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Abstract. Let (Z n ) be a supercritical branching process in a random environment ξ = (ξ n ). We establish a Berry-Esseen bound and a Cramér's type large deviation expansion for log Z n under the annealed law P. We also improve some earlier results about the harmonic moments of the limit variable W = lim n→∞ W n , where W n = Z n /E ξ Z n is the normalized population size. Introduction and main resultsA branching process in a random environment (BPRE) is a natural and important generalisation of the Galton-Watson process, where the reproduction law varies according to a random environment indexed by time. It was introduced for the first time in Smith and Wilkinson [24] to modelize the growth of a population submitted to an environment. For background concepts and basic results concerning a BPRE we refer to Athreya and Karlin [4,3]. In the critical and subcritical regime the process goes out and the research interest is concentrated mostly on the survival probability and conditional limit theorems for the branching process, see e.g. [21]. In this article, we complete on these results by giving the Berry-Esseen bound and asymptotics of large deviations of Cramér's type for a supercritical BPRE.A BPRE can be described as follows. The random environment is represented by a sequence ξ = (ξ 0 , ξ 1 , ...) of independent and identically distributed random variables (i.i.d. r.v.'s); each realization of ξ n corresponds to a probability law {p i (ξ n ) : i ∈ N} on N = {0, 1, 2, . . . }, whose probability generating function is

We give an extension of Hoeffding's inequality to the case of supermartingales with differences bounded from above. Our inequality strengthens or extends the inequalities of Freedman, Bernstein, Prohorov, Bennett and Nagaev.

We establish that a non-Gaussian nonparametric regression model is asymptotically equivalent to a regression model with Gaussian noise. The approximation is in the sense of Le Cam's de®ciency distance D; the models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. Our result concerns a sequence of independent but not identically distributed observations with each distribution in the same real-indexed exponential family. The canonical parameter is a value f t i of a regression function f at a grid point t i (nonparametric GLM). When f is in a HoÈ lder ball with exponent b b 1 2 Y we establish global asymptotic equivalence to observations of a signal Cf t in Gaussian white noise, where C is related to a variance stabilizing transformation in the exponential family. The result is a regression analog of the recently established Gaussian approximation for the i.i.d. model. The proof is based on a functional version of the Hungarian construction for the partial sum process.

An expansion of large deviation probabilities for martingales is given, which extends the classical result due to Cramér to the case of martingale differences satisfying the conditional Bernstein condition. The upper bound of the range of validity and the remainder of our expansion is the same as in the Cramér result and therefore are optimal. Our result implies a moderate deviation principle for martingales.