The discounted local limit theorems for large deviations

Large deviations of combinatorial distributions II: Local limit theorems

This paper is a sequel to our paper [17] where we derived a general central limit theorem for probabilities of large deviations1 applicable to many classes of combinatorial structures and arithmetic functions; we consider corresponding local limit theorems in this paper. More precisely, given a sequence of integral random variables {Ωn}n≥1 each of maximal span 1 (see below for definition), we a...

Large Deviations of Combinatorial Distributions I: Central Limit Theorems

We prove a general central limit theorem for probabilities of large deviations for sequences of random variables satisfying certain natural analytic conditions. This theorem has wide applications to combinatorial structures and to the distribution of additive arithmetical functions. The method of proof is an extension of Kubilius’ version of Cramér’s classical method based on analytic moment ge...

Local limit theorem for large deviations and statistical box-tests

Abstract: Let n particles be independently allocated into N boxes, where the l-th box appears with the probability al. Let μr be the number of boxes with exactly r particles and μ = [μr1 , . . . , μrm ]. Asymptotical behavior of such random variables as N tends to infinity was studied by many authors. It was previously known that ifNal are all upper bounded and n/N is upper and lower bounded by...

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...

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