Time inhomogeneity in longest gap and longest run problems

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• 作者：Sø ; ren Asmussen^a ; ^{asmus@math.au.dk} ; Jevgenijs Ivanovs^b ; ^{Jevgenijs.Ivanovs@unil.ch} ; Anders Rø ; nn Nielsen^c ; ^{arnielsen@math.ku.dk}
• 关键词：Bernoulli trials ; Heads runs ; Tail asymptotics ; Poisson point process ; Delay differential equation ; Computer reliability
• 刊名：Stochastic Processes and their Applications
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：127
• 期：2
• 页码：574-589
• 全文大小：283 K
• 卷排序：127

文摘

Consider an inhomogeneous Poisson process and let DD be the first of its epochs which is followed by a gap of size ℓ>0ℓ>0. We establish a criterion for D<∞D<∞ a.s., as well as for DD being long-tailed and short-tailed, and obtain logarithmic tail asymptotics in various cases. These results are translated into the discrete time framework of independent non-stationary Bernoulli trials where the analogue of DD is the waiting time for the first run of ones of length ℓℓ. A main motivation comes from computer reliability, where D+ℓD+ℓ represents the actual execution time of a program or transfer of a file of size ℓℓ in presence of failures (epochs of the process) which necessitate restart.