Rare events for stationary processes

• Published in: Stochastic processes and their applications Vol. 89; no. 1; pp. 141 - 173
• Main Authors: Baccelli, F., McDonald, D.R.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.09.2000; Elsevier Science; Elsevier
• Series: Stochastic Processes and their Applications
• Subjects: Exact sciences and technology; Mathematics; Probability and statistics; Probability theory and stochastic processes; Rare events; Rare events Stationary marked point processes; Sciences and techniques of general use; Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications); Stationary marked point processes; Stochastic processes; secondary 60K25; Rare events; Stationary marked point processes; Primary 60G55; Markov process; Exponential distribution; Asymptotic behavior; Renewal process; Joint distribution; Rare event; Marked process; Event generation; Convergence in law; Point process; Discrete time; Translation operator; Stationary process; Probability measure
• ISSN: 0304-4149; 1879-209X
• DOI: 10.1016/S0304-4149(00)00018-1

Keilson (1979, Markov Chain Models — Rarity and Exponentiality, Springer, New York) and Aldous (1989, Probability approximations via the Poisson Clumping Heuristic, Springer, New York) have given expressions for the asymptotics of the mean time until a rare event occurs. Here we extend these results beyond the Markovian setting using the theory for stationary point processes. We introduce two notions of asymptotic exponentiality in variance and asymptotic independence and we study their implications on the asymptotics of the mean value of this hitting time under various initial probability measures.