On limit uniform distribution
Articles
Jonas Kazys Sunklodas
Institute of Mathematics and Informatics
Published 2008-12-21

## Keywords

uniform distribution
independent random variables

## How to Cite

Sunklodas J. K. (2008) “On limit uniform distribution”, Lietuvos matematikos rinkinys, 48(proc. LMS), pp. 417–421. doi: 10.15388/LMR.2008.18153.

## Abstract

In the first part of the present paper, we estimate the difference \Delta n(1) = sup-∞<x<∞ |P{Yn < x} - P{ξ < x}|, where Yn = Xn/n, Xn is a discrete r.v. with P{Xn = j} = 1/(l-k)n , for j = nk, nk + 1, . . ., nl - 1, as n = 1, 2, . . ., k < l, and k, l are any integers; the absolutely continuous r.v. ξ is uniformly distributed in the interval [k, l]. The upper bound of \Deltan(1) is 1/(l-k)n . In the second part of the present paper, we estimate the difference \Deltan(2) =sup-∞<x<∞ |P{Sn < x} - P{ξ < x}|, where Sn = \sumnj=1 Xj /2j, the r.v.’s X1, . . ., Xn are independent, Xj is a discrete r.v. with P{Xj = -a} = P{Xj = a} = 1/2 for j = 1, . . ., n and any real number a > 0; the absolutely continuous r.v. ξ is uniformly distributed in the interval [-a, a]. The obtained upper bound of \Deltan(2) is C2-n, where C < 4. 