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Denis Denisov

Publications and source records attributed to Denis Denisov.

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

math.PR↗
arXiv2021

Regret Analysis of a Markov Policy Gradient Algorithm for Multi-arm Bandits

We consider a policy gradient algorithm applied to a finite-arm bandit problem with Bernoulli rewards. We allow learning rates to depend on the current state of the algorithm, rather than use a deterministic time-decreasing learning rate. The state of the algorithm forms a Markov chain on the probability simplex. We apply Foster-Lyapunov techniques to analyse the stability of this Markov chain. We prove that if learning rates are well chosen then the policy gradient algorithm is a transient Markov chain and the state of the chain converges on the optimal arm with logarithmic or poly-logarithmic regret.

cs.LG↗
arXiv2020

First-passage times for random walks in the triangular array setting

In this paper we continue our study of exit times for random walks with independent but not necessarily identical distributed increments. Our paper "First-passage times for random walks with non-identically distributed increments" was devoted to the case when the random walk is constructed by a fixed sequence of independent random variables which satisfies the classical Lindeberg condition. Now we consider a more general situation when we have a triangular array of independent random variables. Our main assumption is that the entries of every row are uniformly bounded by a constant, which tends to zero as the number of the row increases.

math.PR↗
arXiv2020

Renewal Theory for Transient Markov Chains with Asymptotically Zero Drift

We solve the problem of asymptotic behaviour of the renewal measure (Green function) generated by a transient Lamperti's Markov chain $X_n$ in $\mathbf R$, that is, when the drift of the chain tends to zero at infinity. Under this setting, the average time spent by $X_n$ in the interval $(x,x+1]$ is roughly speaking the reciprocal of the drift and tends to infinity as $x$ grows. For the first time we present a general approach relying in a diffusion approximation to prove renewal theorems for Markov chains. We apply a martingale type technique and show that the asymptotic behaviour of the renewal measure heavily depends on the rate at which the drift vanishes. The two main cases are distinguished, either the drift of the chain decreases as $1/x$ or much slower than that, say as $1/x^α$ for some $α\in(0,1)$. The intuition behind how the renewal measure behaves in these two cases is totally different. While in the first case $X_n^2/n$ converges weakly to a $Γ$-distribution and there is no law of large numbers available, in the second case a strong law of large numbers holds true for $X_n^{1+α}/n$ and further normal approximation is available.

math.PR↗
arXiv2019

Maximum on a random time interval of a random walk with infinite mean

Let $ξ_1,ξ_2,\ldots$ be independent, identically distributed random variables with infinite mean $\mathbf E[|ξ_1|]=\infty.$ Consider a random walk $S_n=ξ_1+\cdots+ξ_n$, a stopping time $τ=\min\{n\ge 1: S_n\le 0\}$ and let $M_τ=\max_{0\le i\le τ} S_i$. We study the asymptotics for $\mathbf P(M_τ>x),$ as $x\to\infty$.

math.PR↗
arXiv2019

Alternative constructions of a harmonic function for a random walk in a cone

For a random walk killed at leaving a cone we suggest two new constructions of a positive harmonic function. These constructions allow one to remove a quite strong extendability assumption, which has been imposed in our previous paper (Denisov and Wachtel, 2015, Random walks in cones). As a consequence, all the limit results from that paper remain true for cones which are either convex or star-like and $C^2$.

math.PR↗
arXiv2018

Heavy traffic and heavy tails for the maximum of a random walk

Consider a family of random walks $S_n^{(a)}=X_1^{(a)}+\cdots+X_n^{(a)}$ with negative drift $\mathbf E X_1^{(a)}=-a<0$ and finite variance $\mbox{var}(X_1^{(a)})=σ^2<\infty$.Let $M^{(a)}=\max_{n\ge 0} S_n^{(a)}$ be the maximums of the random walks. The exponential asymptotics $\mathbf P(aM^{(a)}>x)\sim e^{-2x/σ^2}$, as $a\to 0$, were found by Kingman and are known as heavy traffic approximation in the queueing theory. For subexponential random variables the large deviation asymptotics for $\mathbf P(M^{(a)}>x)\sim \frac{1}{a}\overline F^I(x)$ hold for fixed $a$ as $x\to\infty$. In this paper we present asymptotics for $\mathbf P(M^{(a)}>x)$, which hold uniformly on the whole positive axis, as $a\to 0$. Thus, these uniform asymptotics include both the regime of normal and large deviations. We identify the regions where exponential or subexponential asymptotics hold. Our approach is based on construction of corresponding super/sub - martingales to obtain sharp upper and lower bounds.

