On the expected time a branching process has K individuals alive

Abstract: Consider a homogeneous time-continuous branching process where individuals have constant birth rate $\delta$, and life length distribution $Q$ having mean $E(Q)=1$. Let $X(u)$ denote the number of individuals alive at time $u$, and assume that $X(0)=1$. Let $K$ be a positive integer and define $A_K:=\int_0^\infty 1_{{X(u)=K}}du$, the accumulated time that the branching process has exactly $K$ individuals alive. In this paper we prove that $E(A_K)=\delta^{K-1}/\left(k(1\vee\delta)^K\right)$, irrespective of the life length distribution $Q$, subject to the normalizing condition $E(Q)=1$.