Modern Stochastics: Theory and Applications

Buraczewski et al. (2023) proved a functional limit theorem (FLT) and a law of the iterated logarithm (LIL) for a random Dirichlet series ${\textstyle\sum _{k\ge 2}}\frac{{(\log k)^{\alpha }}}{{k^{1/2+s}}}{\eta _{k}}$ as $s\to 0+$, where $\alpha \gt -1/2$ and ${\eta _{1}},{\eta _{2}},\dots $ are independent identically distributed random variables with zero mean and finite variance. A FLT and a LIL are proved in a boundary case $\alpha =-1/2$. The boundary case is more demanding technically than the case $\alpha \gt -1/2$. A FLT and a LIL for ${\textstyle\sum _{p}}\frac{{\eta _{p}}}{{p^{1/2+s}}}$ as $s\to 0+$, where the sum is taken over the prime numbers, are stated as the conjectures.

In the Karlin infinite occupancy scheme, balls are thrown independently into an infinite array of boxes $1,2,\dots $ , with probability ${p_{k}}$ of hitting the box k. For $j,n\in \mathbb{N}$, denote by ${\mathcal{K}_{j}^{\ast }}(n)$ the number of boxes containing exactly j balls provided that n balls have been thrown. Small counts are the variables ${\mathcal{K}_{j}^{\ast }}(n)$, with j fixed. The main result is a law of the iterated logarithm (LIL) for the small counts as the number of balls thrown becomes large. Its proof exploits a Poissonization technique and is based on a new LIL for infinite sums of independent indicators ${\textstyle\sum _{k\ge 1}}{1_{{A_{k}}(t)}}$ as $t\to \infty $, where the family of events ${({A_{k}}(t))_{t\ge 0}}$ is not necessarily monotone in t. The latter LIL is an extension of a LIL obtained recently by Buraczewski, Iksanov and Kotelnikova (2023+) in the situation when ${({A_{k}}(t))_{t\ge 0}}$ forms a nondecreasing family of events.