Chapter 4 LOCAL LIMIT THEOREMS § 1. Formulation of the problem Suppose that the independent, identically distributed random variables X1 , X2 ,. . . . have a lattice distribution with interval h, so that the sum Zn = X1 + X2 + . . . + X„ takes values in the arithmetic progression {na + kh ; k = 0, ± 1, . . . } . | Chapter 4 LOCAL LIMIT THEOREMS 1. Formulation of the problem Suppose that the independent identically distributed random variables Xr X2 . have a lattice distribution with interval h so that the sum Zn Xl X2 . Xn takes values in the arithmetic progression na kh k Q 1 The distribution of Z is completely determined by the numbers Pn k P Z na kh . A local limit theorem is an asymptotic expression for P k as n- oo. If the distribution of the Xj belongs to the domain of attraction of a stable law with density g x the natural way to obtain an asymptotic expression is to associate with the stable law a discrete distribution on the lattice khn where hn h Bn and the Bn are the usual normalising constants assigning to khn the probability r k i hn PnW g x dx hng khn . . J k - i hn The theorems of 2 give conditions which ensure that PM-PM. Another sort of local limit theorem arises when the distribution of the Xj belonging to the domain of attraction of a stable law with density g x has a density p x . The problem then is to give asymptotic expressions for the density p x of the normalised sum Zn X X2 . Xn An Bn and in particular to give conditions under which pn x converges in some sej se to g x . These problems are examined in 3. . LOCAL LIMIT THEOREMS FOR LATTICE DISTRIBUTIONS 121 The first local limit theorem to emerge was that of de Moivre and Laplace. In the last fifteen years local limit problems have been studied by many authors notably Gnedenko whose work on the subject was motivated by the work of Khinchin 74 on the analytic foundations of statistical mechanics. 2. Local limit theorems for lattice distributions Let the independent random variables X1 X2 . Xn . have the same distribution concentrated on the arithmetic progression a kh and write Z Xt X2 . Xn P Zn an kh Pn k . Theorem . In order that for some choice of constants An Bn lim sup n- oo k h Bn 0 where g x is the density of some stable distribution G with exponent a. 0 a 2 it is necessary
