On the absolute continuity of the limit random variable in the supercritical Galton-Watson branching process

Abstract

Let { Z n : n ≧ 0 } \{ {Z_n}:n \geqq 0\} be a simple Galton-Watson branching process with offspring distribution { p j } \{ {p_j}\} satisying 1 > ∑ j p j > ∞ 1 > \sum {j{p_j} > \infty } . It is known that there exist constants C n {C_n} such that W n ≡ Z n C n {W_n} \equiv {Z_n}{C_n} converges with probability one to a nondegenerate limit random variable W. Here we show that this W is always absolutely continuous on ( 0 , ∞ ) (0,\infty ) .