Divisors of Bernoulli Sums

Abstract.

Let $B_{n} = b_{1} + \cdots + b_n, n \geq 1$ where $b_1, b_2,\cdots$ are independent Bernoulli random variables. In relation with the divisor problem, we evaluate the almost sure asymptotic order of the sums $\sum\nolimits_{n=1}^{N}{d_{\theta, {\mathcal{D}}}(B_n)}$, where $d_{\theta,{\mathcal{D}}}(B_n)= \# \{d \in \mathcal{D},d \leq n^{\theta}:d\mid B_n\}$ and ${\mathcal{D}}$ is a sequence of positive integers.