arXiv subjects

Alexander Iksanov

Publications and source records attributed to Alexander Iksanov.

At least 19 recordsLinked to original sources

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↗

arXiv2026

On tail behavior of infinite sums of independent indicators

Let $Y=\sum_{k\ge 1} 1_{A_k}$ be an infinite sum of the indicators of independent events. We investigate a precise (as opposed to logarithmic) first-order asymptotic behavior of the tail probabilities $\mathbb{P}\{Y\ge n\}$ and the point probabilities $\mathbb{P}\{Y=n\}$ as $n\to\infty$. Our analysis provides a reasonably complete classification of the asymptotic behaviors covering most cases of practical interest. These general results are then applied to specific examples where the success probabilities $r_k:=\mathbb{P}(A_k)$ decay polynomially $r_k\sim ck^{-β}$ or (sub-, super-) exponentially $r_k\sim ce^{-k^β}$, yielding the asymptotic tail and point probabilities in explicit forms. As briefly discussed in the paper, infinite sums of independent indicators arise naturally in numerous settings as diverse as the range of Poissonized samples, the infinite Ginibre point processes and decoupled renewal processes, and records in the $F^α$ scheme. We also explore connections between our results and the theory of Hayman-admissible functions, total positivity, and the Laguerre-Pólya class of type I.

math.PR↗

arXiv2026

Laws of the iterated logarithm for random Dirichlet series with general weights

For each $s>0$, we consider a random Dirichlet series $X(s)=\sum_{k\geq 1}k^{-1/2-s}a_kη_k$, where $η_1$, $η_2,\ldots$ are independent and identically distributed random variables with mean zero and finite positive variance, and $(a_k)_{k\geq 1}$ is a deterministic sequence of real numbers satisfying $\sum_{k\geq 1}k^{-1-2s}a_k^2<\infty$ for each $s>0$ and $\sum_{k\geq 1}k^{-1}a_k^2=\infty$. We investigate the almost-sure fluctuations of $X(s)$ as $s\to0+$. Under these minimal assumptions, we construct examples exhibiting several non-standard forms of the law of the iterated logarithm (LIL) along suitable sequences: the normalization and the upper and lower limit constants may differ from their classical counterparts. We also show that a regular growth condition of the form $\sum_{k\leq n}k^{-1}a_k^2\sim c(\log n)^β$, where $c,β>0$, is not by itself sufficient to ensure a standard LIL. Finally, under an additional counting condition controlling the frequency of indices at which the weights $a_k$ are comparatively large, we prove that $(2{\rm Var}\,[X(s)]\log\log({\rm Var}\,[X(s)]))^{-1/2}X(s)$ has the almost-sure cluster set $[-1,1]$ as $s\to 0+$. The latter result is applied to several coefficient sequences of number-theoretic origin.

math.PR↗

arXiv2026

On the tails of Dickman-like perpetuities

Using a probabilistic technique based on exponential change of measure, we derive precise tail asymptotics of some perpetuities with distributions close to the Dickman distribution.

math.PR↗

arXiv2026

Limit theorems for random Dirichlet series with summation over primes, with an application to Rademacher random multiplicative functions

It is shown that two conjectures put forward in the recent article Iksanov and Kostohryz (2025) are true. Namely, we prove a functional central limit theorem (FCLT) and a law of the iterated logarithm (LIL) for a random Dirichlet series $\sum_p \frac{η_p}{p^{1/2+s}}$ as $s\to 0+$, where $η_1$, $η_2,\ldots$ are independent identically distributed random variables with zero mean and finite variance, and $\sum_p$ denotes the summation over the prime numbers. As a consequence, an FCLT and an LIL are obtained for $\log \sum_{n\geq 1} \frac{f(n)}{n^{1/2+s}}$ as $s\to 0+$, where $f$ is a Rademacher random multiplicative function.

math.PR↗

arXiv2026

First passage times for decoupled random walks

Motivated by a connection to the infinite Ginibre point process, decoupled random walks were introduced in a recent article Alsmeyer, Iksanov and Kabluchko (2025). The decoupled random walk is a sequence of independent random variables, in which the $n$th variable has the same distribution as the position at time $n$ of a standard random walk with nonnegative increments. We prove distributional convergence in the Skorokhod space equipped with the $J_1$-topology of the running maxima and the first passage times of decoupled random walks. We show that there exist five different regimes, in which distinct limit theorems arise. Rather different functional limit theorems for the number of visits of decoupled standard random walk to the interval $[0,t]$ as $t\to\infty$ were earlier obtained in the aforementioned paper Alsmeyer, Iksanov and Kabluchko (2025). While the limit processes for the first passage times are inverse extremal-like processes, the limit processes for the number of visits are stationary Gaussian.

