Local limit theorems for one class of distributions in probabilistic combinatorics

Abstract

Abstract Let a function f(z) be decomposed into a power series with nonnegative coefficients which converges in a circle of positive radius R. Let the distribution of the random variable ξn, n ∈ {1, 2, …}, be defined by the formula $$\begin{array}{} \displaystyle P\{\xi_n=N\}=\frac{\mathrm{coeff}_{z^n}\left(\frac{\left(f(z)\right)^N}{N!}\right)}{\mathrm{coeff}_{z^n}\left(\exp(f(z))\right)},\,N=0,1,\ldots \end{array} $$ for some ∣z∣ < R (if the denominator is positive). Examples of appearance of such distributions in probabilistic combinatorics are given. Local theorems on asymptotical normality for distributions of ξn are proved in two cases: a) if f(z) = (1 − z)−λ, λ = const ∈ (0, 1] for ∣z∣ < 1, and b) if all positive coefficients of expansion f (z) in a power series are equal to 1 and the set A of their numbers has the form $$\begin{array}{} \displaystyle A = \{m^r \, | \, m \in \mathbb{N} \}, \, \, r = \mathrm {const},\; r \in \{2,3,\ldots\}. \end{array} $$ A hypothetical general local limit normal theorem for random variables ξn is stated. Some examples of validity of the statement of this theorem are given.