In the problem of Example 4.3-1, assume that from state i the process moves to state i + 1 with probability p_i = λ/1+i, and to state 0 with probability q_i = 1− p_i, where a number λ ≤ 1, and i = 0,1,2, ... .

(a) Proceeding from (4.3.3), find the limiting distribution π.

(b) Let T₀ be the return time to state 0 (not counting the initial stay at 0). Find the distribution of T₀ and its mean value.

Example 4.3-1

We revisit Example 1.2-3 concerning success runs with the transition matrix (1.2.4). It is natural to assume q in (1.2.4) to be positive, so Doeblin’s condition holds (why?). In the system of equations π = πP, the first equation may be written as

and for the other equations,