math.PR↗

arXiv2025

Limit theorems for decoupled renewal processes

The decoupled standard random walk is a sequence of independent random variables $(\hat S_n)_{n\geq 1}$, in which $\hat S_n$ has the same distribution as the position at time $n$ of a standard random walk with nonnegative jumps. Denote by $\hat N(t)$ the number of elements of the decoupled standard random walk which do not exceed $t$. The random process $(\hat N(t))_{t\geq 0}$ is called decoupled renewal process. Under the assumption that $t\mapsto \mathbb{P}\{\hat S_1>t\}$ is regularly varying at infinity of nonpositive index larger than $-1$ we prove a functional central limit theorem in the Skorokhod space equipped with the $J_1$-topology for the decoupled renewal processes, properly scaled, centered and normalized. Also, under the assumption that $t\mapsto \mathbb{P}\{\hat S_1>t\}$ is regularly varying at infinity of index $-α$, $α\in [0,1)\cup (1,2)$ or the distribution of $\hat S_1$ belongs to the domain of attraction of a normal distribution we prove a law of the iterated or single logarithm for $\hat N(t)$, again properly normalized and centered. As an application, we obtain a law of the single logarithm for the number of atoms of a determinantal point process with the Mittag-Leffler kernel, which lie in expanding discs.

math.PR↗

arXiv2025

Some central limit theorems for critical beta-splitting tree

We further explore a connection initially unveiled in Iksanov (2025) between critical beta-splitting trees and infinite `balls-in-boxes' schemes. Using the connection, we derive a new joint central limit theorem for components of the height of a leaf chosen uniformly at random in the discrete version of a critical beta-splitting tree. Also, we obtain a joint central limit theorem for the heights in the discrete and continuous versions of a critical beta-splitting tree.

math.PR↗

arXiv2025

Functional limit theorems for random Lebesgue-Stieltjes convolutions

We prove joint functional limit theorems in the Skorokhod space equipped with the $J_1$-topology for successive Lebesgue-Stieltjes convolutions of nondecreasing stochastic processes with themselves. These convolutions arise naturally in coupled branching random walks, where the displacements of individuals relative to their mother's position are given by the underlying point process rather than its copy. Surprisingly, the numbers of individuals in the $j$th generation, with positions less than or equal to $t$, exhibit remarkably similar distributional behavior in both standard branching random walks and coupled branching random walks as $t$ tends to infinity.

math.PR↗

arXiv2025

On intermediate levels of nested occupancy scheme in random environment generated by stick-breaking: the case of heavy tails

We investigate a nested balls-in-boxes scheme in a random environment. The boxes follow a nested hierarchy, with infinitely many boxes in each level, and the hitting probabilities of boxes are random and obtained by iterated fragmentation of a unit mass. The hitting probabilities of the first-level boxes are given by a stick-breaking model $P_k = W_1 W_2\cdot \ldots\cdot W_{k-1}(1- W_k)$ for $k \in \mathbb{N}$, where $W_1$, $W_2,\ldots$ are independent copies of a random variable $W$ taking values in $(0,1)$. The infinite balls-in-boxes scheme in the first level is known as a Bernoulli sieve. We assume that the mean of $|\log W|$ is infinite and the distribution tail of $|\log W|$ is regularly varying at $\infty$. Denote by $K_n(j)$ the number of occupied boxes in the $j$th level provided that there are $n$ balls and call the level $j$ intermediate, if $j = j_n \to \infty$ and $j_n = o((\log n)^a)$ as $n \to \infty$ for appropriate $a>0$. We prove that, for some intermediate levels $j$, finite-dimensional distributions of the process $(K_n(\lfloor j_n u\rfloor))_{u>0}$, properly normalized, converge weakly as $n\to\infty$ to those of a pathwise Lebesgue-Stieltjes integral, with the integrand being an exponential function and the integrator being an inverse stable subordinator. The present paper continues the line of investigation initiated in the articles Buraczewski, Dovgay and Iksanov (2020) and Iksanov, Marynych and Samoilenko (2022) in which the random variable $|\log W|$ has a finite second moment, and Iksanov, Marynych and Rashytov (2022) in which $|\log W|$ has a finite mean and an infinite second moment.

