Tail estimates for sums of variables sampled by a random walk

2000

We analyze several random random walks on one-dimensional lat- tices using spectral analysis and probabilistic methods. Through our analysis, we develop insight into the pre-asymptotic convergence of Markov chains.

2013

This paper is devoted to establishing sharp bounds for deviation probabilities of partial sums Σn i=1f(Xi), where X = (Xn)n2N is a positive recurrent Markov chain and f is a real valued function defined on its state space. Combining the regenerative method to the Esscher transformation, these estimates are shown in particular to generalize probability inequalities proved in the i.i.d. case to the Markovian setting for (not necessarily uniformly) geometrically ergodic chains.

We obtained a ψ 1 estimate for the sum of Rademacher random variables under condition that they are dependent.

Revista Matemática Iberoamericana, 2000

This paper studies the on-and off-diagonal upper estimate and the two-sided transition probability estimate of random walks on weighted graphs.

Random Structures and Algorithms, 1996

Electronic Journal of Probability, 2001

This paper proves upper and lower off-diagonal, sub-Gaussian transition probabilities estimates for strongly recurrent random walks under sufficient and necessary conditions. Several equivalent conditions are given showing their particular role influence on the connection between the sub-Gaussian estimates, parabolic and elliptic Harnack inequality.

Stochastic Processes and their Applications, 2007

The paper presents two results. The first one provides separate conditions for the upper and lower estimate of the distribution of the exit time from balls of a random walk on a weighted graph. The main result of the paper is that the lower estimate follows from the elliptic Harnack inequality. The second result is an off-diagonal lower bound for the transition probability of the random walk.

Mathematische Annalen, 2002

We show that the β-parabolic Harnack inequality for random walks on graphs is equivalent, on one hand, to the sub-Gaussian estimate for the transition probability and, on the other hand, to the conjunction of the elliptic Harnack inequality, the doubling volume property, and the fact that the mean exit time in any ball of radius R is of the order R β . The latter condition can be replaced by a certain estimate of the resistance of annuli.

This paper explores the joint behaviour of the summands of a random walk when their mean value goes to infinity as its length increases. It is proved that all the summands must share the same value, which extends previous results in the context of large exceedances of finite sums of i.i.d. random variables. Some consequences are drawn pertaining to the local behaviour of a random walk conditioned on a large deviation constraint on its end value. It is shown that the sample paths exhibit local oblic segments with increasing size and slope as the length of the random walk increases.

2007

This paper is devoted to establishing sharp bounds for deviation probabilities of partial sums n i=1f(Xi), where X = (Xn)n is a positive recurrent Markov chain and f is a real valued function defined on its state space. Combining the regenerative method to the Esscher transformation, these estimates are shown in particular to gen- eralize probability inequalities proved in the