arXiv subjects

Dariusz Buraczewski

Publications and source records attributed to Dariusz Buraczewski.

At least 19 recordsLinked to original sources

arXiv2026

A self-normalized law of the iterated logarithm for Rademacher random multiplicative functions

Let $f$ be a Rademacher random multiplicative function supported on the squarefree integers, and let $M(x)=\sum_{n\le x}f(n)$. We prove a two-sided law of the iterated logarithm normalized by the full predictable quadratic variation \[ Q(x)=\sum_{p\le x}\bigg(\sum_{\substack{m\le x/p\\ P(m)<p}}f(m)\bigg)^2 \] of the martingale obtained by revealing the prime signs. Almost surely, \[ \limsup_{x\to\infty}\frac{M(x)}{\sqrt{2Q(x)\log\log x}}=1, \qquad \liminf_{x\to\infty}\frac{M(x)}{\sqrt{2Q(x)\log\log x}}=-1. \] The same equalities hold with $\log\log Q(x)$ in the denominator. The almost sure cluster set of the normalized sum is $[-1,1]$. This gives a self-normalized version of the questions of Wintner and Erdős on almost sure fluctuations. We also prove that $Q(x)/q(x)$ cannot converge in probability to $1$ for any positive deterministic function $q$. Moreover, \[ \limsup_{x\to\infty} \frac{|M(x)|}{\sqrt{x}(\log\log x)^{1/4}} \] is either an almost surely finite and positive random variable with nondegenerate distribution or infinite almost surely. We do not determine which alternative holds.

math.PR↗

arXiv2026

A law of the iterated logarithm for random multiplicative functions on integers with a large prime factor

Let $f$ be a Rademacher random multiplicative function supported on the squarefree integers. We investigate the almost sure fluctuations of the restricted sum $M^{>}(x)=\sum_{n\le x;~ P(n)>x^{1/2}}f(n)$, where $P(n)$ is the largest prime factor of $n$. With $Q(x)$ being the conditional variance of $M^{>}(x)$ given the values of $f$ at primes not exceeding $x^{1/2}$, we prove that, almost surely, the upper and lower limits of $M^{>}(x)(2Q(x)\log\log Q(x))^{-1/2}$ as $x\to\infty$ are $1$ and $-1$, respectively. Furthermore, it is shown that the almost sure cluster set of the normalized sum is $[-1,1]$.

math.NT↗

arXiv2026

On local large deviations for decoupled random walks

A decoupled standard random walk is a sequence of independent random variables $(\hat{S}_n)_{n \geq 1}$ such that, for each $n \geq 1$, the distribution of $\hat{S}_n$ is the same as that of $S_n = ξ_1 + \ldots + ξ_n$, where $(ξ_k)_{k \geq 1}$ are independent copies of a nonnegative random variable $ξ$. We consider the counting process $(\hat{N}(t))_{t\geq 0}$ defined as the number of terms $\hat{S}_n$ in the sequence $(\hat{S}_n)_{n \geq 1}$ that lie within the interval $[0, t]$. Under various assumptions on the tail distribution of $ξ$, we derive logarithmic asymptotics for the local large deviation probabilities $\mathbb{P}\{\hat{N}(t) = \lfloor b \, \mathbb{E}[\hat{N}(t)] \rfloor\}$ as $t \to \infty$ for a fixed constant $b > 0$. These results are then applied to obtain a logarithmic local large deviations asymptotic for the counting process associated with the infinite Ginibre ensemble and, more generally, for determinantal point processes with the Mittag-Leffler kernel.

math.PR↗

arXiv2025

Laws of the iterated and single logarithm for sums of independent indicators, with applications to the Ginibre point process and Karlin's occupancy scheme

We prove a law of the iterated logarithm (LIL) for an infinite sum of independent indicators parameterized by $t$ as $t\to\infty$. It is shown that if the expectation $b$ and the variance $a$ of the sum are comparable, then the normalization in the LIL includes the iterated logarithm of $a$. If the expectation grows faster than the variance, while the ratio $\log b/\log a$ remains bounded, then the normalization in the LIL includes the single logarithm of $a$ (so that the LIL becomes a law of the single logarithm). Applications of our result are given to the number of points of the infinite Ginibre point process in a disk and the number of occupied boxes and related quantities in Karlin's occupancy scheme.

