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

Branching processes (Z n ) n≥0 in varying environment generalize the Galton-Watson process, in that they allow time-dependence of the offspring distribution. Our main results concern general criteria for a.s. extinction, square-integrability of the martingale (Z n /E[Z n ]) n≥0 , properties of the martingale limit W and a Yaglom type result stating convergence to an exponential limit distribution of the suitably normalized population size Z n , conditioned on the event Z n > 0. The theorems generalize/unify diverse results from the literature and lead to a classification of the processes.

For a branching process in random environment it is assumed that the offspring distribution of the individuals varies in a random fashion, independently from one generation to the other. Interestingly there is the possibility that the process may at the same time be subcritical and, conditioned on nonextinction, 'supercritical'. This so-called weakly subcritical case is considered in this paper. We study the asymptotic survival probability and the size of the population conditioned on non-extinction.Also a functional limit theorem is proven, which makes the conditional supercriticality manifest. A main tool is a new type of functional limit theorem for conditional random walks.MSC 2000 subject classifications. Primary 60J80, Secondary 60G50, 60F17

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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.