General theorems on rates of convergence in distribution of random variables I. General limit theorems

P. L. Butzer and L. Hahn

Journal of Multivariate Analysis, 1978, vol. 8, issue 2, 181-201

Abstract: Let (Xn)n[epsilon] be a sequence of real, independent, not necessarily identically distributed random variables (r.v.) with distribution functions FXn, and Sn = [Sigma]i=1n Xi. The authors present limit theorems together with convergence rates for the normalized sums [phi](n)Sn, where [phi]: --> +, [phi](n) --> 0, n --> [infinity], towards appropriate limiting r.v. X, the convergence being taken in the weak (star) sense. Thus higher order estimates are given for the expression [is proportional to]f(x) d[F[phi](n)Sn(x) - FX(x)] which depend upon the normalizing function [phi], decomposability properties of X and smoothness properties of the function f under consideration. The general theorems of this unified approach subsume O- and o-higher order error estimates based upon assumptions on associated moments. These results are also extended to multi-dimensional random vectors.

Keywords: General; limit; theorems; convergence; in; distribution; higher; order; rates; of; convergence (search for similar items in EconPapers)
Date: 1978
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Citations: View citations in EconPapers (1)

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