arXiv subjects

Sergey Foss

Publications and source records attributed to Sergey Foss.

At least 19 recordsLinked to original sources

arXiv2026

On extremes of a random walk with positive drift over an intermediate regularly varying time interval

We consider a random walk $\{S_n\}$ with a finite positive drift that is stopped at a random time $\tau$ having an intermediate regularly varying distribution. We assume that the jump distribution is lighter-tailed than the distribution of $\tau$. Under these conditions, we show that the tails of the distributions of $S_{\tau}$ and $M_{\tau} = \max_{k\le \tau} S_k$ are asymptotically equivalent and are determined by the tail of $\tau$, while the random walk $\{S_n\}$ contributes only through the law of large numbers.

math.PR↗

arXiv2026

Only Segmented Heavy Tails Can Produce a Light-Tailed Minimum

A random variable $\xi$ has a {\it light-tailed} distribution (for short: is light-tailed) if it possesses a finite exponential moment, $\E \exp (\lambda \xi) <\infty$ for some $\lambda >0$, and has a {\it heavy-tailed} distribution (is heavy-tailed) if $\E \exp (\lambda\xi) = \infty$, for all $\lambda>0$. In \cite{LSK1}, the authors presented a particular example of a light-tailed random variable that is the minimum of two independent heavy-tailed random variables. In \cite{FKT}, it was shown that any light-tailed random variable with right-unbounded support may be represented as the minimum of two independent heavy-tailed random variables, with further generalisations of the result in a number of directions. We analyse an ``inverse'' question. Namely, we obtain necessary and sufficient conditions on the distribution of a heavy-tailed random variable, say $\xi_1$, that allow to find another independent heavy-tailed random variable, say $\xi_2$, such that their minimum $\min (\xi_1,\xi_2)$ is light-tailed. We also provide a number of extensions of this result

math.PR↗

arXiv2025

On the instability of local learning algorithms: Q-learning can fail in infinite state spaces

We investigate the challenges of applying model-free reinforcement learning algorithms, like online Q-learning, to infinite state space Markov Decision Processes (MDPs). We first introduce the notion of Local Learning Processes (LLPs), where agents make decisions based solely on local information, and we show that Q-learning can be seen as a specific instance of an LLP. Using renewal techniques, we analyze LLPs and demonstrate their instability under certain drift and initial conditions, revealing fundamental limitations in infinite state spaces. In particular, we show that while asymptotically optimal in finite settings, Q-learning can face instability and strict sub-optimality in infinite spaces. Our findings are illustrated through queueing system examples drawn from load balancing and server allocation. The study underscores the need for new theoretical frameworks and suggests future research into nonlocal Q-learning variants.

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arXiv2025

On Recurrence of the Infinite Server Queue

This paper concerns the recurrence structure of the infinite server queue, as viewed through the prism of the maximum dater sequence, namely the time to drain the current work in the system as seen at arrival epochs. Despite the importance of this model in queueing theory, we are aware of no complete analysis of the stability behavior of this model, especially in settings in which either or both the inter arrival and service time distributions have infinite mean. In this paper, we fully develop the analog of the Loynes construction of the stationary version in the context of stationary ergodic inputs, extending earlier work of E.Altman (2005), and then classify the Markov chain when the inputs are independent and identically distributed. This allows us to classify the chain, according to transience, recurrence in the sense of Harris, and positive recurrence in the sense of Harris. We further go on to develop tail asymptotics for the stationary distribution of the maximum dater sequence, when the service times have tails that are asymptotically exponential or Pareto, and we contrast the stability theory for the infinite server queue relative to that for the single server queue.

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arXiv2025

Efficient Remote Monitoring through Noisy Random Access with Retransmissions

We consider a rare event monitoring system consisting of a set of devices and a base station, where devices transmit information about rare events to the base station using a random multiple access scheme. We introduce a model in which the presence of noise in the multiple access channel can cause message loss even in the absence of transmission collisions. The occurrence of events is modeled by a family of independent two-state Markov chains (with states 0 and 1). We analyze how repeated transmissions affect system performance. Two efficiency criteria are proposed and studied: the maximum probability that a message about an event from a fixed device is successfully delivered to the base station and the maximum frequency at which the base station successfully receives updates about the entire system. For each criterion, we determine the optimal number of retransmissions as a function of the system parameters.