math.PR↗
arXiv2018

First-passage times over moving boundaries for asymptotically stable walks

Let $\{S_n, n\geq1\}$ be a random walk wih independent and identically distributed increments and let $\{g_n,n\geq1\}$ be a sequence of real numbers. Let $T_g$ denote the first time when $S_n$ leaves $(g_n,\infty)$. Assume that the random walk is oscillating and asymptotically stable, that is, there exists a sequence $\{c_n,n\geq1\}$ such that $S_n/c_n$ converges to a stable law. In this paper we determine the tail behaviour of $T_g$ for all oscillating asymptotically stable walks and all boundary sequences satisfying $g_n=o(c_n)$. Furthermore, we prove that the rescaled random walk conditioned to stay above the boundary up to time $n$ converges, as $n\to\infty$, towards the stable meander.

math.PR↗
arXiv2016

First-passage times for random walks with non-identically distributed increments

We consider random walks with independent but not necessarily identical distributed increments. Assuming that the increments satisfy the well-known Lindeberg condition, we investigate the asymptotic behaviour of first-passage times over moving boundaries. Furthermore, we prove that a properly rescaled random walk conditioned to stay above the boundary up to time $n$ converges, as $n\to\infty$, towards the Brownian meander.

math.PR↗
arXiv2015

Random walks in cones

We study the asymptotic behavior of a multidimensional random walk in a general cone. We find the tail asymptotics for the exit time and prove integral and local limit theorems for a random walk conditioned to stay in a cone. The main step in the proof consists in constructing a positive harmonic function for our random walk under minimal moment restrictions on the increments. For the proof of tail asymptotics and integral limit theorems, we use a strong approximation of random walks by Brownian motion. For the proof of local limit theorems, we suggest a rather simple approach, which combines integral theorems for random walks in cones with classical local theorems for unrestricted random walks. We also discuss some possible applications of our results to ordered random walks and lattice path enumeration.

math.PR↗
arXiv2015

Limit Theorems for Multifractal Products of Geometric Stationary Processes

We investigate the properties of multifractal products of geometric Gaussian processes with possible long-range dependence and geometric Ornstein-Uhlenbeck processes driven by Lévy motion and their finite and infinite superpositions. We present the general conditions for the $L_q$ convergence of cumulative processes to the limiting processes and investigate their $q$-th order moments and Rényi functions, which are nonlinear, hence displaying the multifractality of the processes as constructed. We also establish the corresponding scenarios for the limiting processes, such as log-normal, log-gamma, log-tempered stable or log-normal tempered stable scenarios.

math.PR↗
arXiv2014

Universality of local times of killed and reflected random walks

In this note we first consider local times of random walks killed at leaving positive half-axis. We prove that the distribution of the properly rescaled local time at point $N$ conditioned on being positive converges towards an exponential distribution. The proof is based on known results for conditioned random walks, which allow to determine the asymptotic behaviour of moments of local times. Using this information we also show that the field of local times of a reflected random walk converges in the sense of finite dimensional distributions. This is in the spirit of the seminal result by Knight(1963) who has shown that for the symmetric simple random walk local times converge weakly towards a squared Bessel process. Our result can be seen as an extension of the second Ray-Knight theorem to all asymptotically stable random walks.

math.PR↗
arXiv2013

Harmonic functions and stationary distributions for asymptotically homogeneous transition kernels on $Z^+$

We suggest a method for constructing positive harmonic functions for a wide class of transition kernels on $Z^+$. We also find natural conditions under which these functions have positive finite limits at infinity. Further, we apply our results on harmonic functions to asymptotically homogeneous Markov chains on $Z^+$ with asymptotically negative drift. More precisely, assuming that Markov chain satisfy Cramér's condition, we study the tail asymptotics of the stationary distribution. In particular, we clarify the influence of the rate of convergence of jumps of the chain towards the limiting distribution.

math.PR↗
arXiv2013

Tail asymptotics for the supercritical Galton-Watson process in the heavy-tailed case

As well known, for a supercritical Galton-Watson process $Z_n$ whose offspring distribution has mean $m>1$, the ratio $W_n:=Z_n/m^n$ has a.s. limit, say $W$. We study tail behaviour of the distributions of $W_n$ and $W$ in the case where $Z_1$ has heavy-tailed distribution, that is, $\E e^{λZ_1}=\infty$ for every $λ>0$. We show how different types of distributions of $Z_1$ lead to different asymptotic behaviour of the tail of $W_n$ and $W$. We describe the most likely way how large values of the process occur.

math.PR↗
arXiv2012

Tail behaviour of stationary distribution for Markov chains with asymptotically zero drift

We consider a Markov chain on $R^+$ with asymptotically zero drift and finite second moments of jumps which is positive recurrent. A power-like asymptotic behaviour of the invariant tail distribution is proven; such a heavy-tailed invariant measure happens even if the jumps of the chain are bounded. Our analysis is based on test functions technique and on construction of a harmonic function.

math.PR↗
arXiv2012

Exit times for integrated random walks

We consider a centered random walk with finite variance and investigate the asymptotic behaviour of the probability that the area under this walk remains positive up to a large time $n$. Assuming that the moment of order $2+δ$ is finite, we show that the exact asymptotics for this probability are $n^{-1/4}$. To show these asymptotics we develop a discrete potential theory for the integrated random walk.

math.PR↗