We study branching processes in an i.i.d. random environment, where the
associated random walk is of the oscillating type. This class of processes
generalizes the classical notion of criticality. The main properties of such
branching processes are developed under a general assumption, known as
Spitzer's condition in fluctuation theory of random walks, and some additional
moment condition. We determine the exact asymptotic behavior of the survival
probability and prove conditional functional limit theorems for the generation
size process and the associated random walk. The results rely on a stimulating
interplay between branching process theory and fluctuation theory of random
walks.Comment: Published at http://dx.doi.org/10.1214/009117904000000928 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Classical results describe the asymptotic behaviour of a Galton–Watson branching process conditioned on non-extinction. We give new proofs of limit theorems in critical and subcritical cases. The proofs are based on the representation of conditioned Galton–Watson generation sizes as a sum of independent increments which is derived from the decomposition of the conditioned Galton–Watson family tree along the line of descent of the left-most particle.
In this paper we determine the asymptotic behavior of the survival probability of a critical branching process in a random environment. In the special case of independent identically distributed geometric offspring distributions, and the somewhat more general case of offspring distributions with linear fractional generating functions, Kozlov proved that, as n → ∞, the probability of nonextinction at generation n is proportional to n −1/2 . We establish Kozlov's asymptotic for general independent identically distributed offspring distributions.
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