I know that for the infinite 1-D random walk, it is persistent (my professor showed us by showing that , so ), which makes me think that that the semi-infinite walk is also persistent, but I'm having a hard time showing it. The problem I was given is below, any help would be greatly appreciated. below is the probability of the first return (or visit) to state j starting in state i
A random walk is defined on the integers {0,1,2,3,...} with the following
transition probabilities:
,
,
Determine whether the walk is transient or persistent. [Hint: relate the walk
to the standard symmetric walk on the integers. The standard symmetric walk is persistent, thus . By conditioning on the first step of the walk starting at 0, use this to compute for the walk on the half-line given above].