Prelimit and limit generalizations of the Pollaczek–Khinchin formula
Анотація
The moment generating function of the nondegenerate distribution of the maximum ξ + = sup 0≤t<∞ ξ(t) of a compound Poisson processwhere ν(t) is a simple Poisson process with intensity λ > 0, is determined via the well-known Pollaczek-Khinchin formula if m = Eξ(1) < 0. We obtain a prelimit generalization of this formula that determines the Laplace-Carson transform of the moment generating function of the maximum ξ + (t) = sup 0≤t ≤t ξ(t ), 0 < t < ∞, and the moment generating function of ξ + = ξ + (∞) under the assumption that m < 0 for homogeneous processes ξ(t) with independent increments and of bounded variation.Relationships of a different type between characteristic functions of ξ + (θ s ) (P{θ s > t} = e -st , s, t > 0) and of ξ + are also obtained by using earlier results presented by the author.This result is often used in queueing theory and in risk theory (see, for example, [1]-[3]).
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