On the longest gap between power-rate arrivals

2017/03/28 by Asmussen, Søren, Ivanovs, Jevgenijs, Segers, Johan
#60G55 #60G70 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1703.09424

Abstract

Let L_t be the longest gap before time t in an inhomogeneous Poisson process with rate function λ_t proportional to t^α-1 for some α∈(0,1). It is shown that λ_tL_t-b_t has a limiting Gumbel distribution for suitable constants b_t and that the distance of this longest gap from t is asymptotically of the form (t/log t)E for an exponential random variable E. The analysis is performed via weak convergence of related point processes. Subject to a weak technical condition, the results are extended to include a slowly varying term in λ_t.

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