arXiv subjects

Seva Shneer

Publications and source records attributed to Seva Shneer.

At least 19 recordsLinked to original sources

arXiv2026

Geometric approximation for the number of returns of a transient Markov chain to its origin

Given a discrete-time, transient Markov chain, we establish explicit total variation error bounds in the approximation of the number of visits to its starting state within the first $n$ time steps by a geometric distribution. Our error bounds are expressed in terms of the heat kernel of the Markov chain. As applications, we consider Galton--Watson processes with a geometric offspring distribution, and random walks on the one- and three-dimensional integer lattices, regular trees, supercritical percolation clusters, and groups with polynomial growth.

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arXiv2026

Waves Everywhere: A Distributional Equation Approach to Front Propagation

We study reaction-diffusion particle systems with several interaction mechanisms. As the number of particles tends to infinity, the system admits a mean-field limit describing the bulk behaviour. We focus on determining the propagation speed and the particle distribution around the centre of mass, which corresponds to the travelling wave of the limiting equation. We introduce a probabilistic method to characterise these waves via tagged particle distributional equations. Our key technique connects these to linear distributional equations solvable using martingale limits from branching processes. We first demonstrate our approach on a general model where particles move via L\'evy processes and synchronise at interaction moments (the lower particle jumps to the position of the higher one). Assuming the mean-field limit holds, we characterise its travelling-wave solutions. We then apply the method to two specific models with established mean-field limits. For Brownian particles, we recover known travelling-wave solutions of the F-KPP equation. For the compound Poisson model studied in \cite{baryshnikov2025large} where particles perform random walks with exponential holding times and copy positions at interactions, we extend previous results beyond exponential jumps to arbitrary jump distributions. Finally, we analyse the power-of-2 growth model, where interactions add a random value to the lower particle. We characterise its travelling waves and discover a surprising connection to the synchronisation models. The distributional equations at the heart of our technique are of independent interest, and we identify connections and differences with related equations studied extensively in the literature.

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arXiv2025

A model of discrete interacting updates

We consider $N$ counters taking integer values which are subject to the following dynamics. At every time, a pair of distinct counters is chosen uniformly at random and their states are updated according to the following rule. If the states are different, then the smaller one is increased by $1$, while if the states are the same, both of them are increased by $1$. We show that, for a fixed $N$, the distances between consecutive ordered counters form a positive recurrent Markov chain and there exists the speed $V(N)$ defined as the average number of counters updated per time step in the stationary regime. We provide non-trivial upper and lower bounds for $V(N)$ as $N\to \infty$. Despite the simple formulation of the problem, its analysis seems to be highly complicated. We also provide a list of open problems and discuss various methods one may want to use, and obstacles one encounters.

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arXiv2025

Stochastic dominance for linear combinations of infinite-mean risks

In this paper, we establish a sufficient condition to compare linear combinations of independent and identically distributed (iid) infinite-mean random variables under usual stochastic order. We introduce a new class of distributions that includes many commonly used heavy-tailed distributions and show that within this class, a linear combination of random variables is stochastically larger when its weight vector is smaller in the sense of majorization order. We proceed to study the case where each random variable is a compound Poisson sum and demonstrate that if the stochastic dominance relation holds, the summand of the compound Poisson sum belongs to our new class of distributions. Additional discussions are presented for stable distributions.

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arXiv2025

Efficiency of Parallel and Restart Exploration Strategies in Model Free Stochastic Simulations

We analyze the efficiency of parallelization and restart mechanisms for stochastic simulations in model-free settings, where the underlying system dynamics are unknown. Such settings are common in Reinforcement Learning (RL) and rare event estimation, where standard variance-reduction techniques like importance sampling are inapplicable. Focusing on the challenge of reaching rare states under a finite computational budget, we model exploration via random walks and L\'evy processes. Based on rigorous probability analysis, our work reveals a phase transition in the success probability as a function of the number of parallel simulations: an optimal number $N^*$ exists, balancing exploration diversity and time allocation per simulation. Beyond this threshold, performance degrades exponentially. Furthermore, we demonstrate that a restart strategy, which reallocates resources from stagnant trajectories to promising regions, can yield an exponential improvement in success probability. In the context of RL, these strategies can improve policy gradient methods by enabling more efficient state-space exploration, leading to more accurate policy gradient estimates.

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arXiv2024

Risk aggregation and stochastic dominance for a class of heavy-tailed distributions

We introduce a new class of heavy-tailed distributions for which any weighted average of independent and identically distributed random variables is larger than one such random variable in (usual) stochastic order. We show that many commonly used extremely heavy-tailed (i.e., infinite-mean) distributions, such as the Pareto, Fr\'echet, and Burr distributions, belong to this class. The established stochastic dominance relation can be further generalized to allow negatively dependent or non-identically distributed random variables. In particular, the weighted average of non-identically distributed random variables dominates their distribution mixtures in stochastic order.

