Local limit theory and large deviations for supercritical Branching processes
نویسندگان
چکیده
منابع مشابه
Local Limit Theory and Large Deviations for Supercritical Branching Processes
In this paper we study several aspects of the growth of a supercritical Galton–Watson process {Zn :n ≥ 1}, and bring out some criticality phenomena determined by the Schröder constant. We develop the local limit theory of Zn, that is, the behavior of P (Zn = vn) as vn ր ∞, and use this to study conditional large deviations of {YZn :n ≥ 1}, where Yn satisfies an LDP, particularly of {Z −1 n Zn+1...
متن کاملCorrections and Acknowledgment for “ Local Limit Theory and Large Deviations for Supercritical Branching Processes
Theorem 1 in [2] is incorrect in the case α ≥ 1. Our error stems from the fact that the lower bound C 1 was determined by an integral expression which we treated as positive, whereas in fact it was zero. This led to an incorrect normalization A n when α ≥ 1. This error was communicated to us by K. Fleischmann and V. Wachtel, and the correction, that A n = p n 1 v (α−1) n for all 0 < α < ∞, appe...
متن کاملLarge Deviations for Supercritical Multi-type Branching Processes
Large deviation results are obtained for the normed limit of a supercritical 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 [i] be the minimum possible population size at generation k. If Z k [i] is bounded in k (bounded minimum growth) then we show that P(L[i] ≤ x) = P(L[i] = 0) + xF ∗[i](x) + o(x) a...
متن کاملLarge deviations for Branching Processes in Random Environment
A branching process in random environment (Zn, n ∈ N) is a generalization of Galton Watson processes where at each generation the reproduction law is picked randomly. In this paper we give several results which belong to the class of large deviations. By contrast to the Galton-Watson case, here random environments and the branching process can conspire to achieve atypical events such as Zn ≤ e ...
متن کاملGenealogy for Supercritical Branching Processes
We study the genealogy of so-called immortal branching processes, i.e. branching processes where each individual upon death is replaced by at least one new individual, and conclude that their marginal distributions are compound geometric. The result also implies that the limiting distributions of properly scaled supercritical branching processes are compound geometric. We exemplify our results ...
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ژورنال
عنوان ژورنال: The Annals of Applied Probability
سال: 2004
ISSN: 1050-5164
DOI: 10.1214/105051604000000242