arXiv subjects

Paweł Hitczenko

Publications and source records attributed to Paweł Hitczenko.

7 recordsLinked to original sources

arXiv2025

Expected number of jumps and the number of active particles in TASEP

For a TASEP on $\mathbb Z$ with the step initial condition we identify limits as $t\to\infty$ of the expected total number of jumps until time $t>0$ and the expected number of active particles at a time $t$. We also connect the two quantities proving that non-asymptotically, that is as a function of $t>0$, the latter is the derivative of the former. Our approach builds on asymptotics derived by Rost and intensive use of the fact that the rightmost particle evolves according to the Poisson process.

math.PR↗

arXiv2024

A class of polynomial recurrences resulting in $(n/\log n, n/\log^2n)$-asymptotic normality

We consider sequences of polynomials that satisfy differential-difference recurrences. Polynomials satisfying such recurrences frequently appear as generating polynomials of integer valued random variables that are of interest in discrete mathematics. It is, therefore, of interest to understand the properties of such polynomials and their probabilistic consequences. We identify a class of polynomial recurrences that lead to a normal law with the expected value and the variance proportional to $n/\log n$ and $n/\log^2n$, respectively. Examples include Stirling number of the second kind and other polynomials concerning set partitions as well as polynomials related to Whitney numbers of Dowling lattices.

math.CO↗

arXiv2019

Asymptotics of the overflow in urn models

Consider a number, finite or not, of urns each with fixed capacity $r$ and balls randomly distributed among them. An overflow is the number of balls that are assigned to urns that already contain $r$ balls. When $r=1$, using analytic methods, Hwang and Janson gave conditions under which the overflow (which in this case is just the number of balls landing in non--empty urns) has an asymptotically Poisson distribution as the number of balls grows to infinity. Our aim here is to systematically study the asymptotics of the overflow in general situation, i.~e. for arbitrary $r$. In particular, we provide sufficient conditions for both Poissonian and normal asymptotics for general $r$, thus extending Hwang--Janson's work. Our approach relies on purely probabilistic methods.

math.PR↗

arXiv2014

On the covariances of outdegrees in random plane recursive trees

In 2005 Janson, extending earlier work of Mahmoud, Smythe, and Szymański, established the joint asymptotic normality of the outdegrees of a random plane recursive tree. In particular, he gave an explicit description of the limiting covariance matrix. Our aim here is to provide substantially simplified expression for the limiting covariance matrix.

math.CO↗

arXiv2011

Renorming divergent perpetuities

We consider a sequence of random variables $(R_n)$ defined by the recurrence $R_n=Q_n+M_nR_{n-1}$, $n\ge1$, where $R_0$ is arbitrary and $(Q_n,M_n)$, $n\ge1$, are i.i.d. copies of a two-dimensional random vector $(Q,M)$, and $(Q_n,M_n)$ is independent of $R_{n-1}$. It is well known that if $E{\ln}|M|<0$ and $E{\ln^+}|Q|<\infty$, then the sequence $(R_n)$ converges in distribution to a random variable $R$ given by $R\stackrel{d}{=}\sum_{k=1}^{\infty}Q_k\prod_{j=1}^{k-1}M_j$, and usually referred to as perpetuity. In this paper we consider a situation in which the sequence $(R_n)$ itself does not converge. We assume that $E{\ln}|M|$ exists but that it is non-negative and we ask if in this situation the sequence $(R_n)$, after suitable normalization, converges in distribution to a non-degenerate limit.

math.ST↗

arXiv2010

Perpetuities with thin tails revisited

We consider the tail behavior of random variables $R$ which are solutions of the distributional equation $R\stackrel{d}{=}Q+MR$, where $(Q,M)$ is independent of $R$ and $|M|\le 1$. Goldie and Grübel showed that the tails of $R$ are no heavier than exponential and that if $Q$ is bounded and $M$ resembles near 1 the uniform distribution, then the tails of $R$ are Poissonian. In this paper, we further investigate the connection between the tails of $R$ and the behavior of $M$ near 1. We focus on the special case when $Q$ is constant and $M$ is nonnegative.

math.PR↗

arXiv1999

A note on sums of independent random variables

In this note a two sided bound on the tail probability of sums of independent, and either symmetric or nonnegative, random variables is obtained. We utilize a recent result by Latała on bounds on moments of such sums. We also give a new proof of Latała's result for nonnegative random variables, and improve one of the constants in his inequality.

math.PR↗