Large deviations for supercritical multitype branching processes

Large deviation results are obtained for the normed limit of a supercrifical multi-type 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]less than or equal tox)=P(L[i]=0)+x(alpha)F*[i](x)+o(x(alpha)) as x-->0. If Z(k)(min)[i] grows exponentially in k (exponential minimum growth), then we show that -logP(L[i]less than or equal tox)=x(-beta/(1-beta))G*[i](x)+o(x(-beta/(1-beta))) as x-->0. If the maximum family size is bounded, then -logP(L[i]>x)=x(delta/(delta-1))H*[i](x)+o(x(delta/(delta-1))) as x-->infinity. Here alpha, beta and delta are constants obtained from combinations of the minimum, maximum and mean growth rates, and F*, G* and H* are multiplicatively periodic functions.