Showing papers in "Journal of Applied Probability in 2020"
TL;DR: A sufficient condition under which the probability that an incoming job is routed to an idle server is 1 asymptotically (as $N \to \infty$) at steady state is established.
Abstract: We study a class of load-balancing algorithms for many-server systems (N servers). Each server has a buffer of size (r a positive integer). The proof of the main result is based on the framework of Stein’s method. A key contribution is to use a simple generator approximation based on state space collapse.
29 citations
TL;DR: This work presents an alternative exact algorithm to sample in discrete spaces that avoids the eigenvalues and the Eigenvectors computation in determinantal point processes.
Abstract: Determinantal point processes (DPPs) enable the modeling of repulsion: they provide diverse sets of points. The repulsion is encoded in a kernel K that can be seen as a matrix storing the similarity between points. The diversity comes from the fact that the inclusion probability of a subset is equal to the determinant of a submatrice of K. The exact algorithm to sample DPPs uses the spectral decomposition of K, a computation that becomes costly when dealing with a high number of points. Here, we present an alternative exact algorithm in the discrete setting that avoids the eigenvalues and the eigenvectors computation. Instead, it relies on Cholesky decompositions. This is a two steps strategy: first, it samples a Bernoulli point process with an appropriate distribution, then it samples the target DPP distribution through a thinning procedure. Not only is the method used here innovative, but this algorithm can be competitive with the original algorithm or even faster for some applications specified here.
22 citations
TL;DR: This contribution approximations for both simultaneous ruin probability and simultaneous ruin time for the two-dimensional Brownian risk model when the initial capital increases to infinity are derived.
Abstract: The ruin probability in the classical Brownian risk model can be explicitly calculated for both finite and infinite time horizon. This is not the case for the simultaneous ruin probability in the two-dimensional Brownian risk model. Relying on asymptotic theory, we derive in this contribution approximations for both simultaneous ruin probability and simultaneous ruin time for the two-dimensional Brownian risk model when the initial capital increases to infinity.
19 citations
TL;DR: Branching processes as discussed by the authors generalize/unify diverse results from the literature and lead to a classification of the processes, which can be classified into three classes: branched processes, branching processes, and branching processes.
Abstract: Branching processes . The theorems generalize/unify diverse results from the literature and lead to a classification of the processes.
18 citations
TL;DR: This work proposes and analyzes a family of spatially inhomogeneous epidemic models to model epidemic dynamics with spatial variations and environmental noise using stochastic partial differential equations (SPDEs).
Abstract: This work proposes and analyzes a family of spatially inhomogeneous epidemic models. This is our first effort to use stochastic partial differential equations (SPDEs) to model epidemic dynamics with spatial variations and environmental noise. After setting up the problem, the existence and uniqueness of solutions of the underlying SPDEs are examined. Then, definitions of permanence and extinction are given, and certain sufficient conditions are provided for permanence and extinction. Our hope is that this paper will open up windows for investigation of epidemic models from a new angle.
16 citations
TL;DR: It is proved that the class of systems with identically distributed (ID) components which have a diagonal-dependent copula is much larger than the class with EXC components, which is used to compare systems with non-EXC components.
Abstract: The signature representation shows that the reliability of the system is a mixture of the reliability functions of the k-out-of-n systems. The first representation was obtained for systems with independent and identically distributed (IID) components and after it was extended to exchangeable (EXC) components. The purpose of the present paper is to extend it to the class of systems with identically distributed (ID) components which have a diagonal-dependent copula. We prove that this class is much larger than the class with EXC components. This extension is used to compare systems with non-EXC components.
15 citations
TL;DR: In this article, the authors considered a critical branching process with immigration in a random environment and investigated the tail distribution of the so-called life period of the process, i.e. the length of the time interval between the moment when the process is initiated by a positive number of particles and the moment of no individuals in the population for the first time.
Abstract: A critical branching process with immigration which evolves in a random environment is considered. Assuming that immigration is not allowed when there are no individuals in the population, we investigate the tail distribution of the so-called life period of the process, i.e. the length of the time interval between the moment when the process is initiated by a positive number of particles and the moment when there are no individuals in the population for the first time.
13 citations
TL;DR: The two processes INGAR+ and GAR+ are shown to be connected via a duality relation and a detailed analysis of the time-dependent and stationary behavior of the INGar+ process is presented.
