mass exodus. All of these results are then generalized to allow for the removal of the entire workload. In particular, we obtain the Laplace transformation of the transition probability for the absorptive M/M/c queue.

In this paper we establish the central limit theorem for a class of stochastic partial differential equations and as an application derive this theorem for two widely studied population models: super-Brownian motion and the Fleming-Viot process.

Consider an absolutely continuous distribution on [0, ∞) with finite mean μ and hazard rate function h(t) ≤ b for all t. For bμ close to 1, we would expect F to be approximately exponential. In this paper we obtain sharp bounds for the Kolmogorov distance between F and an exponential distribution with mean μ, as well as between F and an exponential distribution with failure rate b. We apply these bounds to several examples. Applications are presented to geometric convolutions, birth and death processes, first-passage times, and to decreasing mean residual life distributions.

In this paper we use the Siegert formula to derive alternative expressions for the moments of the first passage time of the Ornstein-Uhlenbeck process through a constant threshold. The expression for the nth moment is recursively linked to the lower-order moments and consists of only n terms. These compact expressions can substantially facilitate (numerical) applications also for higher-order moments.

We introduce a notion of $k$th order stochastic monotonicity and duality that allows one to unify the notion used in insurance mathematics (sometimes refereed to as Siegmund's duality) for the study of ruin probability and the duality responsible for the so-called put - call symmetries in option pricing. Our general $k$th order duality can be financially interpreted as put - call symmetry for powered options. The main objective of the present paper is to develop an effective analytic approach to the analysis of duality leading to the full characterization of $k$th order duality of Markov processes in terms of their generators, which is new even for the well-studied case of put -call symmetries., Comment: To appear in Journal of Applied Probability 52:1 (March 2015)

The asymptotic behaviour of many locally branching epidemic models can, at least to first order, be deduced from the limit theory of two branching processes. The first is Whittle's (1955) branching approximation to the early stages of the epidemic, the phase in which approximately exponential growth takes place. The second is the susceptibility approximation; the backward branching process that approximates the history of the contacts that would lead to an individual becoming infected. The simplest coupling arguments for demonstrating the closeness of these branching process approximations do not keep the processes identical for quite long enough. Thus, arguments showing that the differences are unimportant are also needed. In this paper we show that, for some models, couplings can be constructed that are sufficiently accurate for this extra step to be dispensed with.

The extremes of a univariate Markov chain with regularly varying stationary marginal distribution and asymptotically linear behavior are known to exhibit a multiplicative random walk structure called the tail chain. In this paper we extend this fact to Markov chains with multivariate regularly varying marginal distributions in R d . We analyze both the forward and the backward tail process and show that they mutually determine each other through a kind of adjoint relation. In a broader setting, we will show that even for non-Markovian underlying processes a Markovian forward tail chain always implies that the backward tail chain is also Markovian. We analyze the resulting class of limiting processes in detail. Applications of the theory yield the asymptotic distribution of both the past and the future of univariate and multivariate stochastic difference equations conditioned on an extreme event.

Onwards from the mid-twentieth century, the stochastic filtering problem has caught the attention of thousands of mathematicians, engineers, statisticians, and computer scientists. Its applications span the whole spectrum of human endeavour, including satellite tracking, credit risk estimation, human genome analysis, and speech recognition. Stochastic filtering has engendered a surprising number of mathematical techniques for its treatment and has played an important role in the development of new research areas, including stochastic partial differential equations, stochastic geometry, rough paths theory, and Malliavin calculus. It also spearheaded research in areas of classical mathematics, such as Lie algebras, control theory, and information theory. The aim of this paper is to give a brief historical account of the subject concentrating on the continuous-time framework.

In this paper we consider a one dimensional stochastic system described by an elliptic equation. A spatially varying random coefficient is introduced to account for uncertainty or imprecise measurements. We model the logarithm of this coefficient by a Gaussian process and provide asymptotic approximations of the tail probabilities of the derivative of the solution.

In this paper we work in the framework of a k-dimensional vector of log-linear risks. Under weak conditions on the marginal tails and the dependence structure of a vector of positive risks, we derive the asymptotic tail behaviour of the aggregated risk and present an application concerning log-normal risks with stochastic volatility.

In this paper we discuss an exponential integrator scheme, based on spatial discretization and time discretization, for a class of stochastic partial differential equations. We show that the scheme has a unique stationary distribution whenever the step size is sufficiently small, and that the weak limit of the stationary distribution of the scheme as the step size tends to 0 is in fact the stationary distribution of the corresponding stochastic partial differential equations.

In this paper we consider some nonstandard renewal risk models with some dependent claim sizes and stochastic return, where an insurance company is allowed to invest her/his wealth in financial assets, and the price process of the investment portfolio is described as a geometric Lévy process. When the claim size distribution belongs to some classes of heavy-tailed distributions and a constraint is imposed on the Lévy process in terms of its Laplace exponent, we obtain some asymptotic formulae for the tail probability of discounted aggregate claims and ruin probabilities holding uniformly for some finite or infinite time horizons.