cs.IT↗

arXiv2025

Heavy-tail asymptotics for the length of a busy period in a Generalised Jackson Network

We consider a Generalised Jackson Network with finitely many servers, a renewal input and $i.i.d.$ service times at each queue. We assume the network to be stable and, in addition, the distribution of the inter-arrival times to have unbounded support. This implies that the length of a typical busy period $B$, which is the time between two successive idle periods, is finite a.s. and has a finite mean. We assume that the distributions of the service times with the heaviest tails belong to the class of so-called intermediate regularly varying distributions. We obtain the exact asymptotics for the probability ${\mathbb P} (B>x)$, as $x\to\infty$. For that, we show that the Principle of a Single Big Jump holds: $B$ takes a large value mainly due to a single unusually large service time.

math.PR↗

arXiv2025

Any random variable with right-unbounded distributional support is the minimum of independent and very heavy-tailed random variables

A random variable $\xi$ has a {\it light-tailed} distribution (for short: is light-tailed) if it possesses a finite exponential moment, $\E \exp (\lambda \xi) <\infty$ for some $\lambda >0$, and has a {\it heavy-tailed} distribution (is heavy-tailed) if $\E \exp (\lambda\xi) = \infty$, for all $\lambda>0$. In (Leipus et al., AIMS Mathematics, 2023), the authors presented a particular example of a light-tailed random variable that is the minimum of two independent heavy-tailed random variables. We will show that this phenomenon is universal: {\it any} light-tailed random variable with right-unbounded support may be represented as the minimum of two independent heavy-tailed random variables. Moreover, a more general fact holds: these two independent random variables may have as heavy-tailed distributions as one wishes. Further, we will extend the latter result onto the minimum of any finite number of independent random variables. We will also comment on possible generalizations of our result to the case of dependent random variables.

math.PR↗

arXiv2025

Poisson Hail on a Wireless Ground

This paper defines a new model which incorporates three key ingredients of a large class of wireless communication systems: (1) spatial interactions through interference, (2) dynamics of the queueing type, with users joining and leaving, and (3) carrier sensing and collision avoidance as used in, e.g., WiFi. In systems using (3), rather than directly accessing the shared resources upon arrival, a customer is considerate and waits to access them until nearby users in service have left. This new model can be seen as a missing piece of a larger puzzle that contains such dynamics as spatial birth-and-death processes, the Poisson-Hail model, and wireless dynamics as key other pieces. It is shown that, under natural assumptions, this model can be represented as a Markov process on the space of counting measures. The main results are then two-fold. The first is on the shape of the stability region and, more precisely, on the characterization of the critical value of the arrival rate that separates stability from instability. The second is of a more qualitative or perhaps even ethical nature. There is evidence that for natural values of the system parameters, the implementation of sensing and collision avoidance stabilizes a system that would be unstable if immediate access to the shared resources would be granted. In other words, for these parameters, renouncing greedy access makes sharing sustainable, whereas indulging in greedy access kills the system.

cs.IT↗

arXiv2024

Compressibility and Stochastic Stability of Monotone Markov Chain

For a stochastically monotone Markov chain taking values in a Polish space, we present a number of conditions for existence and for uniqueness of its stationary regime, as well as for closeness of its transient trajectories. In particular, we generalise a basic result by Bhattacharya and Majumdar (2007) where a certain form of mixing, or swap condition was assumed uniformly over the state space. We do not rely on continuity properties of transition probabilities.

math.PR↗

arXiv2023

Last passage percolation and limit theorems in Barak-Erd\H{o}s directed random graphs and related models

We consider directed random graphs, the prototype of which being the Barak-Erd\H{o}s graph $\vec G(\mathbb Z, p)$, and study the way that long (or heavy, if weights are present) paths grow. This is done by relating the graphs to certain particle systems that we call Infinite Bin Models (IBM). A number of limit theorems are shown. The goal of this paper is to present results along with techniques that have been used in this area. In the case of $\vec G(\mathbb Z, p)$ the last passage percolation constant $C(p)$ is studied in great detail. It is shown that $C(p)$ is analytic for $p>0$, has an interesting asymptotic expansion at $p=1$ and that $C(p)/p$ converges to $e$ like $1/(\log p)^2$ as $p \to 0$. The paper includes the study of IBMs as models on their own as well as their connections to stochastic models of branching processes in continuous or discrete time with selection. Several proofs herein are new or simplified versions of published ones. Regenerative techniques are used where possible, exhibiting random sets of vertices over which the graphs regenerate. When edges have random weights we show how the last passage percolation constants behave and when central limit theorems exist. When the underlying vertex set is partially ordered, new phenomena occur, e.g., there are relations with last passage Brownian percolation. We also look at weights that may possibly take negative values and study in detail some special cases that require combinatorial/graph theoretic techniques that exhibit some interesting non-differentiability properties of the last passage percolation constant. We also explain how to approach the problem of estimation of last passage percolation constants by means of perfect simulation.