Abstract: We introduce two general classes of reflected autoregressive processes, INGAR+ and GAR+. Here, INGAR+ can be seen as the counterpart of INAR(1) with general thinning and reflection being imposed to keep the process non-negative; GAR+ relates to AR(1) in an analogous manner. The two processes INGAR+ and GAR+ are shown to be connected via a duality relation. We proceed by presenting a detailed analysis of the time-dependent and stationary behavior of the INGAR+ process, and then exploit the duality relation to obtain the time-dependent and stationary behavior of the GAR+ process.
12 citations
TL;DR: In this article, a stationary Gibbs particle process with deterministically bounded particles on Euclidean space defined in terms of an activity parameter and non-negative interaction potentials was studied.
Abstract: We study a stationary Gibbs particle process with deterministically bounded particles on
Euclidean space defined in terms of an activity parameter and non-negative interaction
potentials of finite range. Using disagreement percolation we prove exponential decay of
the correlation functions, provided a dominating Boolean model is subcritical. We also
prove this property for the weighted moments of a U-statistic of the process. Under the
assumption of a suitable lower bound on the variance, this implies a central limit theorem
for such U-statistics of the Gibbs particle process. A byproduct of our approach is a new
uniqueness result for Gibbs particle processes.
11 citations
TL;DR: It is proved that it suffices to connect every point to c d,1 log log n points chosen randomly among its cd,2 log n-nearest neighbors to ensure a giant component of size n - o(n) with high probability.
Abstract: If we pick n random points uniformly in [0, 1] d and connect each point to its c d log n-nearest neighbors, where d ≥ 2 is the dimension and c d is a constant depending on the dimension, then it is well known that the graph is connected with high probability. We prove that it suffices to connect every point to c d,1 log log n points chosen randomly among its c d,2 log n-nearest neighbors to ensure a giant component of size n - o(n) with high probability. This construction yields a much sparser random graph with ~ n log log n instead of ~ n log n edges that has comparable connectivity properties. This result has nontrivial implications for problems in data science where an affinity matrix is constructed: instead of connecting each point to its k nearest neighbors, one can often pick k' ≪ k random points out of the k nearest neighbors and only connect to those without sacrificing quality of results. This approach can simplify and accelerate computation; we illustrate this with experimental results in spectral clustering of large-scale datasets.
10 citations
TL;DR: This work provides a new proof of the existence of Gibbs point processes with infinite range interactions, based on the compactness of entropy levels, under the so-called intensity regularity assumption.
Abstract: We provide a new proof of the existence of Gibbs point processes with infinite range interactions, based on the compactness of entropy levels. Our main existence theorem holds under two assumptions. The first one is the standard stability assumption, which means that the energy of any finite configuration is superlinear with respect to the number of points. The second assumption is the so-called intensity regularity, which controls the long range of the interaction via the intensity of the process. This assumption is new and introduced here since it is well adapted to the entropy approach. As a corollary of our main result we improve the existence results by Ruelle (1970) for pairwise interactions by relaxing the superstabilty assumption. Note that our setting is not reduced to pairwise interaction and can contain infinite-range multi-body counterparts.
TL;DR: In this paper, the authors define a shot noise process with a random response function (response process) in which shots occur at arbitrary random times and provide sufficient conditions which ensure weak convergence of finite-dimensional distributions of these processes to certain Gaussian processes.
Abstract: By a random process with immigration at random times we mean a shot noise process with a random response function (response process) in which shots occur at arbitrary random times. Such random processes generalize random processes with immigration at the epochs of a renewal process which were introduced in Iksanov et al. (2017) and bear a strong resemblance to a random characteristic in general branching processes and the counting process in a fixed generation of a branching random walk generated by a general point process. We provide sufficient conditions which ensure weak convergence of finite-dimensional distributions of these processes to certain Gaussian processes. Our main result is specialised to several particular instances of random times and response processes.
TL;DR: Stochastic comparisons of parallel systems with respect to the reversed hazard rate and likelihood rate orders for exponentiated generalized gamma and exponentiated Pareto distributions are investigated and the results recover and strengthen some recent results in the literature.
