1.3. Let {Zn, 11 E NO} be positive integer-valued r.v.'s such that Zo = 1 and for each n ~ 1, the conditional distribution of Zn given Zo, ... ,Zn-1 is that of Zn-1 independent r.v.'s with the common distribution {Pk. kENo}, where PI < 1 and 00 o

k=O Then {W n, n E NO}, where W n = Z n 7 mil, IS a martmgale that con verges, the limit being zero if m < 1. [This is from branching process.]