Large Deviation Probabilities for Positive Random Variables

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...

Probabilities and Random Variables

The purpose of probability theory is to model random experiments so that we can draw inferences about them. The fundamental mathematical object is a triple (Ω,F , P ) called the probability space. A probability space is needed for each experiment or collection of experiments that we wish to describe mathematically. The ingredients of a probability space are a sample space Ω, a collection F of e...

ON THE LAWS OF LARGE NUMBERS FOR DEPENDENT RANDOM VARIABLES

In this paper, we extend and generalize some recent results on the strong laws of large numbers (SLLN) for pairwise independent random variables [3]. No assumption is made concerning the existence of independence among the random variables (henceforth r.v.’s). Also Chandra’s result on Cesàro uniformly integrable r.v.’s is extended.

Subgaussian random variables, Hoeffding’s inequality, and Cramér’s large deviation theorem

Large-deviation Probabilities for Maxima of Sums of Independent Random Variables with Negative Mean and Subexponential Distribution∗

Abstract. We consider the sums Sn = ξ1 + · · · + ξn of independent identically distributed random variables with negative mean value. In the case of subexponential distribution of the summands, the asymptotic behavior is found for the probability of the event that the maximum of sums max(S1, . . . , Sn) exceeds high level x. The asymptotics obtained describe this tail probability uniformly with...