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Alexander Iksanov

Publications and source records attributed to Alexander Iksanov.

At least 37 records · Page 2Linked to original sources
arXiv2024

Asymptotic fluctuations in supercritical Crump-Mode-Jagers processes

Consider a supercritical Crump--Mode--Jagers process $(\mathcal Z_t^φ)_{t \geq 0}$ counted with a random characteristic $φ$. Nerman's celebrated law of large numbers [Z. Wahrsch. Verw. Gebiete 57, 365--395, 1981] states that, under some mild assumptions, $e^{-αt} \mathcal Z_t^φ$ converges almost surely as $t \to \infty$ to $aW$. Here, $α>0$ is the Malthusian parameter, $a$ is a constant and $W$ is the limit of Nerman's martingale, which is positive on the survival event. In this general situation, under additional (second moment) assumptions, we prove a central limit theorem for $(\mathcal Z_t^φ)_{t \geq 0}$. More precisely, we show that there exist a constant $k \in \mathbb N_0$ and a function $H(t)$, a finite random linear combination of functions of the form $t^j e^{λt}$ with $α/2 \leq \mathrm{Re}(λ)<α$, such that $(\mathcal Z_t^φ- a e^{αt}W -H(t))/\sqrt{t^k e^{αt}}$ converges in distribution to a normal random variable with random variance. This result unifies and extends various central limit theorem-type results for specific branching processes.

math.PR↗
arXiv2024

On decoupled standard random walks

Let $S_{n}=\sum_{k=1}^{n}ξ_{k}$, $n\in\mathbb{N}$, be a standard random walk with i.i.d. nonnegative increments $ξ_{1},ξ_{2},\ldots$ and associated renewal counting process $N(t)=\sum_{n\ge 1}1_{\{S_{n}\le t\}}$, $t\ge 0$. A decoupling of $(S_{n})_{n\ge 1}$ is any sequence $\hat{S}_{1}$, $\hat{S}_{2},\ldots$ of independent random variables such that, for each $n\in\mathbb{N}$, $\hat{S}_{n}$ and $S_{n}$ have the same law. Under the assumption that the law of $\hat{S}_{1}$ belongs to the domain of attraction of a stable law with finite mean, we prove a functional limit theorem for the \emph{decoupled renewal counting process} $\hat{N}(t)=\sum_{n\ge 1}1_{\{\hat{S}_{n}\le t\}}$, $t\ge 0$, after proper scaling, centering and normalization. We also study the asymptotics of $\log \mathbb{P}\{\min_{n\ge 1}\hat{S}_{n}>t\}$ as $t\to\infty$ under varying assumptions on the law of $\hat{S}_{1}$. In particular, we recover the assertions which were previously known in the case when $\hat{S}_{1}$ has an exponential law. These results, which were formulated in terms of an infinite Ginibre point process, served as an initial motivation for the present work. Finally, we prove strong law of large numbers type results for the sequence of decoupled maxima $M_{n}=\max_{1\le k\le n}\hat{S}_{k}$, $n\in\mathbb{N}$, and the related first passage time process $\hatτ(t)=\inf\{n\in\mathbb{N}: M_{n}>t\}$, $t\ge 0$. In particular, we provide a tail condition on the law of $\hat{S}_{1}$ in the case when the latter has finite mean but infinite variance that implies $\lim_{t\to\infty}t^{-1}\hatτ(t)=\lim_{t\to\infty}t^{-1}\mathbb{E}\hatτ(t)=0$. In other words, $t^{-1}\hatτ(t)$ may exhibit a different limit behavior than $t^{-1}τ(t)$, where $τ(t)$ denotes the level-$t$ first passage time of $(S_{n})_{n\ge 1}$.

math.PR↗
arXiv2023

On multidimensional locally perturbed standard random walks

Let $d$ be a positive integer and $A$ a set in $\mathbb{Z}^d$, which contains finitely many points with integer coordinates. We consider $X$ a standard random walk perturbed on the set $A$, that is, a Markov chain whose transition probabilities from the points outside $A$ coincide with those of a standard random walk on $\mathbb{Z}^d$, whereas the transition probabilities from the points inside $A$ are different. We investigate the impact of the perturbation on a scaling limit of $X$. It turns out that if $d\geq 2$, then in a typical situation the scaling limit of $X$ coincides with that of the underlying standard random walk. This is unlike the case $d=1$ in which the scaling limit of $X$ is usually a skew Brownian motion, a skew stable Lévy process or some other `skew' process. The distinction between the one-dimensional and the multidimensional cases under comparable assumptions may simply be caused by transience of the underlying standard random walk in $\mathbb{Z}^d$ for $d\geq 3$. More interestingly, in the situation where the standard random walk in $\mathbb{Z}^2$ is recurrent, the preservation of its Donsker scaling limit is secured by the fact that the number of visits of $X$ to the set $A$ is much smaller than in the one-dimensional case. As a consequence, the influence of the perturbation vanishes upon the scaling. On the other edge of the spectrum is the situation in which the standard random walk admits a Donsker's scaling limit, whereas its locally perturbed version does not because of huge jumps from the set $A$ which occur early enough.

