On limit theorems for the generalised allocation scheme

Abstract

For sums of independent identically distributed non-negative integer-valuedrandom variables we give a series of limit theorems which may be of use in probabilistic combinatorics in the framework of the generalised allocation scheme.

On transition of distributions of sums of independent identically distributed random variables from one lattice to another in the generalised allocation scheme

“…Some of limit theorems proved in this paper and in [6] can likely be obtained by checking the general conditions for validity of local limit theorems given in [12] (see [3]), but the authors believe that the presence of simple direct proofs may be of advantage in future studies.…”

“…In [6], the most simple cases were studied. First, b 0 and b 1 were assumed to be positive; second, the values of the parameter  did not approach the convergence radius of the series B.Â/.…”

“…In [6], the condition A 1 is assumed to be fulfilled. It appears that in the proof of the integral limit theorem on weak convergence of the sums S N to the standard normal law no arithmetic constraints need to be imposed on the sequence b 0 ; b 1 ; : : : Theorem 1.…”

Some limit theorems for generalized allocation schemes

“…Further properties of scheme (1.1) can be found in the papers of V. F. Kolchin and A. V. Kolchin (see e.g. [10,11]). In [5] a strong law of large numbers (SLLN) was proved for Model 1.…”

A theorem on probabilities of large deviations for decomposable statistics which do not satisfy the Cramér condition

“…This is equivalent to the fact that the random variables are distributed by the same parametric law …. /, each with its own parameter  [3].…”