Chapter 8 CRAMER'S INTEGRAL THEOREM AND ITS REFINEMENT BY PETROV 1 . Statement of the theorem The first general result in the theory of large deviations was the integral theorem of Cramer [19], published in 1938, which has considerable computational and analytical usefulness . | Chapter 8 CRAMER S INTEGRAL THEOREM AND ITS REFINEMENT BY PETROV 1. Statement of the theorem The first general result in the theory of large deviations was the integral theorem of Cramer 19 published in 1938 which has considerable computational and analytical usefulness. It was refined and generalised by Petrov 133 in 1954. In this chapter we discuss the work of Petrov keeping for simplicity to the case of identically distributed variables Xj. It should be remarked that the most natural method for proving integral theorems under Cramer s condition is the method of steepest descents whose use for local theorems was described in the last chapter. In the case in which the distributions of the Xj are different such an approach encounters however considerable difficulty. Let the Xj satisfy and Cramer s condition and let 2 z denote Cramer s series . Write 0 x 27i -i f e- 2di 00 Z X1 X2 . X mi V y P X y . Then Petrov s refinement of Cramer s theorem in the identically distributed case has the following form. Theorem . For x l x o ni we have P Zn -x - X3 . x exp k r yr l o 4 172 CRAMER S INTEGRAL THEOREM ITS REFINEMENT BY PETROV Chap. 8 2. The introduction of auxiliary random variables Since E expa A 00 we may write for z a R R h i e dT y . J 00 Let Xj be independent random variables with the distribution function F x A-1 i ehy6V y J co and write W x P X1 . X x JF x P X1 . X x . Then m E Xj R 1 i xefadK x J oo R h d z . 4 dAIOg and R iR 2 d2 logJ . Write Fn x P XY . Xn arfix Fn x P Xr . Xn inn a x . We prove by induction on n the fundamental relation 1T x .R T e hydWn y . J 00 When n l this follows trivially from . Suppose it is true for a particular value of n. Then . INTRODUCTION OF AUXILIARY RANDOM VARIABLES 173 r co h i x r x-z dw z J 00 K 1i 0 R x V x z e hzdWn z J CO Rn i i dW z e hz V dF . J - 00 co Making the substitution tj z becomes Rn X dWn z e hz P e-h -z d 7 i-z J QO J X n l i e-Md
