arXiv subjects

Denis Denisov

Publications and source records attributed to Denis Denisov.

At least 19 recordsLinked to original sources

arXiv2026

Fluctuations of Discrete-Time Random Walks

These notes are devoted to fluctuations of one-dimensional random walks. We discuss various approaches to first-passage times and to the corresponding conditional distributions. After discussion of some classical methods, such as reflection principle for simple random walks and Wiener-Hopf factorisation, we proceed to the universality approach, which has been developed in recent past. Considering one-dimensional case allows us to avoid some technical obstacles and to present the core of this method in a more transparent way. It turns out that the universality method is much more robust than the Wiener-Hopf factorisation and allows one to consider walks with non-identically distributed or even dependent increments.

math.PR↗

arXiv2026

Corrected diffusion approximation for random walks conditioned to stay positive

Let $S_n$ be a random walk with i.i.d. increments which have zero mean and finite variance. For every $x\ge0$ we define the stopping time $\tau_x:=\inf\{n\ge1:x+S_n\le0\}$ and consider the probabilities $\mathbb{P}(x+S_n\ge y,\tau_x>n)$. We study the quality of the normal approximation for these probabilities and derive a Berry-Esseen-type inequality for $\mathbb{P}(x+S_n\ge y|\tau_x>n)$. Our Theorem 1 is an extension of the results in our previous paper (arXiv:2412.08502) where we have considered the special case $x=0$. It is also worth mentioning that Theorem 1 complements the results of Siegmund and Yuh (1982) on the corrected diffusion approximation.

math.PR↗

arXiv2026

Harmonic polynomials and other exactly computable characteristics for $2$-dimensional random walks in cones

In this note we consider $2$-dimensional lattice random walks killed at leaving a wedge with opening $\alpha\in(0,\pi]$. Assuming that the walk cannot jump over the boundary of the wedge we prove that there exists a harmonic polynomial if and only if $\alpha=\pi/m$ with some integer $m$. Our proof is constructive and allows one to give exact expressions for harmonic polynomials for every integer $m$. Furthermore, we give exact expressions for all finite moments of the exit time, this result is valid for all angles $\alpha$.

math.PR↗

arXiv2025

A model of discrete interacting updates

We consider $N$ counters taking integer values which are subject to the following dynamics. At every time, a pair of distinct counters is chosen uniformly at random and their states are updated according to the following rule. If the states are different, then the smaller one is increased by $1$, while if the states are the same, both of them are increased by $1$. We show that, for a fixed $N$, the distances between consecutive ordered counters form a positive recurrent Markov chain and there exists the speed $V(N)$ defined as the average number of counters updated per time step in the stationary regime. We provide non-trivial upper and lower bounds for $V(N)$ as $N\to \infty$. Despite the simple formulation of the problem, its analysis seems to be highly complicated. We also provide a list of open problems and discuss various methods one may want to use, and obstacles one encounters.

math.PR↗

arXiv2025

Random walks in Weyl chambers

We study a $d$-dimensional random walk with zero mean and finite variance in the Weyl chambers of type C and D. Under optimal moment assumptions we construct positive harmonic functions for random walks killed on exiting Weyl chambers. We also find the tail asymptotics for the exit time of the random walk from Weyl chambers.

math.PR↗

arXiv2025

Random walks with square-root boundaries: the case of exact boundaries $g(t)=c\sqrt{t+b}-a$

Let $S(n)$ be a real valued random walk with i.i.d. increments which have zero mean and finite variance. We are interested in the asymptotic properties of the stopping time $T(g):=\inf\{n\ge1: S(n)\le g(n)\}$, where $g(t)$ is a boundary function. In the present paper we deal with the parametric family of boundaries $\{g_{a,b}(t)=c\sqrt{t+b}-a, b\ge0, a>c\sqrt{b}\}$. First, assuming that sufficiently many moments of increments of the walk are finite, we construct a positive space-time harmonic function $W(a,b)$. Then we show that there exist $p(c)>0$ and a constant $\varkappa(c)$ such that $\mathbf{P}(T_{g_{a,b}}>n)\sim \varkappa(c)\frac{W(a,b)}{n^{p(c)/2}}$ as $n\to\infty$.

math.PR↗

arXiv2024

Asymptotic expansions for normal deviations of random walks conditioned to stay positive

We consider a one-dimensional random walk $S_n$ having i.i.d. increments with zero mean and finite variance. We continue our study of asymptotic expansions for local probabilities $\mathbf P(S_n=x,\tau_0>n)$, which has been started in \cite{DTW23}. Obtained there expansions make sense in the zone $x=o(\frac{\sqrt{n}}{\log^{1/2} n})$ only. In the present paper we derive an alternative expansion, which deals with $x$ of order $\sqrt{n}$.

math.PR↗

arXiv2024

Berry-Esseen inequality for random walks conditioned to stay positive

We consider random walks conditioned to stay positive. When the mean of increments is zero and variance is finite it is known that they converge to the Rayleigh distribution. In the present paper we derive a Berry-Esseen type estimate and show that the rate of convergence is of order $n^{-1/2}$.

math.PR↗

arXiv2024

Ordered random walks and the Airy line ensemble

The Airy line ensemble is a random collection of continuous ordered paths that plays an important role within random matrix theory and the Kardar-Parisi-Zhang universality class. The aim of this paper is to prove a universality property of the Airy line ensemble. We study growing numbers of i.i.d. continuous-time random walks which are then conditioned to stay in the same order for all time using a Doob h-transform. We consider a general class of increment distributions; a sufficient condition is the existence of an exponential moment and a log-concave density. We prove that the top particles in this system converge in an edge scaling limit to the Airy line ensemble in a regime where the number of random walks is required to grow slower than a certain power (with a non-optimal exponent 3/50) of the expected number of random walk steps. Furthermore, in a similar regime we prove that the law of large numbers and fluctuations of linear statistics agree with non-intersecting Brownian motions.

