Deviation inequalities for random walks

Isoperimetric inequalities for Random Walks

Large Deviation Results for Random Walks Conditioned to Stay Positive

LetX1, X2, ... denote independent, identically distributed random variables with common distribution F , and S the corresponding random walk with ρ := limn→∞ P (Sn > 0) and τ := inf{n ≥ 1 : Sn ≤ 0}. Under the assumption that X is in the domain of attraction of an α-stable law with αρ < 1, we find an estimate for P (Sn > x|τ > n) which holds uniformly as x/cn →∞, where cn denote the norming cons...

Deviation inequalities via coupling for stochastic processes and random fields

We present a new and simple approach to deviation inequalities for non-product measures, i.e., for dependent random variables. Our method is based on coupling. We illustrate our abstract results with chains with complete connections and Gibbsian random fields, both at high and low temperature.

Harnack inequalities and sub-Gaussian estimates for random walks

We show that a-parabolic Harnack inequality for random walks on graphs is equivalent, on one hand, to so called-Gaussian estimates 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. The latter condition can be replaced by a cert...

Convex hulls of random walks: Large-deviation properties.

We study the convex hull of the set of points visited by a two-dimensional random walker of T discrete time steps. Two natural observables that characterize the convex hull in two dimensions are its perimeter L and area A. While the mean perimeter 〈L〉 and the mean area 〈A〉 have been studied before, analytically and numerically, and exact results are known for large T (Brownian motion limit), li...