Probability Seminar presents "Geometric law for numbers of returns until a hazard"

Geometric law for numbers of returns until a hazard
Monday, November 11, 2019 - 3:15pm
Sequoia Hall Room 200
Yuri Kifer (Hebrew University)
Abstract / Description: 

For a ψ-mixing sequence of identically distributed random variables X0, X1, X2, . . . and pairs of shrinking disjoint sets VN , WN , N = 1, 2, . . . , we count the number NN of returns to VN by the sequence until its first arrival to WN (hazard time). Let µ be the distribution of X0. It turns out that if µ(VN ), µ(WN ) → 0 as N → ∞ with the same speed then NN tends in distribution to a geometric random variable. A somewhat different setup deals with a ψ− or φ-mixing stationary process with a countable state space A where for a fixed pair of sequences ξ, η ∈ AN we count the number Nξ,η(n, m) of i's for which (Xi, Xi+1, . . . , Xi+n−1) coincides with (ξ0, ξ1, . . . , ξn−1) until the first j for which (Xj , Xj+1, . . . , Xj+m−1) coincides with (η0, η1, . . . , ηn−1). It turns out that for almost all pairs ξ, η if ratios of probabilities of cylinder sets [ξ0, . . . , ξn−1] and [η0, . . . , ηm(n)−1] converges as n, m(n) → ∞, then Nξ,η(n, m(n)) tends in distribution to a geometric random variable. Motivations, connections, and several generalizations of these results will be discussed as well.