A limit theorem for point processes by applications

Let 0 ≦ T 1 ≦ T 2 ≦ ·· · represent the epochs in time of occurrences of events of a point process N ( t ) with N ( t ) = sup{ k : T k ≦ t }, t ≧ 0. Besides certain mild conditions on the process N ( t ) (see Conditions (A 1 )– (A 3 ) in the text) we assume that for every k ≧ 1, as t →∞, the vector ( t – T N ( t ) , t – T N ( t )–1 , · ··, t – T N ( t )– k +1 ) converges in law to a k -dimensional distribution which coincides with that of a random vector ξ k = ( ξ 1 , · ··, ξ k ) necessarily satisfying P (0 ≦ ξ 1 ≦ ξ 2 ≦ ·· ·≦ ξ k ) = 1. Let R ( t ) be an arbitrary function defined for t ≧ 0, satisfying 0 ≦ R ( t ) ≦ 1, ∀ 0 ≦ t <∞, and certain mild conditions (see Conditions (B 1 )– (B 4 ) 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 of R ( 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 the GI / G /∞ queue apparently not studied before.