Large deviations for supercritical multitype branching processes

Abstract

Large deviation results are obtained for the normed limit of a supercritical multitype branching process. Starting from a single individual of type i, let L[i] be the normed limit of the branching process, and let be the minimum possible population size at generation k. If is bounded in k (bounded minimum growth), then we show that P(L[i] ≤ x) = P(L[i] = 0) + x α F *[i](x) + o(x α ) as x → 0. If grows exponentially in k (exponential minimum growth), then we show that −log P(L[i] ≤ x) = x −β/(1−β) G*[i](x) + o (x −β/(1−β)) as x → 0. If the maximum family size is bounded, then −log P(L[i] > x) = x δ/(δ−1) H *[i](x) + o(x δ/(δ−1)) as x → ∞. Here α, β and δ are constants obtained from combinations of the minimum, maximum and mean growth rates, and F *, G * and H * are multiplicatively periodic functions.