WEAK CONVERGENCE OF SEQUENCES FROM FRACTIONAL PARTS OF RANDOM VARIABLES AND APPLICATIONS

We prove results concerning the weak convergence to the uniform distribution on [0,1] of sequences (Z(n))(n >= 1) of the form Z(n) = Y-n (mod 1)= {Y-n}, where (Y-n)(n >= 1) is a general sequence of real random variables. Applications are given: (1) to the case of partial sums of (i.i.d.) random variables having a distribution belonging to the domain of attraction of a stable law: (ii) to the case of sample maxima of i.i.d random variables.