On the Asymptotic Behavior of Recurrent and "Almost Recurrent" Events

Citation

Lamperti, John Williams (1957) On the Asymptotic Behavior of Recurrent and "Almost Recurrent" Events. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/ZEZC-ZN21. https://resolver.caltech.edu/CaltechETD:etd-07142004-143902

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. In connection with a recurrent event E, random variables of interest include [...], the time since the last occurrence of E, [...], the total time during which E occurs, and [...], the time between occurrences. In this thesis theorems are given concerning the limiting distributions of these and other quantities as n [approaches infinity]. It is shown that the same distributions apply in certain cases where E is not a recurrent event; this suggests the concept of an "almost recurrent event" which is shown to have the same asymptotic behavior as an "associated" recurrent event. These results extend and correlate previous work of Feller, Dynkin, Spitzer, and Darling and Kac. Finally, an occupation time theorem is proved for processes whose states comprise two classes separated by the occurrence of a recurrent event.

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