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 Z_k^min[i] be the minimum possible population size at generation k. If Z_k^min[i] is bounded in k (bounded minimum growth), then we show that P(L[i] ≤ x) = P(L[i] = 0) + x^αF^*[i](x) + o(x^α) as x → 0. If Z_k^min[i] 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.