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Independent And Stationary Sequences Of Random Variables - Chapter 3

Chapter 3 REFINEMENTS OF THE LIMIT THEOREMS FOR NORMAL CONVERGENCE § 1 . Introduction In this chapter we consider a sequence X 1 , X2 , . . . of independent, identically distributed random variables belonging to the domain of attraction of the normal law. As shown in § 2 .6, the X; necessarily have a finite variance a 2 . | Chapter 3 REFINEMENTS OF THE LIMIT THEOREMS FOR NORMAL CONVERGENCE 1. Introduction In this chapter we consider a sequence Xr X2 . of independent identically distributed random variables belonging to the domain of attraction of the normal law. As shown in 2.6 the Xj necessarily have a finite variance a2. We shall assume that E 2Q 0 then necessarily the distribution F of Zn X F X2 X an 3.1.1 converges to the normal distribution F with zero mean and unit variance. Indeed with 4 x i e iz2dz J 00 we have K x F x -i x - 0 as n- oo uniformly in x. In 3 we give an asymptotic formula for Rn x in terms of n In the later sections the behaviour of sup R x for large n is the object of study. The symbols f fn and v will denote the characteristic functions corresponding to the distributions F the common distribution of the 2Q Fn and F respectively so that t i e-i 2 . We shall also write 72 E X as E XJ S E X . 2. Some auxiliary theorems This section is devoted to some important properties of the characteristic functions 3.2. SOME AUXILIARY THEOREMS 95 Theorem 3.2.1. If fi3 is finite then 1 for t Tn r3ni 5 3 3.2.1 2 for t Tn3 o3ni 24f3. IXW-9n t n- d n t 3 11 6 e it2 3.2.2 where g t l n i Pt it Pi it a3o- 3 ii 3 and lim d n 0 n- oo 3 for t Tn3 fW 9n t 5i n i 4H- 7 e f2 3.2.3 where lim 5x n 0 . n oo Proof 1 Using the expression rz 1 zi 3 f00 j f t on 1 - exp t- dF x 2n 6 rn2 an J where 0 1 it is easy to show that in t Tn f t an jj. Therefore in this domain t3 n log fit on o3 r d3 - f 6aW dz3 g z _ z 0t anl 2 where by Lyapunov s inequality 3 10g z 3 3 1A2 2fft l z l3 7 24 25 3 P3 Since t exp n log t crn1 it follows that 96 LIMIT THEOREMS FOR NORMAL CONVERGENCE REFINEMENTS Chap. 3 fn t -vW e 2 2 - 1 6cr n2 -7 f 3 c P 7 fl3 6cr3n2 2 6 73n J where we have used the obvious inequality ex 1 x e x . Finally if 111 T then _ Z l 3 2 6 T3n2 4 3-2.4 and 3.2.1 is proved. 2 Since 2n 6 7 n2 ri we have log t T772 - 2n 6 t3 rfi Using 3.2.4 again we have for t Tn3 fn t -gn t gn t 1 - exp no t 3 n o n