LARGE DEVIATIONS OF COMBINATORIAL DISTRIBUTIONS II. LOCAL LIMIT THEOREMS

We derive a general local limit theorem for probabilities of large deviations for a sequence of random variables by means of the saddlepoint method on Laplace-type integrals. This result is applicable to parameters in a number of combinatorial structures and the distribution of additive arithmetical functions. 1. Main result. This paper is a sequel to our paper [18] where we derived a general central limit theorem for probabilities of large deviations applicable to many classes of combinatorial structures and arithmetic functions. The ranges of large deviations treated here are usually referred to as "moderate deviations" in probability literature (cf., e.g., [23]); we follow the same terms as in Part I [18] to keep consistency. In this paper we consider corresponding local limit theorems. More precisely, given a sequence of integral random variables

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