math.PR↗

arXiv2023

Maxima over random time intervals for heavy-tailed compound renewal and L\'evy processes

We derive subexponential tail asymptotics for the distribution of the maximum of a compound renewal process with linear component and of a L\'evy process, both with negative drift, over random time horizon $\tau$ that does not depend on the future increments of the process. Our asymptotic results are uniform over the whole class of such random times. Particular examples are given by stopping times and by $\tau$ independent of the processes. We link our results with random walk theory.

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arXiv2022

Migration-Contagion Processes

Consider a migration process based on a closed network of N stations with K_N customers. Each station is a ./M/\infty queue with service (migration) rate mu. Upon departure, a customer is routed at random to another station. In addition to migration, these customers are subject to an SIS (Susceptible, Infected, Susceptible) dynamics: customers are either I for infected, or S for susceptible. They can swap their state either from I to S or from S to I only in stations. At any station, each S customer becomes I with rate alpha Y if there are Y infected customers in the station, and each I customer recovers and becomes S with rate beta. We let N tend to infinity and assume that lim_{N\to infty} K_N/N= eta>0. The main problem is about the set of parameters for which there exists a stationary regime where the epidemic survives in the thermodynamic limit. We establish several structural properties of the system, which allow us to give the phase transition diagram of this thermodynamic limit w.r.t. eta. The analysis of the SIS model reduces to that of a wave-type PDE for which we found no explicit solution. This SIS model is one among several companion stochastic processes with migration and contagion. Two of them are discussed as they provide some bounds and approximations to SIS. These two variants are the DOCS (Departure On Change of State) and the AIR (Averaged Infection Rate), which both admit closed-form solutions. The AIR system is a mean-field model where the infection mechanism is based on the empirical average of the number of infected customers in all stations. The latter admits a product-form solution. DOCS features accelerated migration in that each change of SIS state implies an immediate departure. It leads to another wave-type PDE that admits a closed-form solution.

math.PR↗

arXiv2021

The probability of reaching a receding boundary by branching random walk with fading branching and heavy-tailed jump distribution

Foss and Zachary (2003) and Foss, Palmowski and Zachary (2005) studied the probability of achieving a receding boundary on a time interval of random length by a random walk with a heavy-tailed jump distribution. They have proposed and developed a new approach that allows to generalise results of Asmussen (1998) onto the case of arbitrary stopping times and a wide class of nonlinear boundaries, and to obtain uniform results over all stopping times. In this paper, we consider a class of branching random walks with fading branching and obtain results on the tail asymptotics for the maximum of a branching random walk on a time interval of random (possibly unlimited) length, as well as uniform results within a class of bounded random time intervals.

math.PR↗

arXiv2021

Estimation of the last passage percolation constant in a charged complete directed acyclic graph via perfect simulation

Our object of study is the asymptotic growth of heaviest paths in a charged (weighted with signed weights) complete directed acyclic graph. Edge charges are i.i.d. random variables with common distribution $F$ supported on $[-\infty,1]$ with essential supremum equal to $1$ (a charge of $-\infty$ is understood as the absence of an edge). The asymptotic growth rate is a constant that we denote by $C(F)$. Even in the simplest case where $F=p\delta_1 + (1-p)\delta_{-\infty}$, corresponding to the longest path in the Barak-Erd\H{o}s random graph, there is no closed-form expression for this function, but good bounds do exist. In this paper we construct a Markovian particle system that we call "Max Growth System" (MGS), and show how it is related to the charged random graph. The MGS is a generalization of the Infinite Bin Model that has been the object of study of a number of papers. We then identify a random functional of the process that admits a stationary version and whose expectation equals the unknown constant $C(F)$. Furthermore, we construct an effective perfect simulation algorithm for this functional which produces samples from the random functional.

