Showing papers in "Journal of Applied Probability in 2021"

A product form for the general stochastic matching model

Abstract: We consider a stochastic matching model with a general compatibility graph, as introduced by Mairesse and Moyal (2016). We show that the natural necessary condition of stability of the system is also sufficient for the natural ‘first-come, first-matched’ matching policy. To do so, we derive the stationary distribution under a remarkable product form, by using an original dynamic reversibility property related to that of Adan, Busic, Mairesse, and Weiss (2018) for the bipartite matching model.

Exact simulation of Ornstein-Uhlenbeck tempered stable processes

Abstract: There are two types of tempered stable (TS) based Ornstein–Uhlenbeck (OU) processes: (i) the OU-TS process, the OU process driven by a TS subordinator, and (ii) the TS-OU process, the OU process with TS marginal law. They have various applications in financial engineering and econometrics. In the literature, only the second type under the stationary assumption has an exact simulation algorithm. In this paper we develop a unified approach to exactly simulate both types without the stationary assumption. It is mainly based on the distributional decomposition of stochastic processes with the aid of an acceptance–rejection scheme. As the inverse Gaussian distribution is an important special case of TS distribution, we also provide tailored algorithms for the corresponding OU processes. Numerical experiments and tests are reported to demonstrate the accuracy and effectiveness of our algorithms, and some further extensions are also discussed.

On classes of life distributions based on the mean time to failure function

Abstract: The performance and effectiveness of an age replacement policy can be assessed by its mean time to failure (MTTF) function. We develop shock model theory in different scenarios for classes of life distributions based on the MTTF function where the probabilities of surviving the first k shocks are assumed to have discrete DMTTF, IMTTF and IDMTTF properties. The cumulative damage model of A-Hameed and Proschan [1] is studied in this context and analogous results are established. Weak convergence and moment convergence issues within the IDMTTF class of life distributions are explored. The preservation of the IDMTTF property under some basic reliability operations is also investigated. Finally we show that the intersection of IDMRL and IDMTTF classes contains the BFR family and establish results outlining the positions of various non-monotonic ageing classes in the hierarchy.

Risk-sensitive average continuous-time Markov decision processes with unbounded transition and cost rates

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.

Optimal allocation of relevations in coherent systems

Abstract: In this paper we study the allocation problem of relevations in coherent systems. The optimal allocation strategies are obtained by implementing stochastic comparisons of different policies according to the usual stochastic order and the hazard rate order. As special cases of relevations, the load-sharing and minimal repair policies are further investigated. Sufficient (and necessary) conditions are established for various stochastic orderings. Numerical examples are also presented as illustrations.

Stochastic comparisons and ageing properties of an extended gamma process

Abstract: Extended gamma processes have been seen as a flexible extension of standard gamma processes in the recent reliability literature, for the purpose of cumulative deterioration modeling. The probabilistic properties of the standard gamma process have been well explored since the 1970s, whereas those of its extension remain largely unexplored. In particular, stochastic comparisons between degradation levels modeled by standard gamma processes and ageing properties for the corresponding level-crossing times are now well understood. The aim of this paper is to explore similar properties for extended gamma processes and see which ones can be broadened to this new context. As a by-product, new stochastic comparisons for convolutions of gamma random variables are also obtained.

Subgeometric ergodicity and β-mixing

Abstract: It is well known that stationary geometrically ergodic Markov chains are -mixing (absolutely regular) with geometrically decaying mixing coefficients. Furthermore, for initial distributions other than the stationary one, geometric ergodicity implies -mixing under suitable moment assumptions. In this note we show that similar results hold also for subgeometrically ergodic Markov chains. In particular, for both stationary and other initial distributions, subgeometric ergodicity implies -mixing with subgeometrically decaying mixing coefficients. Although this result is simple, it should prove very useful in obtaining rates of mixing in situations where geometric ergodicity cannot be established. To illustrate our results we derive new subgeometric ergodicity and -mixing results for the self-exciting threshold autoregressive model.

Diffusion approximations for randomly arriving expert opinions in a financial market with Gaussian drift

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.

