Abstract
In this paper, we consider the sum Snξ = ξ1 + ... + ξn of possibly dependent and nonidentically distributed real-valued random variables ξ1, ... , ξn with consistently varying distributions. By assuming that collection {ξ1, ... , ξn} follows the dependence structure, similar to the asymptotic independence, we obtain the asymptotic relations for E((Snξ)α1(Snξ > x)) and E((Snξ – x)+)α, where α is an arbitrary nonnegative real number. The obtained results have applications in various fields of applied probability, including risk theory and random walks.
Highlights
Let n ∈ N := {1, 2, . . . } and let {ξ1, . . . , ξn} be a collection of possibly dependent real-valued random variables (r.v.s) with heavy-tailed distributions
Throughout the paper, we assume that random summands have consistently varying distributions
Real-valued random variables ξ1, . . . , ξn with distributions supported on R are called pairwise quasi-asymptotically independent if for all pairs of indices k, l ∈ {1, 2, . . . , n}, k = l, it holds that lim x→∞
Summary
Let n ∈ N := {1, 2, . . . } and let {ξ1, . . . , ξn} be a collection of possibly dependent real-valued random variables (r.v.s) with heavy-tailed distributions. Ξn} be a collection of possibly dependent real-valued random variables (r.v.s) with heavy-tailed distributions. Throughout the paper, we assume that random summands have consistently varying distributions. This is a subclass of heavy-tailed distributions. The following two indices are important to the determination whether d.f. F belongs to the aforementioned heavy-tailed distribution classes. The definitions of the aforementioned heavy-tailed distribution classes imply that. Ξn with distributions supported on R are called pairwise quasi-asymptotically independent (pQAI) if for all pairs of indices k, l ∈ {1, 2, . The statement provides the asymptotic results for tail probability of sums of pQAI r.v.s having distributions from class C.
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