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Abstract
Let {ξ1, ξ2,...} be a sequence of independent real-valued and possibly nonidentically distributed random variables. Suppose that η is a nonnegative, nondegenerate at 0 and integer-valued random variable, which is independent of {ξ1, ξ2,...}. In this paper, we consider conditions for {ξ1, ξ2,...} and η under which the distributions of the randomly stopped maxima and minima as well as randomly stopped maxima of sums and randomly stopped minima of sums belong to the class of exponential distributions.
Highlights
Let {ξ1, ξ2, . . .} be a sequence of independent random variables (r.v.s) with distribution functions (d.f.s) {Fξ1, Fξ2, . . .}, and let η be a counting random variable (c.r.v.), i.e. a nonnegative, nondegenerate at 0 and integer-valued r.v
We suppose that the r.v. η and the sequence {ξ1, ξ2, . . .} are independent
We denote by Fξ(η), Fξ(η), FSη, FS(η) and FS(η) the d.f.s of ξ(η), ξ(η), Sη, S(η) and S(η), respectively
Summary
Let {ξ1, ξ2, . . .} be a sequence of independent random variables (r.v.s) with distribution functions (d.f.s) {Fξ1 , Fξ2 , . . .}, and let η be a counting random variable (c.r.v.), i.e. a nonnegative, nondegenerate at 0 and integer-valued r.v. A d.f. from the class L(γ) can be stepped, i.e. there is such a function that describes a distribution of a discrete r.v. To be more precise, if γ > 0, we define F (x) = exp(−γ log ex ) for all x 0, where a denotes the integer part of a real number a. . .} consists of independent but not necessarily identically distributed r.v.s, the following generalization of Theorem 2 was obtained by Danilenko et al (see [13, Thm. 4]). Motivated by the presented assertions and the results of papers [3, 5, 13,14,15,16,17, 22], we continue to consider conditions under which the d.f.s Fξ(η) , Fξ(η) , FS(η) or FS(η) belong to either of the following classes: L, L(γ) with some γ 0, L∞ or L∞ +.
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