On series of discrete random variables. I: Real trinomial distributions with fixed probabilities. (English) Zbl 0945.60011

Deshouillers, Jean-Marc; Freiman, Gregory A.; Moran, William

Deshouillers, Jean-Marc (ed.) et al., Structure theory of set addition. Paris: Société Mathématique de France, Astérisque. 258, 411-423 (1999).

Summary: This paper begins the study of the local limit behaviour of triangular arrays of independent random variables \((\zeta_{n,k})_{1 \leq k\leq n}\) where the law of \(\zeta_{n,k}\) depends on \(n\). We consider the case when \(\zeta_{n,1}\) takes three integral values \(0< a_1(n)< a_2(n)\) with respective probabilities \(p_0, p_1, p_2\) which do not depend on \(n\). We show three types of limit behaviours for the sequence of r.v. \(\eta_n=\zeta_{n,1}+\cdots +\zeta_{n,n} \), according as \(a_2(n)/\text{gcd}(a_1(n),a_2(n))\) tends to infinity slower, quicker or at the same speed as \(\sqrt{n}\).
For the entire collection see [Zbl 0919.00044].

MSC:

60F05	Central limit and other weak theorems
60E10	Characteristic functions; other transforms
11P55	Applications of the Hardy-Littlewood method

Keywords:

sums of discrete random variables; local limit theorems; characteristic function; number theoretic methods; circle method