math.PR↗

arXiv2024

Stable random walks in cones

In this paper we consider a multidimensional random walk killed on leaving a right circular cone with a distribution of increments belonging to the normal domain of attraction of an $\alpha$-stable and rotationally-invariant law with $\alpha \in (0,2)\setminus \{1\}$. Based on Bogdan et al. (2018) describing the tail behaviour of the exit time of $\alpha$-stable process from a cone and using some properties of Martin kernel of the isotropic $\alpha$-stable process, in this paper we construct a positive harmonic function of the discrete time random walk under consideration. Then we find the asymptotic tail of the distribution of the exit time of this random walk from the cone. We also prove the corresponding conditional functional limit theorem.

math.PR↗

arXiv2024

Expansions for random walks conditioned to stay positive

We consider a one-dimensional random walk $S_n$ with i.i.d. increments with zero mean and finite variance. We study the asymptotic expansion for the tail distribution $\mathbf P(\tau_x>n)$ of the first passage times $\tau_x:=\inf\{n\ge1:x+S_n\le0\}$ for $\ x\ge0.$ We also derive asymptotic expansion for local probabilities $\mathbf P(S_n=x,\tau_0>n)$. Studying the asymptotic expansions we obtain a sequence of discrete polyharmonic functions and obtain analogues of renewal theorem for them.

math.PR↗

arXiv2023

Probabilistic approach to risk processes with level-dependent premium rate

We study risk processes with level dependent premium rate. Assuming that the premium rate converges, as the risk reserve increases, to the critical value in the net-profit condition, we obtain upper and lower bounds for the ruin probability. In contrast to existing in the literature results, our approach is purely probabilistic and based on the analysis of Markov chains with asymptotically zero drift.

math.PR↗

arXiv2023

Markov chains in the domain of attraction of Brownian motion in cones

We consider a multidimensional Markov Chain $X$ converging to a multidimensional Brownian Motion. We construct a positive harmonic function for $X$ killed on exiting the cone. We show that its asymptotic behavior is similar to that of to the harmonic function of Brownian motion. We use the harmonic function to study the asymptotic behaviour of the tail distribution of the exit time $\tau$ of $X$ from a cone.

math.PR↗

arXiv2022

Harmonic measure in a multidimensional gambler's problem

We consider a random walk in a truncated cone $K_N$, which is obtained by slicing cone $K$ by a hyperplane at a growing level of order $N$. We study the behaviour of the Green function in this truncated cone as $N$ increases. Using these results we also obtain the asymptotic behaviour of the harmonic measure. The obtained results are applied to a multidimensional gambler's problem studied by Diaconis and Ethier (2022). In particular we confirm their conjecture that the probability of eliminating players in a particular order has the same exact asymptotic behaviour as for the Brownian motion approximation. We also provide a rate of convergence of this probability towards this approximation.

math.PR↗

arXiv2022

Green function for an asymptotically stable random walk in a half space

We consider an asymptotically stable multidimensional random walk $S(n)=(S_1(n),\ldots, S_d(n) )$. Let $τ_x:=\min\{n>0: x_{1}+S_1(n)\le 0\}$ be the first time the random walk $S(n)$ leaves the upper half-space. We obtain the asymptotics of $p_n(x,y):= P(x+S(n) \in y+Δ, τ_x>n)$ as $n$ tends to infinity, where $Δ$ is a fixed cube. From that we obtain the local asymptotics for the Green function $G(x,y):=\sum_n p_n(x,y)$, as $|y|$ and/or $|x|$ tend to infinity.

math.PR↗

arXiv2022

Ordered exponential random walks

We study a $d$-dimensional random walk with exponentially distributed increments conditioned so that the components stay ordered (in the sense of Doob). We find explicitly a positive harmonic function $h$ for the killed process and then construct an ordered process using Doob's $h$-transform. Since these random walks are not nearest-neighbour, the harmonic function is not the Vandermonde determinant. The ordered process is related to the departure process of M/M/1 queues in tandem. We find asymptotics for the tail probabilities of the time until the components in exponential random walks become disordered and a local limit theorem. We find the distribution of the processes of smallest and largest particles as Fredholm determinants.

math.PR↗

arXiv2022

Persistence of autoregressive sequences with logarithmic tails

We consider autoregressive sequences $X_n=aX_{n-1}+ξ_n$ and $M_n=\max\{aM_{n-1},ξ_n\}$ with a constant $a\in(0,1)$ and with positive, independent and identically distributed innovations $\{ξ_k\}$. It is known that if $\mathbf P(ξ_1>x)\sim\frac{d}{\log x}$ with some $d\in(0,-\log a)$ then the chains $\{X_n\}$ and $\{M_n\}$ are null recurrent. We investigate the tail behaviour of recurrence times in this case of logarithmically decaying tails. More precisely, we show that the tails of recurrence times are regularly varying of index $-1-d/\log a$. We also prove limit theorems for $\{X_n\}$ and $\{M_n\}$ conditioned to stay over a fixed level $x_0$. Furthermore, we study tail asymptotics for recurrence times of $\{X_n\}$ and $\{M_n\}$ in the case when these chains are positive recurrent and the tail of $\logξ_1$ is subexponential.

math.PR↗

arXiv2021

Random walks in cones revisited

In this paper we continue our study of a multidimensional random walk with zero mean and finite variance killed on leaving a cone. We suggest a new approach that allows one to construct a positive harmonic function in Lipschitz cones under minimal moment conditions. This approach allows one also to obtain more accurate information about the behaviour of the harmonic function not far from the boundary of the cone. We also prove limit theorems under new moment conditions.

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