Geodesic
Limit Theorems for Regenerative Excursion Processes
Serdica Mathematical Journal, Tome 25 (1999) no. 1, pp. 19-40.

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The regenerative excursion process Z(t), t = 0, 1, 2, . . . is constructed by two independent sequences X = {Xi , i ≥ 1} and Z = {Ti , (Zi (t), 0 ≤ t Ti ), i ≥ 1}. For the embedded alternating renewal process, with interarrival times Xi – the time for the installation and Ti – the time for the work, are proved some limit theorems for the spent worktime and the residual worktime, when at least one of the means of Xi and Ti is infinite. Limit theorems for the process Z(t) are proved, too. Finally, some applications to the branching processes with state-dependent immigration are given.
Keywords: Alternating Renewal Processes, Regenerative Processes, Limit Theorems, Branching Processes, State-Dependent Immigration 
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Mitov, Kosto. Limit Theorems for Regenerative Excursion Processes. Serdica Mathematical Journal, Tome 25 (1999) no. 1, pp. 19-40. http://geodesic.mathdoc.fr/item/SMJ2_1999_25_1_a3/