Abstract: We investigate stochastic comparisons of parallel systems (corresponding to the largest-order statistics) with respect to the reversed hazard rate and likelihood ratio orders for the proportional reversed hazard rate (PRHR) model. As applications of the main results, we obtain the equivalent characterizations of stochastic comparisons with respect to the reversed hazard rate and likelihood rate orders for the exponentiated generalized gamma and exponentiated Pareto distributions. Our results recover and strengthen some recent results in the literature.
TL;DR: Techniques based on pathwise properties of the Brownian bridge and martingale methods of optimal stopping theory are developed, which allow to find the optimal stopping rule and to show the regularity of the value function.
Abstract: We study the problem of stopping a Brownian bridge X in order to maximise the expected value of an exponential gain function. The problem was posed by Ernst and Shepp (2015), and was motivated by bond selling with non-negative prices.
Due to the non-linear structure of the exponential gain, we cannot rely on methods used in the literature to find closed-form solutions to other problems involving the Brownian bridge. Instead, we must deal directly with a stopping problem for a time-inhomogeneous diffusion. We develop techniques based on pathwise properties of the Brownian bridge and martingale methods of optimal stopping theory, which allow us to find the optimal stopping rule and to show the regularity of the value function.
TL;DR: In this article, it was shown that a general shot noise process satisfies a functional limit theorem in the Skorokhod space with a locally Holder continuous Gaussian limit process, and that the response function is regularly varying at infinity.
Abstract: By a general shot noise process we mean a shot noise process in which the counting process of shots is arbitrary locally finite. Assuming that the counting process of shots satisfies a functional limit theorem in the Skorokhod space with a locally Holder continuous Gaussian limit process, and that the response function is regularly varying at infinity, we prove that the corresponding general shot noise process satisfies a similar functional limit theorem with a different limit process and different normalization and centering functions. For instance, if the limit process for the counting process of shots is a Brownian motion, then the limit process for the general shot noise process is a Riemann–Liouville process. We specialize our result for five particular counting processes. Also, we investigate Holder continuity of the limit processes for general shot noise processes.
TL;DR: Weighted recursive trees as mentioned in this paper generalize both uniform recursive trees and Hoppe trees, providing diversity among the nodes and making the model more flexible for applications, and they have been shown to be more robust than uniform trees in terms of number of leaves, height, depth, number of branches, and the size of the largest branch.
Abstract: A uniform recursive tree on n vertices is a random tree where each possible labelled recursive rooted tree is selected with equal probability. We introduce and study weighted trees, a non-uniform recursive tree model departing from the recently introduced Hoppe trees. This class generalizes both uniform recursive trees and Hoppe trees, providing diversity among the nodes and making the model more flexible for applications. We analyse the number of leaves, the height, the depth, the number of branches, and the size of the largest branch in these weighted trees.
TL;DR: The limit theorem for the classical urn models with finitely many colors is reprove and almost sure convergence of the urn configuration under uniform ergodicity assumption is proved.
Abstract: We consider the generalization of the Polya urn scheme with possibly infinitely many colors, as introduced in [37], [4], [5], and [6]. For countably many colors, we prove almost sure convergence of the urn configuration under the uniform ergodicity assumption on the associated Markov chain. The proof uses a stochastic coupling of the sequence of chosen colors with a branching Markov chain on a weighted random recursive tree as described in [6], [31], and [26]. Using this coupling we estimate the covariance between any two selected colors. In particular, we re-prove the limit theorem for the classical urn models with finitely many colors.
TL;DR: In this paper, the authors derived new representations of Hermite processes with multiple Wiener-Ito integrals, whose integrands involve the local time of intersecting stationary stable regenerative sets.
Abstract: Hermite processes are a class of self-similar processes with stationary increments. They often arise in limit theorems under long-range dependence. We derive new representations of Hermite processes with multiple Wiener–Ito integrals, whose integrands involve the local time of intersecting stationary stable regenerative sets. The proof relies on an approximation of regenerative sets and local times based on a scheme of random interval covering.
TL;DR: It is shown that, in general, similar results may not hold for hazard rate, reversehazard rate, and likelihood ratio orderings for coherent systems with independent and identically distributed components.
Abstract: We consider coherent systems with independent and identically distributed components. While it is clear that the system’s life will be stochastically larger when the components are replaced with stochastically better components, we show that, in general, similar results may not hold for hazard rate, reverse hazard rate, and likelihood ratio orderings. We find sufficient conditions on the signature vector for these results to hold. These results are combined with other well-known results in the literature to get more general results for comparing two systems of the same size with different signature vectors and possibly with different independent and identically distributed component lifetimes. Some numerical examples are also provided to illustrate the theoretical results.
