Convergence Rates for a Supercritical Branching Process in a Random Environment

Let (Z(n)) be a supercritical branching process in a stationary and ergodic random environment xi.We study the convergence rates of the martingale W-n = Zn/E[Zn vertical bar xi] to its limit W. The following results about the convergence almost sure (a.s.), in law or in probability, are shown. (1) Under a moment condition of order p is an element of (1, 2), W - W-n = o(e(n-a)) a.s. for some alpha > 0 that we find explicitly; assuming only EW1 log W-1(alpha+1) < oo for some alpha > 0, we have W W = o(n) a.s.; similar conclusions hold for a branching process in a varying environment. (2) Under a second moment condition, there are norming constants a(e) (that we calculate explicitly) such that W-n = Zn/E[Zn vertical bar xi] converges in law to a non-degenerate distribution. (3) For a branching process in a finite state random environment, if W1 has a finite exponential moment, then so does W, and the decay rate of W-n = Zn/E[Zn vertical bar xi] is supergeometric.