Local Limit Theorems for Large Deviations

Abstract

Let $(X_j ),j = 1,2, \cdots $, be a sequence of independent random variables with the distribution functions $V_j (x)$. We assume the existence of ${\bf D}X_j = \sigma _j^2 ,s_n^2 = \sum\nolimits_{j = 1}^n {\sigma _j^2 } ,{\bf E}X_j = 0,j = 1,2, \cdots $. We put \[ Z_n = \sum\limits_{j = 1}^n X_j /s_n . \] With the aid of the saddlepoint method of function theory several local limit theorems are derived, in complete analogy to the previously known integral limit theorems for large deviations of H. Cramér [1] and V. Petrov [5]. These authors considered the behavior of the function ${\bf P}\{ Z_n < x\} = F_n (x)$ for $n \to \infty $, where x together with n becomes infinite (“large deviations”). V. Petrov generalized Cramér’s theorem from the case of identically distributed $X_j $ to the general case and at the same time improved the remainder term and the growth of x. The present work shows that their method of proof, namely the introduction of a definite transformation of the distribution laws of the $X_j $, was very natural. The present work makes consistent use of the function theoretic possibilities that are given by the assumption that the functions \[ M_j (z) = {\bf E}_{e^{zX_j } } = \int_{ - \infty }^\infty {e^{zy} } dv_j (y) \] are analytic in a strip $| {\operatorname{Re} z} | < A$. Theorem 1.Let conditions A—C be fulfilled. Then for sufficiently largeneach$Z_n $possesses a distribution density$p_{z_n } (x)$. Assume further that$x > 1 $and$x = o(\sqrt n )$for$n \to \infty $. Then one has\[ \frac{{p_{z_n } (x)}}{{\varphi (x)}} = e^{(x/\sqrt n )\lambda _n (x/\sqrt n )} \left[ {1 + O\left( {\frac{x}{{\sqrt n }}} \right)} \right], \]where$\lambda _n (t)$is a power series converging, uniformly inn, for sufficiently small values$| t |$, and$\varphi (x)$is the density o f the normal distribution. For negative x there is a similar relation. For identically distributed $X_j $ the condition C can be considerably weakened. In this case Theorem 2 holds. Also in the case of a lattice-like distribution of the random variables $X_j $ an analogous limit relation holds (Theorem 3).