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Large deviations for supercritical multitype branching processes
Published online by Cambridge University Press: 14 July 2016
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.
MSC classification
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- Research Papers
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- Copyright © Applied Probability Trust 2004
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