TAILIEUCHUNG - Independent And Stationary Sequences Of Random Variables - Chapter 6

Chapter 6 LIMIT THEOREMS FOR LARGE DEVIATIONS § 1 . Introduction and examples In this and succeeding chapters we shall examine the simplest problems in the theory of large deviations . Let X1 , X2 ,. . . be independent, identically distributed random variables, with E(X1) = 0 | Chapter 6 LIMIT THEOREMS FOR LARGE DEVIATIONS 1. Introduction and examples In this and succeeding chapters we shall examine the simplest problems in the theory of large deviations. Let X2 . be independent identically distributed random variables with E y 0 V X a2 and let Z denote the normalised sum Z X1 X2 . X 7n Then for any x0 P Zn x - 27t T e i 2dt- 0 J oo as m oo uniformly in x x0. If the Xj have a probability density p x then the results of show that under weak conditions the density pn x of Z satisfies p x - 2ir e 0 as x oo uniformly in x x0. In many problems encountered in such different branches of science as mathematical statistics 18 24 information theory 185 the statistical physics of polymers 181 and even the analytic arithmetic of the hypercomplex numbers 103 more precise information about the distribution of Z is required than is contained in the classical theorems. In particular such problems require the estimation of P Z x . INTRODUCTION AND EXAMPLES 155 when both n and x are large. Such problems constitute the theory of large deviations. Since the probability will in general be small the usual methods of establishing limit theorems via characteristic functions and partial differential equations are too crude for the derivation of sufficiently general results and most of the theorems about large deviations are proved under very stringent conditions. Before formulating the problem in general we consider some simple but characteristic special results. Consider a Bernoulli scheme of n independent trials with a probability p Q of success. Write Yj 1 if thej th trial results in a success and Y - 0 otherwise. If b m n p P X Yj m j i then of course n b m .p P - m n m where q 1 p. If Xj Yj p then E X 0 F Xj P9 and Z takes only the values xm m-np npq i m 0 1 2 . n with respective probabilities b m n p . If we apply Stirling s formula to we obtain the following local limit theorem if xm o ni as n- oo then b m n

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