Weighted moments and sharp convergence rates for supercritical multitype branching processes

Abstract

Let $Z_n^i=(Z_n^i(1),\cdots,Z_n^i(d)), n\geqslant0$, be a supercritical $d$-type branching process starting from a single particle of type $i\in\{1,\cdots,d\}$, where $d$ is a positive integer. The Kesten-Stigum theorem states that if the mean matrix $M$ of the offspring distribution has spectral radius $ρ>1$ with the associated positive left and right eigenvectors $u$ and $v$, then for all $1\leqslant i \leqslant d$, the sequence $W_n^i=\langle Z_n^i,u\rangle/ρ^n u(i)$ forms a non-negative martingale, and hence converges almost surely to a random variable $W^i$ as $n\rightarrow\infty$. In this paper, we investigate the asymptotic properties of the martingale $\{W_n^i\}$, focusing on weighted moments and sharp rates of convergence. We establish necessary and sufficient conditions for the finiteness of $ϕ$-moments of $W^i$, where $ϕ(x)=x^α\ell(x)$ with $α\geqslant1$ and $\ell$ is a positive function slowly varying at infinity. We also determine the sharp exponential rate of convergence of $W_n^i$ to $W^i$ in the $ϕ$-moment sense, identify a phase transition at the critical rate $ρ^{1/2},$ and characterize the behavior at the critical value. Our proofs rely on the double martingale structure of the fundamental martingale and on convex inequalities for martingales.