ON-%E & CONVERGENCE RATE IN THE CENTRAL LIMIT THEOWEM OF SOME FUNmONS OF THE AVERAGE-

Abstract. This note gives the convergence rate in the central limit theorem and the random central limit theorem of some functions of the average of independent random variables. lection and preliminaries. Let {X,, k 2 1) be a sequence of random vakiables and put A s, = C x,. k = 1 We are interested in finding estimates of the convergence rate in where g denotes a real function, c, and d,> ' O are normalizing constants depending on g, D denotes weak convergence, and Na,, a normal random variable with mean a and standard deviation b. To give our results, which are an extension of some considerations in [6], we need the following notation, lemmas, and theorems. Let C ( R) be the class of all continuous functions and let C,(R) stand for the dass of all bounded and continuous functions. Pu