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.