We study large partial sums, localized with respect to the sums of variances, of a sequence of centered random variables. An application is given to the distribution of prime factors of typical integers.

A simple proof is given for the monotonicity of entropy and Fisher information associated to sums of i.i.d. random variables. The proof relies on a characterization of maximal correlation for partial sums due to Dembo, Kagan and Shepp.

In this note, we obtain a Chung's integral test for self-normalized sums of i.i.d. random variables. Furthermore, we obtain a convergence rate of Chung law of the iterated logarithm for self-normalized sums.

Analogous to the Donsker theorem on partial cumulative sums of independent random variables, for one-sample rank-order statistics, weak convergence to Brownian motion processes is studied here. This yields a simple proof of the asymptotic normality of the related rank statistics for random sample sizes. Analogous to the Donsker theorem on partial cumulative sums of independent random variables,...

Operator geometric stable laws are the weak limits of operator normed and centered geometric random sums of independent, identically distributed random vectors. They generalize operator stable laws and geometric stable laws. In this work we characterize operator geometric stable distributions, their divisibility and domains of attraction, and present their application to finance. Operator geome...

2 Preliminaries 7 2.1 Matrices and their singular values . . . . . . . . . . . . . . . . . . 7 2.2 Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Sub-gaussian random variables . . . . . . . . . . . . . . . . . . . 9 2.4 Sub-exponential random variables . . . . . . . . . . . . . . . . . . 14 2.5 Isotropic random vectors . . . . . . . . . . . . . . . . . . . . . . ...