math.PR↗

arXiv2024

Convolution powers of unbounded measures on the positive half-line

For a right-continuous nondecreasing and unbounded function $V$ of at most exponential growth, which vanishes on the negative halfline, we investigate the asymptotic behavior of the Lebesgue-Stieltjes convolution powers $V^{\ast(j)}(t)$ as both $j$ and $t$ tend to infinity. We obtain a comprehensive asymptotic formula for $V^{\ast(j)}(t)$, which is valid across different regimes of simultaneous growth of $j$ and $t$. Our main technical tool is an exponential change of measure, which is a standard technique in the large deviations theory. Various applications of our result are given.

math.PR↗

arXiv2023

Solutions of kinetic-type equations with perturbed collisions

We study a class of kinetic-type differential equations $\partial ϕ_t/\partial t+ϕ_t=\widehat{\mathcal{Q}}ϕ_t$, where $\widehat{\mathcal{Q}}$ is an inhomogeneous smoothing transform and, for every $t\geq 0$, $ϕ_t$ is the Fourier--Stieltjes transform of a probability measure. We show that under mild assumptions on $\widehat{\mathcal{Q}}$ the above differential equation possesses a unique solution and represent this solution as the characteristic function of a certain stochastic process associated with the continuous time branching random walk pertaining to $\widehat{\mathcal{Q}}$. Establishing limit theorems for this process allows us to describe asymptotic properties of the solution, as $t\to\infty$.

math.PR↗

arXiv2023

Critical branching processes in a sparse random environment

We introduce a branching process in a sparse random environment as an intermediate model between a Galton--Watson process and a branching process in a random environment. In the critical case we investigate the survival probability and prove Yaglom-type limit theorems, that is, limit theorems for the size of population conditioned on the survival event.

math.PR↗

arXiv2023

Weak quenched limit theorems for a random walk in a sparse random environment

We study the quenched behaviour of a perturbed version of the simple symmetric random walk on the set of integers. The random walker moves symmetrically with an exception of some randomly chosen sites where we impose a random drift. We show that if the gaps between the marked sites are i.i.d. and regularly varying with a sufficiently small index, then there is no strong quenched limit laws for the position of the random walker. As a consequence we study the quenched limit laws in the context of weak convergence of random measures.

math.PR↗

arXiv2022

Limit theorems for random Dirichlet series

We prove a functional limit theorem in a space of analytic functions for the random Dirichlet series $D(α;z)=\sum_{n\geq 2}(\log n)^α(η_n+{\rm i} θ_n)/n^z$, properly scaled and normalized, where $(η_n,θ_n)_{n\in\mathbb{N}}$ is a sequence of independent copies of a centered $\mathbb{R}^2$-valued random vector $(η,θ)$ with a finite second moment and $α>-1/2$ is a fixed real parameter. As a consequence, we show that the point processes of complex and real zeros of $D(α;z)$ converge vaguely, thereby obtaining a universality result. In the real case, that is, when $\mathbb{P}\{θ=0\}=1$, we also prove a law of the iterated logarithm for $D(α;z)$, properly normalized, as $z\to (1/2)+$.

math.PR↗

arXiv2021

Asymptotically linear iterated function systems on the real line

Given a sequence of i.i.d. random functions $Ψ_{n}:\mathbb{R}\to\mathbb{R}$, $n\in\mathbb{N}$, we consider the iterated function system and Markov chain which is recursively defined by $X_{0}^{x}:=x$ and $X_{n}^{x}:=Ψ_{n-1}(X_{n-1}^{x})$ for $x\in\mathbb{R}$ and $n\in\mathbb{N}$. Under the two basic assumptions that the $Ψ_{n}$ are a.s. continuous at any point in $\mathbb{R}$ and asymptotically linear at the "endpoints" $\pm\infty$, we study the tail behavior of the stationary laws of such Markov chains by means of Markov renewal theory. Our approach provides an extension of Goldie's implicit renewal theory and can also be viewed as an adaptation of Kesten's work on products of random matrices to one-dimensional function systems as described. Our results have applications in quite different areas of applied probability like queuing theory, econometrics, mathematical finance and population dynamics. Our results have applications in quite different areas of applied probability like queuing theory, econometrics, mathematical finance and population dynamics, e.g. ARCH models and random logistic transforms.

