Large deviations for random walks with nonidentically distributed jumps having infinite variance

Abstract

Let ξ 1 , ξ 2 , . . . be independent random variables with distributions F 1 , F 2 , . . . in a triangular scheme (F i may depend on some parameter), Eξ i = 0, and put S n = n i=1 ξ i , S n = max k≤n S k . Assuming that some regularly varying functions majorize and minorize F = 1 n n i=1 F i , we find upper and lower bounds for the probabilities P(S n > x) and P(S n > x). These bounds are precise enough to yield asymptotics. We also study the asymptotics of the probability that a trajectory {S k } crosses the remote boundary {g(k)}; i.e., the asymptotics of P max k≤n (S k − g(k)) > 0 . The case n = ∞ is not excluded. We also estimate the distribution of the first crossing time.

Asymptotic Analysis for Random Walks with Nonidentically Distributed Jumps Having Finite Variance

“…where α 0 and α 0 are independent of n. If α 0 = α 0 = α then the "individual" version of conditions [ < ] U which was stated above implies the "averaged" version of (1.1) and (1.2) (see [13]). If α j are distinct then the fulfillment of the averaged conditions [ < ] U becomes unclear.…”

“…(compare with [13]). Without loss of generality the functions T δ in [U 1 ] and [U 2 ] can be assumed identical.…”

“…The first version of conditions. In this subsection the basic conditions will be imposed as in [13] on the averaged distribution F defined in (0.1). The first of these conditions may be regarded as the condition of existence of uniformly regular varying majorants for the tails of the averaged distribution F .…”

“…Alongside conditions [U] on the averaged majorant, as in [13] we may consider the alternative conditions under which each majorant V 1 , . .…”

“…If α j are distinct then the fulfillment of the averaged conditions [ < ] U becomes unclear. Motivated by this observation, as in [13] we may consider the two versions of conditions [ < ] U : (i)averaged and (ii) individual. In contrast to [13] here we do not introduce in the main theorem conditions [J] which state that the tails do not decay very fast as n → ∞; this condition will be introduced later.…”