math.PR↗

arXiv2021

Moments of the first descending epoch for a random walk with negative drift

We consider the first exit time $\tau = \min \{n\ge 1 : S_n\le 0\}$ from the positive halfline of a random walk $S_n = \sum_1^n \xi_i, n\ge 1$ with i.d.d. summands having a negative drift ${\mathbb E} \xi = -a< 0$. Let $\xi^+ = \max (0, \xi_1)$. It is well-known that, for any $c>1$, the finiteness of ${\mathbb E}(\xi^+)^{c}$ implies the finiteness of ${\mathbb E} \tau^c$ and, for any $c>0$, the finiteness of ${\mathbb E} \exp({c\xi^+})$ implies that of ${\mathbb E} \exp({c'\tau})$ where $c'>0$ is, in general, another constant that depends on $c$ and on the distribution of $\xi_1$. We consider the intermediate case, assuming that ${\mathbb E} \exp({g(\xi^+)})<\infty$ for a positive increasing function $g$ such that $\liminf_{x\to\infty} g(x)/\log x = \infty$ and $\limsup_{x\to\infty} g(x)/x =0$, and that ${\mathbb E} \exp({c\xi^+})=\infty$, for all $c>0$. Assuming a few further technical assumptions, we show that then ${\mathbb E} \exp({(1-\varepsilon){g}((1-\varepsilon)a\tau)})<\infty$, for any $\varepsilon \in (0,1)$.

math.PR↗

arXiv2020

Non-standard limits for a family of autoregressive stochastic sequences

We consider a family of multivariate autoregressive stochastic sequences that restart when hit a neighbourhood of the origin, and study their distributional limits when the autoregressive coefficient tends to one, the noise scaling parameter tends to zero, and the neighbourhood size varies. We obtain a non-standard limit theorem where the limiting distribution is a mixture of an atomic distribution and an absolutely continuous distribution whose marginals, in turn, are mixtures of distributions of signed absolute values of normal random variables. In particular, we provide conditions for the limiting distribution to be normal, like in the case without restart mechanism. The main theorem is accompanied by a number of examples and auxiliary results of their own interest.

math.PR↗

arXiv2020

Branching processes with immigration in atypical random environment

Motivated by a seminal paper of Kesten et al. (1975) we consider a branching process with a geometric offspring distribution with i.i.d. random environmental parameters $A_n$, $n\ge 1$ and size -1 immigration in each generation. In contrast to above mentioned paper we assume that the environment is long-tailed, that is that the distribution $F$ of $\xi_n := \log ((1-A_n)/A_n)$ is long-tailed. We prove that although the offspring distribution is light-tailed, the environment itself can produce extremely heavy tails of the distribution of the population size in the n-th generation which becomes even heavier with increase of n. More precisely, we prove that, for any n, the distribution tail $\mathbb{P}(Z_n > m)$ of the $n$-th population size $Z_n$ is asymptotically equivalent to $n\overline{F}(\log m)$ as $m$ grows. In this way we generalize Bhattacharya and Palmowski (2019) who proved this result in the case $n=1$ for regularly varying environment $F$ with parameter $\alpha >1$. Further, for a subcritical branching process with subexponentially distributed $\xi_n$, we provide the asymptotics for the distribution tail $\mathbb{P}(Z_n>m)$ which are valid uniformly for all $n$, and also for the stationary tail distribution. Then we establish the "principle of a single atypical environment" which says that the main cause for the number of particles to be large is a presence of a single very small environmental parameter $A_k$.

math.PR↗

arXiv2020

Structural Properties of Conditioned Random Walks on Integer Lattices with Random Local Constraints

We consider a random walk on a multidimensional integer lattice with random bounds on local times, conditioned on the event that it hits a high level before its death. We introduce an auxiliary "core" process that has a regenerative structure and plays a key role in our analysis. We obtain a number of representations for the distribution of the random walk in terms of the similar distribution of the "core" process. Based on that, we prove a number of limiting results by letting the high level to tend to infinity. In particular, we generalise results for a simple symmetric one-dimensional random walk obtained earlier in the paper by Benjamini and Berestycki (2010).

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