# Time inhomogeneity in longest gap and longest run problems

Søren Asmussen
(Department of Mathematics, Aarhus University)
ASE-event
Torsdag, 17 september, 2015, at 13:15-14:00, in Koll. D (1531-211)
Abstrakt:

Let D be the time the first gap of length $\ell$ starts in an inhomogeneous Poisson process with rate function $\mu(t)$. We give an integral test for D to be finite a.s., which in particular shows that the critical rate of increase of $\mu(t)$ is $\ell\log t$. Asymptotic properties of the tail $P(D>t)$ are studied and compared to the exponential decay in the homogeneous case.

The discrete time analogue of the setting is runs of length $\ell$ of ones in Bernoulli 0-1 sequence, the study of which is a classical topic in the i.i.d. case but for which time inhomogeneity seems little developed.

Kontaktperson: Søren Asmussen