Let $\sum _{n=1}^{\infty }{a}_n$ be a convergent series of positive real numbers. L. Olivier proved that if the sequence $\left(a_n\right)$ is non-increasing, then $\underset{n\to \infty }{lim}n{a}_n=0$. In the present paper: (a) We formulate and prove a necessary and sufficient condition for having $\underset{n\to \infty }{lim}n{a}_n=0$; Olivier’s theorem is a consequence of our Theorem . (b) We prove properties analogous to Olivier’s property when the usual convergence is replaced by the $ℐ$-convergence, that is a convergence according to an ideal $ℐ$ of subsets of $\mathbb{N}$. Again, Olivier’s theorem is a consequence of our Theorem , when one takes as $ℐ$ the ideal of all finite subsets of $\mathbb{N}$.

In this paper the ideas of three types of statistical convergence of a sequence of random variables, namely, statistical convergence in probability, statistical convergence in mean of order $r$ and statistical convergence in distribution are introduced and the interrelation among them is investigated. Also their certain basic properties are studied.

We study several kinds of statistical convergence of sequences of functions with values in semi-uniform spaces. Particularly, we generalize to statistical convergence the classical results of C. Arzelà, Dini and P.S. Alexandroff, as well as their statistical versions studied in [Caserta A., Di Maio G., Kočinac L.D.R., {Statistical convergence in function spaces},. Abstr. Appl. Anal. 2011, Art. ID 420419, 11 pp.] and [Caserta A., Kočinac L.D.R., {On statistical exhaustiveness}, Appl. Math. Lett....