Growth rates for pure birth Markov chains

A pure birth Markov chain is a continuous time Markov chain {Z(t):t≥0} with state space S≡{0,1,2,…} such that for each i≥0 the chain stays in state i for a random length of time that is exponentially distributed with mean λi−1 and then jumps to (i+1). Suppose b(⋅) is a function from (0,∞)→(0,∞) that is nondecreasing and ↑∞. This paper addresses the two questions: (1) Given {λi}i≥0 what is the growth rate of Z(t)? (2) Given b(⋅) does there exist {λi} such that Z(t) grows at rate b(t)?