Limit theorems for distributions of sums reduced modulo a

Let X1, X2,... be independent random variables and define . Let partition of into intervals , of any fixed length a > 0 be given. We ask for the distribution of the remainder Sn-kna where[kna (kn+1)a) is the random interval con Sn. Necessary and sufficient conditions are given for this distribution to converge to the uniform distribution in [0, a] as n-->[infinity].renewal processes sums of random variables reduced modulo a limit theorems convergence to uniform distribution