A series concerning subcritical multitype branching processes

Abstract

Let ( Z n ) n ⩾ 0 be a p-type positively regular Galton-Watson process whose mean matrix has maximal eigenvalue ϱ < 1 and associated left eigenvector v > 0. Let f n ( s), s ϵ C, be the probability generating function of Z n , where C is the p-dimensional unit cube. It is known that ϱ − n v · (1 − f n ( s)) decreases to a limit γ(s) ⩾ 0 as n → ∞, and γ(s) > 0 on C − {1} iff E(∥Z 1∥ log∥Z 1∥ ¦ Z 0 = e α) < ∞, 1 ⩽ α ⩽ p , where e α denotes the basis p-dimensional vector whose αth coordinate is 1, and ∥ Z 1∥ = Z 11 + … + Z 1 p . We prove that if γ( s) > 0 on C − {1}, then the slightly stronger condition E(∥Z 1∥ ( log∥Z 1∥) 2¦ Z 0 = e α) < ∞, 1 ⩽ α ⩽ p , is necessary and sufficient for the convergence of the series ∑ n = 1 ∞ ( ϱ − n v · (1 − f n ( s)) − γ( s)).