TL;DR: A model for the spreading of fake news in a community of size n, where there are active gullible persons who are willing to believe and spread the fake news but the rest do not react to it is introduced.
Abstract: We introduce a model for the spreading of fake news in a community of size n. There are active gullible persons who are willing to believe and spread the fake news, the rest do not react to it. We address the question ‘How long does it take for persons to become spreaders?’ (The perturbation functions and are o(n), and .) The setup has a straightforward representation as a convolution of geometric random variables with quadratic probabilities. However, asymptotic distributions require delicate analysis that gives a somewhat surprising outcome. Normalized appropriately, the waiting time has three main phases: (a) away from the depletion of active gullible persons, when , the normalized variable converges in distribution to a Gumbel random variable; (b) near depletion, when , with , the normalized variable also converges in distribution to a Gumbel random variable, but the centering function gains weight with increasing perturbations; (c) at almost complete depletion, when , for integer , the normalized variable converges in distribution to a convolution of two independent generalized Gumbel random variables. The influence of various perturbation functions endows the three main phases with an infinite number of phase transitions at the seam lines.
TL;DR: A class of non-uniform random recursive trees grown with an attachment preference for young age is introduced, and it is found that the outdegree of a node is characterized in the limit by 'perturbed' Poisson laws, and the perturbation diminishes as the node index increases.
Abstract: We introduce a class of non-uniform random recursive trees grown with an attachment preference for young age. Via the Chen–Stein method of Poisson approximation, we find that the outdegree of a node is characterized in the limit by ‘perturbed’ Poisson laws, and the perturbation diminishes as the node index increases. As the perturbation is attenuated, a pure Poisson limit ultimately emerges in later phases. Moreover, we derive asymptotics for the proportion of leaves and show that the limiting fraction is less than one half. Finally, we study the insertion depth in a random tree in this class. For the insertion depth, we find the exact probability distribution, involving Stirling numbers, and consequently we find the exact and asymptotic mean and variance. Under appropriate normalization, we derive a concentration law and a limiting normal distribution. Some of these results contrast with their counterparts in the uniform attachment model, and some are similar.
TL;DR: The behaviour of the failure rate and reversed failure rate of an n-component coherent system is studied, where it is assumed that the lifetimes of the components are independent and have a common cumulative distribution function F.
Abstract: In this paper the behaviour of the failure rate and reversed failure rate of an n-component coherent system is studied, where it is assumed that the lifetimes of the components are independent and have a common cumulative distribution function F. Sufficient conditions are provided under which the system failure rate is increasing and the corresponding reversed failure rate is decreasing. We also study the stochastic and ageing properties of doubly truncated random variables for coherent systems.
TL;DR: This work provides novel explicit results on the conditional distribution of the total sum of X_i given that a subset sum x exceeds a certain threshold value $t>0$ and investigates the characteristic tail behavior of these conditional distributions for $t\to\infty$.
Abstract: For independent exponentially distributed random variables , , with distinct rates we consider sums for which follow generalized exponential mixture distributions. We provide novel explicit results on the conditional distribution of the total sum given that a subset sum exceeds a certain threshold value , and vice versa. Moreover, we investigate the characteristic tail behavior of these conditional distributions for . Finally, we illustrate how our probabilistic results can be applied in practice by providing examples from both reliability theory and risk management.
TL;DR: 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 is presented.
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.
TL;DR: This paper investigates the random horizon optimal stopping problem for measure-valued piecewise deterministic Markov processes (PDMPs) and proves that the value function of the problems can be obtained by iterating some dynamic programming operator.
Abstract: This paper investigates the random horizon optimal stopping problem for measure-valued piecewise deterministic Markov processes (PDMPs). This is motivated by population dynamics applications, when one wants to monitor some characteristics of the individuals in a small population. The population and its individual characteristics can be represented by a point measure. We first define a PDMP on a space of locally finite measures. Then we define a sequence of random horizon optimal stopping problems for such processes. We prove that the value function of the problems can be obtained by iterating some dynamic programming operator. Finally we prove via a simple counter-example that controlling the whole population is not equivalent to controlling a random lineage.