math.PR↗

arXiv2021

Limit theorems for supercritical branching processes in random environment

We consider the branching process in random environment $\{Z_n\}_{n\geq 0}$, which is a~population growth process where individuals reproduce independently of each other with the reproduction law randomly picked at each generation. We focus on the supercritical case, when the process survives with a positive probability and grows exponentially fast on the nonextinction set. Our main is goal is establish Fourier techniques for this model, which allow to obtain a number of precise estimates related to limit theorems. As a consequence we provide new results concerning central limit theorem, Edgeworth expansions and renewal theorem for $\log Z_n$.

math.PR↗

arXiv2020

Moderate parts in regenerative compositions: the case of regular variation

A regenerative random composition of integer $n$ is constructed by allocating $n$ standard exponential points over a countable number of intervals, comprising the complement of the closed range of a subordinator $S$. Assuming that the Lévy measure of $S$ is infinite and regularly varying at zero of index $-α$, $α\in(0,\,1)$, we find an explicit threshold $r=r(n)$, such that the number $K_{n,\,r(n)}$ of blocks of size $r(n)$ converges in distribution without any normalization to a mixed Poisson distribution. The sequence $(r(n))$ turns out to be regularly varying with index $α/(α+1)$ and the mixing distribution is that of the exponential functional of $S$. The result is derived as a consequence of a general Poisson limit theorem for an infinite occupancy scheme with power-like decay of the frequencies. We also discuss asymptotic behavior of $K_{n,\,w(n)}$ in cases when $w(n)$ diverges but grows slower than $r(n)$. Our findings complement previously known strong laws of large numbers for $K_{n,\,r}$ in case of a fixed $r\in\mathbb{N}$. As a key tool we employ new Abelian theorems for Laplace--Stiletjes transforms of regularly varying functions with the indexes of regular variation diverging to infinity.

math.PR↗

arXiv2020

On intermediate levels of nested occupancy scheme in random environment generated by stick-breaking I

Consider a weighted branching process generated by the lengths of intervals obtained by stick-breaking of unit length (a.k.a. the residual allocation model) and associate with each weight a `box'. Given the weights `balls' are thrown independently into the boxes of the first generation with probability of hitting a box being equal to its weight. Each ball located in a box of the $j$th generation, independently of the others, hits a daughter box in the $(j+1)$th generation with probability being equal the ratio of the daughter weight and the mother weight. This is what we call nested occupancy scheme in random environment. Restricting attention to a particular generation one obtains the classical Karlin occupancy scheme in random environment. Assuming that the stick-breaking factor has a uniform distribution on $[0,1]$ and that the number of balls is $n$ we investigate occupancy of intermediate generations, that is, those with indices $\lfloor j_n u\rfloor$ for $u>0$, where $j_n$ diverges to infinity at a sublogarithmic rate as $n$ becomes large. Denote by $K_n(j)$ the number of occupied (ever hit) boxes in the $j$th generation. It is shown that the finite-dimensional distributions of the process $(K_n(\lfloor j_n u\rfloor))_{u>0}$, properly normalized and centered, converge weakly to those of an integral functional of a Brownian motion. The case of a more general stick-breaking is also analyzed.

math.PR↗

arXiv2020

On uniqueness of invariant measures for random walks on HOMEO(R)