math.PR↗
arXiv2023

A law of the iterated logarithm for small counts in Karlin's occupancy scheme

In the Karlin infinite occupancy scheme, balls are thrown independently into an infinite array of boxes $1$, $2,\ldots$, with probability $p_k$ of hitting the box $k$. For $j,n\in\mathbb{N}$, denote by $\mathcal{K}^*_j(n)$ the number of boxes containing exactly $j$ balls provided that $n$ balls have been thrown. We call $\textit{small counts}$ the variables $\mathcal{K}^*_j(n)$, with $j$ fixed. Our main result is a law of the iterated logarithm (LIL) for the small counts as the number of balls thrown becomes large. Its proof exploits a Poissonization technique and is based on a new LIL for infinite sums of independent indicators $\sum_{k\geq 1}\Bbb{1}_{A_k(t)}$ as $t\to\infty$, where the family of events $(A_k(t))_{t\geq 0}$ is not necessarily monotone in $t$. The latter LIL is an extension of a LIL obtained recently by Buraczewski, Iksanov and Kotelnikova (2023+) in the situation that $(A_k(t))_{t\geq 0}$ forms a nondecreasing family of events.

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arXiv2023

Precise tail behavior of some Dirichlet series

Let $η_1$, $η_2,\ldots$ be independent copies of a random variable $η$ with zero mean and finite variance which is bounded from the right, that is, $η\leq b$ almost surely for some $b>0$. Considering different types of the asymptotic behaviour of the probability $\mathbb{P}\{η\in[b-x,b]\}$ as $x\to 0+$ we derive precise tail asymptotics of the random Dirichlet series $\sum_{k\geq 1}k^{-α}η_k$ for $α\in (1/2, 1]$.

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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↗
arXiv2022

A law of the iterated logarithm for iterated random walks, with application to random recursive trees

Consider a Crump-Mode-Jagers process generated by an increasing random walk whose increments have finite second moment. Let $Y_k(t)$ be the number of individuals in generation $k\in \mathbb N$ born in the time interval $[0,t]$. We prove a law of the iterated logarithm for $Y_k(t)$ with fixed $k$, as $t\to +\infty$. As a consequence, we derive a law of the iterated logarithm for the number of vertices at a fixed level $k$ in a random recursive tree, as the number of vertices goes to $\infty$.

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↗
arXiv2022

Limit theorems for discounted convergent perpetuities II

Let $(ξ_1, η_1)$, $(ξ_2, η_2),\ldots$ be independent identically distributed $\mathbb{R}^2$-valued random vectors. Assuming that $ξ_1$ has zero mean and finite variance and imposing three distinct groups of assumptions on the distribution of $η_1$ we prove three functional limit theorems for the logarithm of convergent discounted perpetuities $\sum_{k\geq 0}e^{ξ_1+\ldots+ξ_k-ak}η_{k+1}$ as $a\to 0+$. Also, we prove a law of the iterated logarithm which corresponds to one of the aforementioned functional limit theorems. The present paper continues a line of research initiated in the paper Iksanov, Nikitin and Samoillenko (2022), which focused on limit theorems for a different type of convergent discounted perpetuities.

math.PR↗
arXiv2022

Functional limit theorems for random walks perturbed by positive alpha-stable jumps

Let $ξ_1$, $ξ_2,\ldots$ be i.i.d. random variables of zero mean and finite variance and $η_1$, $η_2,\ldots$ positive i.i.d. random variables whose distribution belongs to the domain of attraction of an $α$-stable distribution, $α\in (0,1)$. The two collections are assumed independent. We consider a Markov chain with jumps of two types. If the present position of the Markov chain is positive, then the jump $ξ_k$ occurs; if the present position of the Markov chain is nonpositive, then its next position is $η_j$. We prove a functional limit theorem for this Markov chain under Donsker's scaling. The weak limit is a nonnegative process $(X(t))_{t\geq 0}$ satisfying a stochastic equation ${\rm d}X(t)={\rm d}W(t)+ {\rm d}U_α(L_X^{(0)}(t))$, where $W$ is a Brownian motion, $U_α$ is an $α$-stable subordinator which is independent of $W$, and $L_X^{(0)}$ is a local time of $X$ at $0$. Also, we explain that $X$ is a Feller Brownian motion with a `jump-type' exit from $0$.