Bounds for expected supremum of fractional Brownian motion with drift

Abstract: We provide upper and lower bounds for the mean M (H) of supp t≥0{BH (t)} , with BH (.) a zero-mean, variance-normalized version of fractional Brownian motion with Hurst parameter H ∈ (o,1). We find bounds in (semi-) closed form, distinguishing between H ∈ (0, ½] and H ∈ [½, 1) , where in the former regime a numerical procedure is presented that drastically reduces the upper bound. For H ∈ (0, ½] , the ratio between the upper and lower bound is bounded, whereas for H ∈ [½, 1) the derived upper and lower bound have a strongly similar shape. We also derive a new upper bound for the mean of sup t∈[0,1] BH (t), H ∈ (0, ½] , which is tight around H = ½.

Variations of the elephant random walk

Abstract: In the classical simple random walk the steps are independent, that is, the walker has no memory. In contrast, in the elephant random walk, which was introduced by Schutz and Trimper [19] in 2004, the next step always depends on the whole path so far. Our main aim is to prove analogous results when the elephant has only a restricted memory, for example remembering only the most remote step(s), the most recent step(s), or both. We also extend the models to cover more general step sizes.

Scale-free percolation in continuous space: quenched degree and clustering coefficient

Abstract: Spatial random graphs capture several important properties of real-world networks. We prove quenched results for the continuous-space version of scale-free percolation introduced in [14]. This is an undirected inhomogeneous random graph whose vertices are given by a Poisson point process in . Each vertex is equipped with a random weight, and the probability that two vertices are connected by an edge depends on their weights and on their distance. Under suitable conditions on the parameters of the model, we show that, for almost all realizations of the point process, the degree distributions of all the nodes of the graph follow a power law with the same tail at infinity. We also show that the averaged clustering coefficient of the graph is self-averaging. In particular, it is almost surely equal to the annealed clustering coefficient of one point, which is a strictly positive quantity.

Stochastic orderings of multivariate elliptical distributions

Abstract: For two n-dimensional elliptical random vectors X and Y, we establish an identity for satisfies some regularity conditions. Using this identity we provide a unified method to derive sufficient and necessary conditions for classifying multivariate elliptical random vectors according to several main integral stochastic orders. As a consequence we obtain new inequalities by applying the method to multivariate elliptical distributions. The results generalize the corresponding ones for multivariate normal random vectors in the literature.

Multitype branching process with non-homogeneous Poisson and contagious Poisson immigration

Abstract: In a multitype branching process, it is assumed that immigrants arrive according to a non-homogeneous Poisson or a contagious Poisson process (both processes are formulated as a non-homogeneous birth process with an appropriate choice of transition intensities). We show that the normalized numbers of objects of the various types alive at time t for supercritical, critical, and subcritical cases jointly converge in distribution under those two different arrival processes. Furthermore, we provide some transient expectation results when there are only two types of particles.

Ergodic control of diffusions with random intervention times

Abstract: We study an ergodic singular control problem with constraint of a regular one-dimensional linear diffusion. The constraint allows the agent to control the diffusion only at the jump times of an independent Poisson process. Under relatively weak assumptions, we characterize the optimal solution as an impulse-type control policy, where it is optimal to exert the exact amount of control needed to push the process to a unique threshold. Moreover, we discuss the connection of the present problem to ergodic singular control problems, and illustrate the results with different well-known cost and diffusion structures.

General drawdown of general tax model in a time-homogeneous Markov framework

Abstract: Drawdown/regret times feature prominently in optimal stopping problems, in statistics (CUSUM procedure), and in mathematical finance (Russian options). Recently it was discovered that a first passage theory with more general drawdown times, which generalize classic ruin times, may be explicitly developed for spectrally negative Levy processes [9, 20]. In this paper we further examine the general drawdown-related quantities in the (upward skip-free) time-homogeneous Markov process, and then in its (general) tax process by noticing the pathwise connection between general drawdown and the tax process.

