Divisors of Bernoulli sums (Q5900091)

Let \(\beta_1,\beta_2,...\) be independent Bernoulli random variables, and \(B_n=\beta_1+\beta_2+ ... +\beta_n\) be the sequence of associated partial sums. The almost sure asymptotic order is obtained for the sums \[ \sum_{n\leq N} d_{\theta,\mathcal{D}}\left(B_n\right),\quad \sum_{n\leq N\atop n\in \mathcal{N}} d_{\eta,\mathcal{D}}\left(B_n\right). \] Here \(d_{\theta,\mathcal{D}}\) and \(d_{\eta,\mathcal{D}}\) are generalized divisor functions, i.e. \[ d_{\theta,\mathcal{D}}\left(B_n\right)=\sum_{d\in\mathcal{D}\atop d\leq n^\theta}\mathbb{I}_{\{d|B_n\}},\quad d_{\eta,\mathcal{D}}\left(B_n\right)=\sum_{d\in\mathcal{D}\atop d<\eta\sqrt{\frac{n}{\log n}}}\mathbb{I}_{\{d|B_n\}}, \] and \(\mathcal{D},\mathcal{N}\) are subsets of the natural numbers. In addition, the sequence \(\mathcal{N}\) satisfies some growth condition.