math.PR↗
arXiv2022

Asymptotics of arithmetic functions of GCD and LCM of random integers in hyperbolic regions

We prove limit theorems for the greatest common divisor and the least common multiple of random integers. While the case of integers uniformly distributed on a hypercube with growing size is classical, we look at the uniform distribution on sublevel sets of multivariate symmetric polynomials, which we call hyperbolic regions. Along the way of deriving our main results, we obtain some asymptotic estimates for the number of integer points in these hyperbolic domains, when their size goes to infinity.

math.NT↗
arXiv2022

Small counts in nested Karlin's occupancy scheme generated by discrete Weibull-like distributions

A nested Karlin's occupancy scheme is a symbiosis of classical Karlin's balls-in-boxes scheme and a weighted branching process. To define it, imagine a deterministic weighted branching process in which weights of the first generation individuals are given by the elements of a discrete probability distribution. For each positive integer $j$, identify the $j$th generation individuals with the $j$th generation boxes. The collection of balls is one and the same for all generations, and each ball starts at the root of the weighted branching process tree and moves along the tree according to the following rule: transition from a mother box to a daughter box occurs with probability given by the ratio of the daughter and mother weights. Assume that there are $n$ balls and that the discrete probability distribution responsible for the first generation is Weibull-like. Denote by $\mathcal{K}_n^{(j)}(l)$ and $\mathcal{K}_n^{*(j)}(l)$ the number of the $j$th generation boxes which contain at least $l$ balls and exactly $l$ balls, respectively. We prove functional limit theorems (FLTs) for the matrix-valued processes $\big(\mathcal{K}_{[{\rm e}^{T+\cdot}]}^{(j)}(l)\big)_{j,l\in\mathbb{N}}$ and $\big(\mathcal{ K}_{[{\rm e}^{T+\cdot}]}^{*(j)}(l)\big)_{j,l\in\mathbb{N}}$, properly normalized and centered, as $T\to \infty$. The present FLTs are an extension of a FLT proved by Iksanov, Kabluchko and Kotelnikova (2022) for the vector-valued process $\big(\mathcal{K}_{[{\rm e}^{T+\cdot}]}^{(j)}(1)\big)_{j\in\mathbb{N}}$. While the rows of each of the limit matrix-valued processes are independent and identically distributed, the entries within each row are stationary Gaussian processes with explicitly given covariances and cross-covariances. We provide an integral representation for each row. The results obtained are new even for Karlin's occupancy scheme.

math.PR↗
arXiv2022

Stable fluctuations of iterated perturbed random walks in intermediate generations of a general branching process tree

Consider a general branching process, a.k.a. Crump-Mode-Jagers process, generated by a perturbed random walk $η_1$, $ξ_1+η_2$, $ξ_1+ξ_2+η_3,\ldots$. Here, $(ξ_1,η_1)$, $(ξ_2, η_2),\ldots$ are independent identically distributed random vectors with arbitrarily dependent positive components. Denote by $N_j(t)$ the number of the $j$th generation individuals with birth times $\leq t$. Assume that $j=j(t)\to\infty$ and $j(t)=o(t^a)$ as $t\to\infty$ for some explicitly given $a>0$ (to be specified in the paper). The corresponding $j$th generation belongs to the set of intermediate generations. We provide sufficient conditions under which finite-dimensional distributions of the process $(N_{\lfloor j(t)u\rfloor}(t))_{u>0}$, properly normalized and centered, converge weakly to those of an integral functional of a stable Lévy process with finite mean.

math.PR↗
arXiv2021

On a skew stable Lévy process

The skew Brownian motion is a strong Markov process which behaves like a Brownian motion until hitting zero and exhibits an asymmetry at zero. We address the following question: what is a natural counterpart of the skew Brownian motion in the situation that the noise is a stable Lévy process with finite mean and infinite variance. We define a skew stable Lévy process $X$ as the limit of a sequence of stable Lévy processes which are perturbed at zero. We point out a formula for the resolvent of $X$ and show that $X$ is a solution to a stochastic differential equation with a local time. Also, we provide a representation of $X$ in terms of Itô`s excursion theory.