math.PR↗

arXiv2025

Limit theorems for globally perturbed random walks

Let $(ξ_1, η_1)$, $(ξ_2, η_2),\ldots$ be independent copies of an $\mathbb{R}^2$-valued random vector $(ξ, η)$ with arbitrarily dependent components. Put $T_n:= ξ_1+\ldots+ξ_{n-1} + η_n $ for $n\in\mathbb{N}$ and define $τ(t) := \inf\{n\geq 1: T_n>t\}$ the first passage time into $(t,\infty)$, $N(t) :=\sum_{n\geq 1}1_{\{T_n\leq t\}}$ the number of visits to $(-\infty, t]$ and $ρ(t):=\sup\{n\geq 1: T_n \leq t\}$ the associated last exit time for $t\in\mathbb{R}$. The standing assumption of the paper is $\mathbb{E}[ξ]\in (0,\infty)$. We prove a weak law of large numbers for $τ(t)$ and strong laws of large numbers for $τ(t)$, $N(t)$ and $ρ(t)$. The strong law of large numbers for $τ(t)$ holds if, and only if, $\mathbb{E}[η^+]<\infty$. In the complementary situation $\mathbb{E}[η^+]=\infty$ we prove functional limit theorems in the Skorokhod space for $(τ(ut))_{u\geq 0}$, properly normalized without centering. Also, we provide sufficient conditions under which finite dimensional distributions of $(τ(ut))_{u\geq 0}$, $(N(ut))_{u\geq 0}$ and $(ρ(ut))_{u\geq 0}$, properly normalized and centered, converge weakly as $t\to\infty$ to those of a Brownian motion. Quite unexpectedly, the centering needed for $(N(ut))$ takes in general a more complicated form than the centering $ut/\mathbb{E}[ξ]$ needed for $(τ(ut))$ and $(ρ(ut))$. Finally, we prove a functional limit theorem in the Skorokhod space for $(N(ut))$ under optimal moment conditions.

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↗

arXiv2025

Almost periodic stochastic processes with applications to analytic number theory

A classical fact of the theory of almost periodic functions is the existence of their asymptotic distributions. In probabilistic terms, this means that if $f$ is a Besicovitch almost periodic function and $V$ is a random variable uniformly distributed on $[-1,1]$, then the random variables $f(L\cdot V)$ converge in distribution, as $L\to\infty$, to a proper non-degenerate random variable. We prove a functional extension of this result for the random processes $(f(L\cdot V+t))_{t\in\mathbb{R}}$ in the space of Besicovitch almost periodic functions, and also in the sense of weak convergence of finite-dimensional distributions. We further investigate the properties of the limiting stationary process and demonstrate applications in analytic number theory by extending the one-dimensional results of [Limiting distributions of the classical error terms of prime number theory, Quart. J. Math. 65 (2014), 743--780] and earlier works.

math.PR↗

arXiv2024

A comment on `The harmonic descent chain' by D.J. Aldous, S. Janson and X. Li

An alternative proof is given for the main result of the article referred to in the title and published in ECP (2024). The proof exploits the theory of regenerative composition structures due to Gnedin and Pitman. The present article is a slight revision of my note written up in May 2024 as a reaction to the preprint arxiv.org version of the paper by Aldous, Janson and Li.

math.PR↗

arXiv2024

Limit theorems for random Dirichlet series: boundary case

Buraczewski et al (2023) proved a functional limit theorem (FLT) and a law of the iterated logarithm (LIL) for a random Dirichlet series $\sum_{k\geq 2}(\log k)^αk^{-1/2-s}η_k$ as $s\to 0+$, where $α>-1/2$ and $η_1$, $η_2,\ldots$ are independent identically distributed random variables with zero mean and finite variance. We prove a FLT and a LIL in a boundary case $α=-1/2$. The boundary case is more demanding technically than the case $α>-1/2$.

math.PR↗

arXiv2024

A law of the iterated logarithm for the number of blocks in regenerative compositions generated by gamma-like subordinators

The points of the closed range of a drift-free subordinator with no killing are used for separating into blocks the elements of a sample of size $n$ from the standard exponential distribution. This gives rise to a random composition of $n$. Assuming that the subordinator has the Lévy measure, which behaves near zero like the gamma subordinator, we prove a law of the iterated logarithm for the number of blocks in the composition as $n$ tends to infinity. Along the way we prove a law of the iterated logarithm for the Lebesgue convolution of a standard Brownian motion and a deterministic regularly varying function. This result may be of independent interest.

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↗

arXiv2024

Multinomial random combinatorial structures and $r$-versions of Stirling, Eulerian and Lah numbers

We introduce multinomial and $r$-variants of several classic objects of combinatorial probability, such as the random recursive and Hoppe trees, random set partitions and compositions, the Chinese restaurant process, Feller's coupling, and some others. Just as various classic combinatorial numbers - like Stirling, Eulerian and Lah numbers - emerge as essential ingredients defining the distributions of the mentioned processes, the so-called $r$-versions of these numbers appear in exact distributional formulas for the multinomial and $r$-counterparts. This approach allows us to offer a concise probabilistic interpretation for various identities involving $r$-versions of these combinatorial numbers, which were either unavailable or meaningful only for specific values of the parameter $r$. We analyze the derived distributions for fixed-size structures and establish distributional limit theorems as the size tends to infinity. Utilizing the aforementioned generalized Stirling numbers of both kinds, we define and analyze $(r,s)$-Lah distributions, which have arisen in the existing literature on combinatorial probability in various contexts.

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