A limit theorem for point processes by applications

Publisher Website
Google Scholar
Abstract
Let 0 ≦T1≦T2≦ ·· · represent the epochs in time of occurrences of events of a point processN(t) withN(t) = sup{k:Tk≦t},t≧ 0. Besides certain mild conditions on the processN(t) (see Conditions (A1)– (A3) in the text) we assume that for everyk≧ 1, ast→∞, the vector (t – TN(t),t – TN(t)–1, · ··,t–TN(t)–k+1) converges in law to ak-dimensional distribution which coincides with that of a random vectorξk= (ξ1, · ··,ξk) necessarily satisfyingP(0 ≦ξ1≦ξ2≦ ·· ·≦ξk) = 1. LetR(t) be an arbitrary function defined fort≧ 0, satisfying 0 ≦R(t) ≦ 1, ∀0≦t<∞, and certain mild conditions (see Conditions (B1)– (B4) in the text). Then among other results, it is shown that The paper also deals with conditions under which the limit (∗) will be positive. The results are applied to several point processes and to the situations where the role ofR(t) is taken over by an appropriate transform such as a probability generating function, where conditions are given under which the limit (∗) itself will be a transform of an honest distribution. Finally the results are applied to the study of certain characteristics of theGI/G/∞ queue apparently not studied before.