Branching processes in simple random walk

Abstract

Let N ( a ) N(a) be the number of overcrossings of height a a in a simple random walk. For p > 1 / 2 p > 1/2 , the process N ( 0 ) , N ( 1 ) , … N(0),N(1), \ldots is a branching process which eventually becomes extinct. For 1 / 2 > p , N ( 0 ) , N ( 1 ) , … 1/2 > p,N(0),N(1), \ldots is a stationary process which is a branching process with immigration.