STA 290 Seminar Series
DATE: Thursday, February 9th 2017, 4:10pm
LOCATION: MSB 1147, Colloquium Room. Refreshments at 3:30pm in MSB 4110
SPEAKER: Jim Pitman, UC Berkeley
TITLE: “Hidden symmetries, limit laws and Bessel processes in the extreme order statistics of random walks”
ABSTRACT: Consider the order statistics generated generated by n steps of a random walk on the line with i.i.d. increments, shifted relative to the minimal value of the walk, so the shifted values are all non-negative. For each fixed K the joint distribution of the lowest K of these shifted order statistics converges in total variation norm as n tends to infinity. The limit law is that of the order statistics formed by merging values of two independent conditioned versions of the random walk, one conditioned to be strictly positive for all time, and the other conditioned to be non-negative for all time. These two conditioned walks can be constructed from the original random walk by an increment splitting construction due to Feller, and developed in various ways by Tanaka, Bertoin and others. Various limit theorems involving the gaps between extreme order statistics of random walks, found by Schehr and Majumbdar, can be understood and extended in terms of a limiting construction involving the square of a four-dimensional Bessel process, due to the Williams–Denisov path decomposition of Brownian motion a minimum time, and the Ray-Knight description of Brownian local time processes.