Large deviations of combinatorial distributions. II: Local limit theorems
[For part I see ibid. 6, No. 1, 297-319 (1996; Zbl 0863.60013).] Let \(\{\Omega _n\}_{n\geq 1} \) be a sequence of random variables taking integral values and of maximal span 1 for \(n\geq n_0\); \(\mu_n={\mathbb{E}} \Omega_n\); \(\sigma_n^2=\text{ Var} \Omega_n\). The asymptotic formulas for \(\text{ Pr} \{\Omega_n=m\}\) for \(m\) lying in the interval \(\mu_n + O(\sigma_n^2)\) are obtained under conditions on the moment generating function of \(\Omega_n\). The result is derived by means of the saddlepoint method on Laplace-type integrals and may be applicable considering parameters of combinatorial structures, as well as the distribution of arithmetical functions.
scientific article; zbMATH DE number 1319848
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