On limit uniform distribution - Sunklodas, Jonas Kazys (Lietuvos matematikos rinkinys, 2008)
In the first part of the present paper, we estimate the difference Δ n(1) = sup-∞<x<∞ |PYn < x - Pξ < x|, where Yn = Xn/n, Xn is a discrete r.v. with PXn = 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 (1) is 1/(l-k)n . In the second part of the present paper, we estimate the difference (2) =sup-∞<x<∞ |PSn < x - Pξ < x|, where Sn = j=1 Xj /2j, the r.v.’s X1, . . ., Xn are independent, Xj is a discrete r.v. with PXj = -a = PXj = 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 (2) is C2-n, where C < 4.