Limit theorems for distributions of sums reduced modulo a (Q1085514)

Let \(X_ 1,X_ 2,..\). be independent random variables and define \(S_ n:=\sum^{n}_{i=1}X_ i\), \(n=1,2,... \). Let a partition of \({\mathbb{R}}\) into intervals \([ka,(k+1)a)\), \(k\in {\mathbb{Z}}\), of any fixed length \(a>0\) be given. We ask for the distribution of the remainder \(S_ n-k_ na\) where \([k_ na,(k_ n+1)a)\) is the random interval containing \(S_ n\). Necessary and sufficient conditions are given for this distribution to converge to the uniform distribution in [0,a] as \(n\to \infty\).