Two poisson limit theorems for the coupon collector’s problem with group drawings

Abstract: In the collector’s problem with group drawings, s out of n different types of coupon are sampled with replacement. In the uniform case, each s-subset of the types has the same probability of being sampled. For this case, we derive a Poisson limit theorem for the number of types that are sampled at most times, where is fixed. In a specified approximate nonuniform setting, we prove a Poisson limit theorem for the special case . As corollaries, we obtain limit distributions for the waiting time for c complete series of types in the uniform case and a single complete series in the approximate nonuniform case.

Sequential metric dimension for random graphs

Abstract: In the localization game on a graph, the goal is to find a fixed but unknown target node with the least number of distance queries possible. In the jth step of the game, the player queries a single node and receives, as an answer to their query, the distance between the nodes and . The sequential metric dimension (SMD) is the minimal number of queries that the player needs to guess the target with absolute certainty, no matter where the target is. The term SMD originates from the related notion of metric dimension (MD), which can be defined the same way as the SMD except that the player’s queries are non-adaptive. In this work we extend the results of Bollobas, Mitsche, and Pralat [4] on the MD of Erdős–Renyi graphs to the SMD. We find that, in connected Erdős–Renyi graphs, the MD and the SMD are a constant factor apart. For the lower bound we present a clean analysis by combining tools developed for the MD and a novel coupling argument. For the upper bound we show that a strategy that greedily minimizes the number of candidate targets in each step uses asymptotically optimal queries in Erdős–Renyi graphs. Connections with source localization, binary search on graphs, and the birthday problem are discussed.

On transform orders for largest claim amounts

Abstract: This paper investigates the ordering properties of largest claim amounts in heterogeneous insurance portfolios in the sense of some transform orders, including the convex transform order and the star order. It is shown that the largest claim amount from a set of independent and heterogeneous exponential claims is more skewed than that from a set of independent and homogeneous exponential claims in the sense of the convex transform order. As a result, a lower bound for the coefficient of variation of the largest claim amount is established without any restrictions on the parameters of the distributions of claim severities. Furthermore, sufficient conditions are presented to compare the skewness of the largest claim amounts from two sets of independent multiple-outlier scaled claims according to the star order. Some comparison results are also developed for the multiple-outlier proportional hazard rates claims. Numerical examples are presented to illustrate these theoretical results.

Extinction time of the logistic process

Abstract: The logistic birth and death process is perhaps the simplest stochastic population model that has both density-dependent reproduction and a phase transition, and a lot can be learned about the process by studying its extinction time, , as a function of system size n. A number of existing results describe the scaling of as for various choices of reproductive rate and initial population as a function of n. We collect and complete this picture, obtaining a complete classification of all sequences and for which there exist rescaling parameters and such that converges in distribution as , and identifying the limits in each case.

Large-deviation asymptotics of condition numbers of random matrices

Abstract: Let be a random matrix whose entries are independent and identically distributed real random variables with zero mean and unit variance. We study the limiting behaviors of the 2-normal condition number k(p,n) of in terms of large deviations for large n, with p being fixed or with . We propose two main ingredients: (i) to relate the large-deviation probabilities of k(p,n) to those involving n independent and identically distributed random variables, which enables us to consider a quite general distribution of the entries (namely the sub-Gaussian distribution), and (ii) to control, for standard normal entries, the upper tail of k(p,n) using the upper tails of ratios of two independent random variables, which enables us to establish an application in statistical inference.

Double hypergeometric Lévy processes and self-similarity

Abstract: Motivated by a recent paper (Budd (2018)), where a new family of positive self-similar Markov processes associated to stable processes appears, we introduce a new family of Levy processes, called the double hypergeometric class, whose Wiener–Hopf factorisation is explicit, and as a result many functionals can be determined in closed form.

Couplings for determinantal point processes and their reduced Palm distributions with a view to quantifying repulsiveness.

Abstract: For a determinantal point process (DPP) X with a kernel K whose spectrum is strictly less than one, Andre Goldman has established a coupling to its reduced Palm process , including Ginibre point processes and other specific parametric models for DPPs.