math.PR↗
arXiv2021

Gaussian fluctuations and a law of the iterated logarithm for Nerman's martingale in the supercritical general branching process

In his, by now, classical work from 1981, Nerman made extensive use of a crucial martingale $(W_t)_{t \geq 0}$ to prove convergence in probability, in mean and almost surely, of supercritical general branching processes (a.k.a. Crump-Mode-Jagers branching processes) counted with a general characteristic. The martingale terminal value $W$ figures in the limits of his results. We investigate the rate at which the martingale, now called Nerman's martingale, converges to its limit $W$. More precisely, assuming the existence of a Malthusian parameter $α> 0$ and $W_0\in L^2$, we prove a functional central limit theorem for $(W-W_{t+s})_{s\in\mathbb{R}}$, properly normalized, as $t\to\infty$. The weak limit is a randomly scaled time-changed Brownian motion. Under an additional technical assumption, we prove a law of the iterated logarithm for $W-W_t$.

math.PR↗
arXiv2021

Late levels of nested occupancy scheme in random environment

Consider a weighted branching process generated by a point process on $[0,1]$, whose atoms sum up to one. Then the weights of all individuals in any given generation sum up to one, as well. We define a nested occupancy scheme in random environment as the sequence of balls-in-boxes schemes (with random probabilities) in which boxes of the $j$th level, $j=1,2,\ldots$ are identified with the $j$th generation individuals and the hitting probabilities of boxes are identified with the corresponding weights. The collection of balls is the same for all generations, and each ball starts at the root and moves along the tree of the weighted branching process according to the following rule: transition from a mother box to a daughter box occurs with probability given by the ratio of the daughter and mother weights. Assuming that there are $n$ balls, we give a full classification of regimes of the a.s.\ convergence for the number of occupied (ever hit) boxes in the $j$th level, properly normalized, as $n$ and $j=j_n$ grow to $\infty$. Here, $(j_n)_{n\in\mathbb N}$ is a sequence of positive numbers growing proportionally to $\log n$. We call such levels late, for the nested occupancy scheme gets extinct in the levels which grow super-logarithmically in $n$ in the sense that each occupied box contains one ball. Also, in some regimes we prove the strong laws of large numbers (a) for the number of the $j$th level boxes which contain at least $k$ balls, $k\geq 2$ and (b) under the assumption that the mean number of the first level boxes is finite, for the number of empty boxes in the $j$th level.

math.PR↗
arXiv2021

Recurrence and transience of random difference equations in the critical case

For i.i.d. random vectors $(M_{1},Q_{1}),(M_{2},Q_{2}),\ldots$ such that $M>0$ a.s., $Q\geq 0$ a.s. and $\mathbb{P}(Q=0)<1$, the random difference equation $X_{n}=M_{n}X_{n-1}+Q_{n}$, $n=1,2,\ldots$, is studied in the critical case when the random walk with increments $\log M_{1},\log M_{2}$ is oscillating. We provide conditions for the null-recurrence and transience of the Markov chain $(X_{n})_{n\ge 0}$ by inter alia drawing on techniques developed in the related article Alsmeyer et al (2017) for another case exhibiting the null-recurrence/transience dichotomy.

math.PR↗
arXiv2021

Renewal theory for iterated perturbed random walks on a general branching process tree: early generations

Let $(ξ_k,η_k)_{k\in\mathbb{N}}$ be independent identically distributed random vectors with arbitrarily dependent positive components. We call a (globally) perturbed random walk a random sequence $T:=(T_k)_{k\in\mathbb{N}}$ defined by $T_k:=ξ_1+\ldots+ξ_{k-1}+η_k$ for $k\in\mathbb{N}$. Consider a general branching process generated by $T$ and denote by $N_j(t)$ the number of the $j$th generation individuals with birth times $\leq t$. We treat early generations, that is, fixed generations $j$ which do not depend on $t$. In this setting we prove counterparts for $\mathbb{E}N_j$ of the Blackwell theorem and the key renewal theorem, prove a strong law of large numbers for $N_j$, find the first-order asymptotics for the variance of $N_j$. Also, we prove a functional limit theorem for the vector-valued process $(N_1(ut),\ldots, N_j(ut))_{u\geq 0}$, properly normalized and centered, as $t\to\infty$. The limit is a vector-valued Gaussian process whose components are integrated Brownian motions.

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