Searcharxiv⌕ Search

arXiv subjects

Congzao Dong

Publications and source records attributed to Congzao Dong.

12 recordsLinked to original sources
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 $-\alpha$, $\alpha\in [0,1)\cup (1,2)$ 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

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 $\sum_p \frac{\eta_p}{p^{1/2+s}}$ as $s\to 0+$, where $\eta_1$, $\eta_2,\ldots$ 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↗
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 L\'{e}vy 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

Set-valued recursions arising from vantage-point trees

We study vantage-point trees constructed using an independent sample from the uniform distribution on a fixed convex body $K$ in $(\mathbb{R}^d,\|\cdot\|)$, where $\|\cdot\|$ is an arbitrary norm on $\mathbb{R}^d$. We prove that a sequence of sets, associated with the left boundary of a vantage-point tree, forms a recurrent Harris chain on the space of convex bodies in $(\mathbb{R}^d,\|\cdot\|)$. The limiting object is a ball polyhedron, that is, an a.s.~finite intersection of closed balls in $(\mathbb{R}^d,\|\cdot\|)$ of possibly different radii. As a consequence, we derive a limit theorem for the length of the leftmost path of a vantage-point tree.

math.PR↗
arXiv2023

Some functionals for random walks and critical branching processes in extreme random environment

Let $\left\{ S_{n},n\geq 0\right\} $ be a random walk whose increment distribution belongs without centering to the domain of attraction of an $% \alpha $-stable law, i.e., there are some scaling constants $a_{n}$ such that the sequence $S_{n}/a_{n},n=1,2,...,$ weakly converges, as $% n\rightarrow \infty $ to a random variable having an $\alpha $-stable distribution. Let $S_{0}=0,$% \begin{equation*} L_{n}:=\min \left( S_{1},...,S_{n}\right) ,\tau _{n}:=\min \left\{ 0\leq k\leq n:S_{k}=\min (0,L_{n})\right\} . \end{equation*}% Assuming that $S_{n}\leq h(n),$ where $h(n)$ is $o(a_{n})$ and $% \lim_{n\rightarrow \infty }h(n)\in \lbrack -\infty ,+\infty ]$ exists we prove several limit theorems describing the asymptotic behavior of the functionals \begin{equation*} \mathbf{E}\left[ e^{S_{\tau _{n}}};S_{n}\leq h(n)\right] \end{equation*}% as $n\rightarrow \infty $. The obtained results are applied for studying the survival probability of a critical branching process evolving in an extremely unfavorable random environment. Key words: random walk, branching processes, random environment, survival probability, unfavorable environment

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

Random walks conditioned to stay non-negative and branching processes in non-favorable random environment

Let $\{S_n,n\geq 0\} $ be a random walk whose increments belong without centering to the domain of attraction of an $\alpha$-stable law $\{Y_t,t\geq 0\}$, i.e. $S_{nt}/a_n\Rightarrow Y_t,t\geq 0,$ for some scaling constants $a_n$. Assuming that $S_0=o(a_{n})$ and $S_n\leq \varphi (n)=o(a_n),$ we prove several conditional limit theorems for the distribution of $S_{n-m}$ given $m=o(n)$ and $\min_{0\leq k\leq n}S_k\geq 0$. These theorems complement the statements established by F. Caravenna and L. Chaumont in 2013. The obtained results are applied for studying the population size of a critical branching process evolving in non-favorable environment.

math.PR↗
arXiv2023

On a discrete approximation of a skew stable L\'{e}vy process

Iksanov and Pilipenko (2023) defined a skew stable L\'{e}vy process as a scaling limit of a sequence of perturbed at $0$ symmetric stable L\'{e}vy processes (continuous-time processes). Here, we provide a simpler construction of the skew stable L\'{e}vy process as a scaling limit of a sequence of perturbed at $0$ standard random walks (random sequences).

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↗
arXiv2019

Critical branching processes in random environment and Cauchy domain of attraction

We are interested in the survival probability of a population modeled by a critical branching process in an i.i.d. random environment. We assume that the random walk associated with the branching process is oscillating and satisfies a Spitzer condition $\mathbf{P}(S_{n}>0)\rightarrow ρ,\ n\rightarrow \infty $, which is a standard condition in fluctuation theory of random walks. Unlike the previously studied case $ρ\in (0,1)$, we investigate the case where the offspring distribution is in the domain of attraction of a stable law with parameter $1$, which implies that $ρ=0$ or $1$. We find the asymptotic behaviour of the survival probability of the population in these two cases.

math.PR↗
arXiv2019

Weak convergence of random processes with immigration at random times

By a random process with immigration at random times we mean a shot noise process with a random response function (response process) in which shots occur at arbitrary random times. The so defined random processes generalize random processes with immigration at the epochs of a renewal process which were introduced in [Iksanov et al. (2017). Bernoulli, 23, 1233--1278] and bear a strong resemblance to a random characteristic in general branching processes and the counting process in a fixed generation of a branching random walk generated by a general point process. We provide sufficient conditions which ensure weak convergence of finite-dimensional distributions of these processes to certain Gaussian processes. Our main result is specialised to several particular instances of random times and response processes.

math.PR↗