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arXiv2023

First passage percolation on Erd\H{o}s-R\'{e}nyi graphs with general weights

We consider first passage percolation on the Erd\H{o}s--R\'{e}nyi graph with $n$ vertices in which each pair of distinct vertices is connected independently by an edge with probability $\lambda/n$ for some $\lambda>1$. The edges of the graph are given non-negative i.i.d. weights with a non-degenerate distribution such that the probability of zero is not too large. We consider the paths with small total weight between two distinct typical vertices and analyse the joint behaviour of the numbers of edges on such paths, the so-called hopcounts, and the total weights of these paths. For $n\to\infty$, we show that, after a suitable transformation, the pairs of hopcounts and total weights of these paths converge in distribution to a Cox process, i.e., a Poisson process with a random intensity measure. The random intensity measure is controlled by two independent random variables, whose distribution is the solution of a distributional fixed point equation and is related to branching processes. For non-arithmetic and arithmetic edge-weight distributions we observe different behaviour. In particular, we derive the limiting distribution for the minimal total weight and the corresponding hopcount(s). Our results generalise earlier work of Bhamidi, van der Hofstad and Hooghiemstra, who assume that edge weights have an absolutely continuous distribution. The main tool we employ is the method of moments.

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arXiv2023

An instability condition for queuing systems with state-dependent departure rates

In this paper, we present a condition to obtain instability for a class of queueing networks where the arrival rates in each server are constant and the departure rate in each server is a decreasing function of the queue lengths of other servers. Under a stronger assumption, that the departure rates are proportional to the queue length in each server, we obtain a characterization of the stability region through a system of equations. We start by defining the mathematical model and the queueing discipline we will study. We then obtain irreducibility and monotonicity for the dynamics, which allow us to state our two main results. We then use this result to obtain instability conditions for two queueing networks for which stability is known: a medium access algorithm and an interference queueing network.

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arXiv2023

It is hard to kill fake news

We study a model for the spread of fake news, where first a piece of fake news is spread from a location in a network, followed by a correction to the news. We assume that both the fake as well as correct news travel as first-passage percolations or SI epidemics with i.i.d.\ traversal times, possibly with different distributions and dependence. We make the (hopeful) assumption that once a vertex in the network hears the correct news, they are immediately convinced that this is indeed correct, and continue to spread the correct news. Even in this optimistic scenario, our main results show that it is very difficult to remove the fake news from the network, even when the correct news would spread much faster than the fake news. We show this on the configuration model, a model that has gained enormous popularity as a simple, yet flexible, model for real-world networks. The crux of the proof is the realization that this problem on a branching process tree (which is known to be the local limit of the configuration model) can be reformulated in terms of the maximum of a branching random walk, a topic that has attracted considerable attention in the past decade. We then extend the results to the graph setting using couplings to branching processes, local convergence and detailed estimates on first-passage percolation on random graphs as derived in \cite{BhaHofHoo17}. Remarkably, despite the fact that our proofs for the configuration model rely on its local branching process structure, the condition for strong survival on a finite number of generations of a branching process tree is {\em different} from that on the configuration model.

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arXiv2021

The Borel Distribution: Approximation and Concentration

We develop the tools necessary to use Stein's method for approximation by a Borel distribution, which we illustrate by considering the approximation of the number of customers served in the busy period of an M/G/1 queue. We further derive concentration inequalities for the Borel distribution. Both these sets of results are based on a representation for the size-biased version of a Borel random variable.

math.PR↗

arXiv2021

Limit theorems and structural properties of the Cat-and-Mouse Markov chain

We revisit the so-called Cat-and-Mouse Markov chain, studied earlier by Litvak and Robert (2012). This is a 2-dimensional Markov chain on the lattice $\mathbb{Z}^2$, where the first component (the cat) is a simple random walk and the second component (the mouse) changes when the components meet. We obtain new results for two generalisations of the model. Firstly, in the 2-dimensional case we consider far more general jump distributions for the components and obtain a scaling limit for the second component. When we let the first component be a simple random walk again, we further generalise the jump distribution of the second component. Secondly, we consider chains of three and more dimensions, where we investigate structural properties of the model and find a limiting law for the last component.