TL;DR: The directed preferential attachment model is revisited and a new exact characterization of the limiting in- and out-degree distribution is given by two independent pure birth processes that are observed at a common exponentially distributed time T to confirm previously derived tail probabilities for the two marginal degree distributions.
Abstract: The directed preferential attachment model is revisited. A new exact characterization of the limiting in- and out-degree distribution is given by two independent pure birth processes that are observed at a common exponentially distributed time T (thus creating dependence between in- and out-degree). The characterization gives an explicit form for the joint degree distribution, and this confirms previously derived tail probabilities for the two marginal degree distributions. The new characterization is also used to obtain an explicit expression for tail probabilities in which both degrees are large. A new generalized directed preferential attachment model is then defined and analyzed using similar methods. The two extensions, motivated by empirical evidence, are to allow double-directed (i.e. undirected) edges in the network, and to allow the probability of connecting an ingoing (outgoing) edge to a specified node to also depend on the out-degree (in-degree) of that node.
TL;DR: In this paper, the authors studied the expected occupancy probabilities on an alphabet, where observations are assumed to follow a regime-switching Markov chain, and gave finite sample bounds on the occupancy probabilities and detailed asymptotics in the case of regularly varying underlying distribution.
Abstract: This article studies the expected occupancy probabilities on an alphabet. Unlike the standard situation, where observations are assumed to be independent and identically distributed, we assume that they follow a regime-switching Markov chain. For this model, we (1) give finite sample bounds on the expected occupancy probabilities, and (2) provide detailed asymptotics in the case where the underlying distribution is regularly varying. We find that in the regularly varying case the finite sample bounds are rate optimal and have, up to a constant, the same rate of decay as the asymptotic result.
TL;DR: In this article, it was shown that the n-step embedded discrete-time Markov chain is positive recurrent under the following additional assumptions: (i) the system is binary, and (ii) for each species, there is a complex (vertex in the associated reaction diagram) that is a multiple of that species.
Abstract: It has been known for nearly a decade that deterministically modeled reaction networks that are weakly reversible and consist of a single linkage class have trajectories that are bounded from both above and below by positive constants (so long as the initial condition has strictly positive components). It is conjectured that the stochastically modeled analogs of these systems are positive recurrent. We prove this conjecture in the affirmative under the following additional assumptions: (i) the system is binary, and (ii) for each species, there is a complex (vertex in the associated reaction diagram) that is a multiple of that species. To show this result, a new proof technique is developed in which we study the recurrence properties of the n-step embedded discrete-time Markov chain.
TL;DR: A revealing elementary proof of a result proved earlier using heavy machinery from Malliavin calculus is given and precise vanishing noise asymptotics are obtained for the tail of the exit time and for the exit distribution conditioned on atypically long exits.
Abstract: For a one-dimensional smooth vector field in a neighborhood of an unstable equilibrium, we consider the associated dynamics perturbed by small noise. We give a revealing elementary proof of a result proved earlier using heavy machinery from Malliavin calculus. In particular, we obtain precise vanishing noise asymptotics for the tail of the exit time and for the exit distribution conditioned on atypically long exits. We also discuss our program on rare transitions in noisy heteroclinic networks.
TL;DR: This multidimensional model aims for the generality of adaptive dynamics and the tractability of population genetics, and provides a natural bridge between two of the most prominent modelling frameworks of biological evolution: population genetics and eco-evolutionary models.
Abstract: We construct a multitype constant-size population model allowing for general selective interactions as well as extreme reproductive events. Our multidimensional model aims for the generality of adaptive dynamics and the tractability of population genetics. It generalises the idea of Krone and Neuhauser [39] and Gonzalez Casanova and Spano [29], who represented the selection by allowing individuals to sample several potential parents in the previous generation before choosing the ‘strongest’ one, by allowing individuals to use any rule to choose their parent. The type of the newborn can even not be one of the types of the potential parents, which allows modelling mutations. Via a large population limit, we obtain a generalisation of -Fleming–Viot processes, with a diffusion term and a general frequency-dependent selection, which allows for non-transitive interactions between the different types present in the population. We provide some properties of these processes related to extinction and fixation events, and give conditions for them to be realised as unique strong solutions of multidimensional stochastic differential equations with jumps. Finally, we illustrate the generality of our model with applications to some classical biological interactions. This framework provides a natural bridge between two of the most prominent modelling frameworks of biological evolution: population genetics and eco-evolutionary models.