Deterministic time intervals on which a class of persistent processes are away from their origins

K Bruce Erickson

doi:10.1214/16-ecp4688

Abstract

There are three results each concerning large but remote deterministic time intervals at which excursions of a process away from the origin must occur. The first result gives a sufficient condition for a persistent random walk with a finite fourth moment. In this instance the aforementioned time intervals include an additional requirement that the walk is far away from the origin. The second result gives a necessary and a sufficient condition for similar excursions in the case of Brownian motion. The third result gives a necessary and a sufficient condition for time intervals to be free of the zeros of a class of persistent natural scale linear diffusions on the line and is equivalent to the determination of recurrent sets at infinity of the inverse local time.

Highlights

There are three results each concerning large but remote deterministic time intervals at which excursions of a process away from the origin must occur
The first result gives a sufficient condition for a persistent random walk with a finite fourth moment
In this instance the aforementioned time intervals include an additional requirement that the walk is far away from the origin

Summary

Times away from the origin

They show that under mild growth conditions on a deterministic sequence of times, {2ni}, a simple random walk (unit step with probability 1/2 in either direction) returns to its starting position at infinitely many of these times, w.p.1, if and only if n−i 1/2 diverges. According to the Skorokhod Embeding Theorem, [2], on some probability space (Ω, A, P ) a standard Brownian motion (B(t), t ≥ 0) and an increasing sequence of ECP 21 (2016), paper 62. Given the position y of the path at time a ( = time a − s = a − a − r for B), the probability of a hit of (−c , c ) during [a , a + b ] is the same as an unconditional hit during [0, b ] for a brand new Brownian starting at place y.

Note that

Kc c