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arXiv2021

Complete resource pooling of a load balancing policy for a network of battery swapping stations

To reduce carbon emission in the transportation sector, there is currently a steady move taking place to an electrified transportation system. This brings about various issues for which a promising solution involves the construction and operation of a battery swapping infrastructure rather than in-vehicle charging of batteries. In this paper, we study a closed Markovian queueing network that allows for spare batteries under a dynamic arrival policy. We propose a provisioning rule for the capacity levels and show that these lead to near-optimal resource utilization, while guaranteeing good quality-of-service levels for Electric Vehicle (EV) users. Key in the derivations is to prove a state-space collapse result, which in turn implies that performance levels are as good as if there would have been a single station with an aggregated number of resources, thus achieving complete resource pooling.

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arXiv2021

Unified approach for solving exit problems for additive-increase and multiplicative-decrease processes

We analyse an additive-increase and multiplicative-decrease (aka growth-collapse) process that grows linearly in time and that experiences downward jumps at Poisson epochs that are (deterministically) proportional to its present position. This process is used for example in modelling of Transmission Control Protocol (TCP) and can be viewed as a particular example of the so-called shot noise model, a basic tool in modeling earthquakes, avalanches and neuron firings. For this process, and also for its reflected versions, we consider one- and two-sided exit problems that concern the identification of the laws of exit times from fixed intervals and half-lines. All proofs are based on a unified first-step analysis approach at the first jump epoch, which allows us to give explicit, yet involved, formulas for their Laplace transforms. All the eight Laplace transforms can be described in terms of two so-called scale functions $Z_{\uparrow}$ and $L_{\uparrow}$. Here $Z_{\uparrow}$ is described in terms of multiple explicit sums, and $L_{\uparrow}$ in terms of an explicit recursion formula. All other Laplace transforms can be obtained from $Z_{\uparrow}$ and $L_{\uparrow}$ by taking limits, derivatives, integrals and combinations of these.

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arXiv2020

Large-scale parallel server system with multi-component jobs

A broad class of parallel server systems is considered, for which we prove the steady-state asymptotic independence of server workloads, as the number of servers goes to infinity, while the system load remains sub-critical. Arriving jobs consist of multiple components. There are multiple job classes, and each class may be of one of two types, which determines the rule according to which the job components add workloads to the servers. The model is broad enough to include as special cases some popular queueing models with redundancy, such as cancel-on-start and cancel-on-completion redundancy. Our analysis uses mean-field process representation and the corresponding mean-field limits. In essence, our approach relies almost exclusively on three fundamental properties of the model: (a) monotonicity, (b) work conservation, (c) the property that, on average, "new arriving workload prefers to go to servers with lower workloads."

math.PR↗

arXiv2020

Stability of two-dimensional Markov processes, with an application to QBD processes with an infinite number of phases

In this paper, we derive a simple drift condition for the stability of a class of two-dimensional Markov processes, for which one of the coordinates (also referred to as the {\em phase} for convenience) has a well understood behaviour dependent on the other coordinate (also referred as {\em level}). The first (phase) component's transitions may depend on the second component and are only assumed to be eventually independent. The second (level) component has partially bounded jumps and it is assumed to have a negative drift given that the first one is in its stationary distribution. The results presented in this work can be applied to processes of the QBD (quasi-birth-and-death) type on the quarter- and on the half-plane, where the phase and level are interdependent. Furthermore, they provide an off-the-shelf technique to tackle stability issues for a class of two-dimensional Markov processes. These results set the stepping stones towards closing the existing gap in the literature of deriving easily verifiable conditions/criteria for two-dimensional processes with unbounded jumps and interdependence between the two components.

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arXiv2020

Asymptotics for cliques in scale-free random graphs

In this paper we establish asymptotics (as the size of the graph grows to infinity) for the expected number of cliques in the Chung--Lu inhomogeneous random graph model in which vertices are assigned independent weights which have tail probabilities $h^{1-\alpha}l(h)$, where $\alpha>2$ and $l$ is a slowly varying function. Each pair of vertices is connected by an edge with a probability proportional to the product of the weights of those vertices. We present a complete set of asymptotics for all clique sizes and for all non-integer $\alpha > 2$. We also explain why the case of an integer $\alpha$ is different, and present partial results for the asymptotics in that case.

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arXiv2020

Local stability in a transient Markov chain

We prove two propositions with conditions that a system, which is described by a transient Markov chain, will display local stability. Examples of such systems include partly overloaded Jackson networks, partly overloaded polling systems, and overloaded multi-server queues with skill based service, under first come first served policy.

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arXiv2020

The empirical mean position of a branching Lévy process

We consider a supercritical branching Lévy process on the real line. Under mild moment assumptions on the number of offspring and their displacements, we prove a second-order limit theorem on the empirical mean position.

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