LIMIT THEOREMS FOR MULTITYPE CONTINUOUS TIME MARKOV BRANCHING PROCESSES .I. CASE OF AN EIGENVECTOR LINEAR FUNCTIONAL

Let X(t)=(X1(t), X2(t), ⋯, Xt(t)) be a k-type (2≦k<∞) continuous time, supercritical, nonsingular, positively regular Markov branching process. Let M(t)=((mij(t))) be the mean matrix where mij(t)=E(Xj(t)|Xr(0)=δir for r=1, 2, ⋯, k) and write M(t)=exp(At). Let ξ be an eigenvector of A corresponding to an eigenvalue λ. Assuming second moments this paper studies the limit behavior as t → ∞ of the stochastic process {Mathematical expression}. It is shown that i) if 2 Re λ>λ1, then ξ · X(t)e{-λt| converges a.s. and in mean square to a random variable. ii) if 2 Re λ≦λ1 then [ξ · X(t)] f(v · X(t)) converges in law to a normal distribution where f(x)=(√x)-1 if 2 Re λ<λ1 and f(x)=(√x log x)-1 if 2 Re λ=λ1, λ1 the largest real eigenvalue of A and v the corresponding right eigenvector. © 1969 Springer-Verlag.