On the forward algorithm for stopping problems on continuous-time Markov chains

Abstract: We revisit the forward algorithm, developed by Irle, to characterize both the value function and the stopping set for a large class of optimal stopping problems on continuous-time Markov chains. Our objective is to renew interest in this constructive method by showing its usefulness in solving some constrained optimal stopping problems that have emerged recently.

Stochastic analysis of average-based distributed algorithms

Abstract: We analyze average-based distributed algorithms relying on simple and pairwise random interactions among a large and unknown number of anonymous agents. This allows the characterization of global properties emerging from these local interactions. Agents start with an initial integer value, and at each interaction keep the average integer part of both values as their new value. The convergence occurs when, with high probability, all the agents possess the same value, which means that they all know a property of the global system. Using a well-chosen stochastic coupling, we improve upon existing results by providing explicit and tight bounds on the convergence time. We apply these general results to both the proportion problem and the system size problem.

Tail Asymptotics of an Infinitely Divisible Space-time Model with Convolution Equivalent Lévy Measure

Abstract: We consider a space-time random field on given as an integral of a kernel function with respect to a Levy basis with a convolution equivalent Levy measure. The field obeys causality in time and is thereby not continuous along the time axis. For a large class of such random fields we study the tail behaviour of certain functionals of the field. It turns out that the tail is asymptotically equivalent to the right tail of the underlying Levy measure. Particular examples are the asymptotic probability that there is a time point and a rotation of a spatial object with fixed radius, in which the field exceeds the level x, and that there is a time interval and a rotation of a spatial object with fixed radius, in which the average of the field exceeds the level x.

On the comparison of Shapley values for variance and standard deviation games

Abstract: Motivated by the problem of variance allocation for the sum of dependent random variables, Colini-Baldeschi, Scarsini and Vaccari (2018) recently introduced Shapley values for variance and standard deviation games. These Shapley values constitute a criterion satisfying nice properties useful for allocating the variance and the standard deviation of the sum of dependent random variables. However, since Shapley values are in general computationally demanding, Colini-Baldeschi, Scarsini and Vaccari also formulated a conjecture about the relation of the Shapley values of two games, which they proved for the case of two dependent random variables. In this work we prove that their conjecture holds true in the case of an arbitrary number of independent random variables but, at the same time, we provide counterexamples to the conjecture for the case of three dependent random variables.

Tail asymptotics for the area under the excursion of a random walk with heavy-tailed increments

Abstract: We study the tail behaviour of the distribution of the area under the positive excursion of a random walk which has negative drift and heavy-tailed increments. We determine the asymptotics for tail probabilities for the area.

Gibbs partitions, Riemann-Liouville fractional operators, Mittag-Leffler functions, and fragmentations derived from stable subordinators

Abstract: Pitman (2003), and subsequently Gnedin and Pitman (2006), showed that a large class of random partitions of the integers derived from a stable subordinator of index -diversity.

Gaussian process approximations for multicolor Pólya urn models

Abstract: Motivated by mathematical tissue growth modelling, we consider the problem of approximating the dynamics of multicolor Polya urn processes that start with large numbers of balls of different colors and run for a long time. Using strong approximation theorems for empirical and quantile processes, we establish Gaussian process approximations for the Polya urn processes. The approximating processes are sums of a multivariate Brownian motion process and an independent linear drift with a random Gaussian coefficient. The dominating term between the two depends on the ratio of the number of time steps n to the initial number of balls N in the urn. We also establish an upper bound of the form for the maximum deviation over the class of convex Borel sets of the step-n urn composition distribution from the approximating normal law.

Bounds on the mean and squared coefficient of variation of phase-type distributions

Abstract: We consider a class of phase-type distributions (PH-distributions), to be called the MMPP class of PH-distributions, and find bounds of their mean and squared coefficient of variation (SCV). As an application, we have shown that the SCV of the event-stationary inter-event time for Markov modulated Poisson processes (MMPPs) is greater than or equal to unity, which answers an open problem for MMPPs. The results are useful for selecting proper PH-distributions and counting processes in stochastic modeling.