Acta Mathematica Academiae Scientiarum Hungaricae 28. (1976)

1976 / 3-4. szám - Basu, A. K.: On the rate of convergence to normality for sums of dependent random variables

Acta Mathematica Academiae Scientiarum Hungaricae Tomus 28 (3-4), (1976), 261-265. ON THE RATE OF CONVERGENCE TO NORMALITY FOR SUMS OF DEPENDENT RANDOM VARIABLES By A. K. BASU1 (Sudbury) I. Introduction. In this paper we attempt to get uniform bounds in the central limit theorem for a class of dependent random variables considered by Dvoretzky [2]. Throughout the sequel we shall consider a double array {Xnj,j ^ k„} of random variables. Relationships of equality or inequality stated between random variables к are to be understood to hold only almost surely. Let S„ k = X Xn } for к = 0, 1,.. k„ i=1 and Fn k = 3!>(S„yk) be the er-field generated by S„yk. Assume цп к = ЩХ„ук | ir„_*_1) = 0 and the conditional variances k — E(Xl k | Fn к^г) exists almost surely. k„ Furthermore X а\ к = 1 a.s. and the Lindberg condition k = l lim £ E[XlkIQ Xnk I > e)] = 0 n-> oo k — 1 for e > 0 will be assumed to hold. (/( • ) is the indicator function of the set within the bracket.) In section three we shall find a rate of convergence under the additional con­dition lim X E[l xnk l2_a] -*■ 0 n—> oo k = 1 for some ö > 0. Of course this Liapaunov condition implies Lindberg condition. kn Let Sn = X In section two we have derived estimate of the form j=l P(Sn ^ an) = exp [- (alj2)(l + o(l)] as n -*■ oo where a„ = (1 + e)(2 log log n)1/2, s > 0 by examining the convergence rate in Trotter’s method of operators. The above large deviation estimate can be used to prove ordinary law of the iterated logarithm for dependent random variables. This motivates our results for section two. But a careful analysis shows that Trotter’s method yields a slower rate of convergence to zero than is actually known to be the case for i.i.d. random variables. This point was noted in the introduction to Pinsky’s note [8] and was pointed out more generally in Feller [4], Chapter IX. J. Chover [1] proved the functional law of the iterated logarithm for i.i.d. random variables. One of the key tools was a Berry-Esseen Theorem. Chover’s idea 1 This work was partly done when the author was at 1973 Carleton Summer Research In­stitute of Canadian Mathematical Congress. Appreciation is extended to N. R. C. of Canada and the Canadian Mathematical Congress for financial support of this work. Acta Mathematica Academiae Scientiarum Hungaricae 28> 1976

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