We consider random walks on the group of orientation-preserving homeomorphisms of the real line ${\mathbb R}$. In particular, the fundamental question of uniqueness of an invariant measure of the generated process is raised. This problem was already studied by Choquet and Deny (1960) in the context of random walks generated by translations of the line. Nowadays the answer is quite well understood in general settings of strongly contractive systems. Here we focus on broader class of systems satisfying the conditions: recurrence, contraction and unbounded action. We prove that under these conditions the random process possesses a unique invariant Radon measure on ${\mathbb R}$. Our work can be viewed as a subsequent paper of Babillot et al. (1997) and Deroin et al. (2013).

math.PR↗

arXiv2020

Precise large deviation estimates for branching process in random environment

We consider the branching process in random environment $\{Z_n\}_{n\geq 0}$, which is a~population growth process where individuals reproduce independently of each other with the reproduction law randomly picked at each generation. We describe precise asymptotics of upper large deviations, i.e. $\mathbb{P}[Z_n > e^{ρn}]$. Moreover in the subcritical case, under the Cramér condition on the mean of the reproduction law, we investigate large deviations-type estimates for the first passage time of the branching process in question and its total population size.

math.PR↗

arXiv2020

Solutions to kinetic-type evolution equations: beyond the boundary case

We study the asymptotic behavior as $t \to \infty$ of a time-dependent family $(μ_t)_{t \geq 0}$ of probability measures on $\mathbb{R}$ solving the kinetic-type evolution equation $\partial_t μ_t + μ_t = Q(μ_t)$ where $Q$ is a smoothing transformation on $\mathbb{R}$. This problem has been investigated earlier, e.g. by Bassetti and Ladelli [Ann. Appl. Probab. 22(5): 1928-1961, 2012] and Bogus, Buraczewski and Marynych [Stochastic Process. Appl. 130(2):677-693, 2020]. Combining the refined analysis of the latter paper, which provides a probabilistic description of the solution $μ_t$ as the law of a suitable random sum related to a continuous-time branching random walk at time $t$, with recent advances in the analysis of the extremal positions in the branching random walk we are able to solve the remaining case that has been left open until now. In the course of our work, we significantly weaken the assumptions in the literature that guarantee the existence (and uniqueness) of a solution to the evolution equation $\partial_t μ_t + μ_t = Q(μ_t)$.

math.PR↗

arXiv2020

A Brownian weak limit for the least common multiple of a random m-tuple of integers

Let $B_n(m)$ be a set picked uniformly at random among all $m$-elements subsets of $\{1,2,\ldots,n\}$. We provide a pathwise construction of the collection $(B_n(m))_{1\leq m\leq n}$ and prove that the logarithm of the least common multiple of the integers in $(B_n(\lfloor mt\rfloor))_{t\geq 0}$, properly centered and normalized, converges to a Brownian motion when both $m,n$ tend to infinity. Our approach consists of two steps. First, we show that the aforementioned result is a consequence of a multidimensional central limit theorem for the logarithm of the least common multiple of $m$ independent random variables having uniform distribution on $\{1,2,\ldots,n\}$. Second, we offer a novel approximation of the least common multiple of a random sample by the product of the elements of the sample with neglected multiplicities in their prime decompositions.

math.PR↗

arXiv2020

On the derivative martingale in a branching random walk

We work under the A\"ıdékon-Chen conditions which ensure that the derivative martingale in a supercritical branching random walk on the line converges almost surely to a nondegenerate nonnegative random variable that we denote by $Z$. It is shown that $\mathbb{E} Z\mathbf{1}_{\{Z\le x\}}=\log x+o(\log x)$ as $x\to\infty$. Also, we provide necessary and sufficient conditions under which $\mathbb{E} Z\mathbf{1}_{\{Z\le x\}}=\log x+{\rm const}+o(1)$ as $x\to\infty$. This more precise asymptotics is a key tool for proving distributional limit theorems which quantify the rate of convergence of the derivative martingale to its limit $Z$. The methodological novelty of the present paper is a three terms representation of a subharmonic function of at most linear growth for a killed centered random walk of finite variance. This yields the aforementioned asymptotics and should also be applicable to other models.

math.PR↗