Martingale approach to subexponential asymptotics for random walks

Consider the random walk $S_n=\xi_1+...+\xi_n$ with independent and identically distributed increments and negative mean $\mathbf E\xi=-m<0$. Let $M=\sup_{0\le i} S_i$ be the supremum of the random walk. In this note we present derivation of asymptotics for $\mathbf P(M>x), x\to\infty$ for long-tailed distributions. This derivation is based on the martingale arguments and does not require any prior knowledge of the theory of long-tailed distributions. In addition the same approach allows to obtain asymptotics for $\mathbf P(M_\tau>x)$, where $M_\tau=\max_{0\le i<\tau}S_i$ and $\tau=\min\{n\ge 1: S_n\le 0 \}$.Comment: 9 page

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.Comment: Published at http://dx.doi.org/10.1214/13-AOP867 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

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 L\'{e}vy 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 R\'{e}nyi 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.Comment: 41 pages(some errors and misprints are corrected

Ordered random walks with heavy tails

This note continues paper of Denisov and Wachtel (2010), where we have constructed a $k$-dimensional random walk conditioned to stay in the Weyl chamber of type $A$. The construction was done under the assumption that the original random walk has $k-1$ moments. In this note we continue the study of killed random walks in the Weyl chamber, and assume that the tail of increments is regularly varying of index $\alpha<k-1$. It appears that the asymptotic behaviour of random walks is different in this case. We determine the asymptotic behaviour of the exit time, and, using thisinformation, construct a conditioned process which lives on a partial compactification of the Weyl chamber.Comment: 20 page

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^{\lambda Z_1}=\infty for every $\lambda>0$. 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

Limit theorems for a random directed slab graph

We consider a stochastic directed graph on the integers whereby a directed edge between $i$ and a larger integer $j$ exists with probability $p_{j-i}$ depending solely on the distance between the two integers. Under broad conditions, we identify a regenerative structure that enables us to prove limit theorems for the maximal path length in a long chunk of the graph. The model is an extension of a special case of graphs studied by Foss and Konstantopoulos, Markov Process and Related Fields, 9, 413-468. We then consider a similar type of graph but on the `slab' $\Z \times I$, where $I$ is a finite partially ordered set. We extend the techniques introduced in the in the first part of the paper to obtain a central limit theorem for the longest path. When $I$ is linearly ordered, the limiting distribution can be seen to be that of the largest eigenvalue of a $|I| \times |I|$ random matrix in the Gaussian unitary ensemble (GUE).Comment: 26 pages, 3 figure

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.Comment: 27 page