Weak convergence of sequences from fractional parts of random variables and applications

Abstract

We prove results concerning the weak convergence to the uniform distribution on $[0,1]$ of sequences $(Z_n)_{n \geq 1}$ of the form $Z_n = Y_n \pmod 1= \{Y_n \}$, where $(Y_n)_{n \geq 1}$ is a general sequence of real random variables. Applications are given: (i) 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.