On strong and almost sure local limit theorems for a probabilistic model of the Dickman distribution

Abstract

Let $\{Z_k{\}}_{k⩾1}$ denote a sequence of independent Bernoulli random variables defined by $\mathbb{P}(Z_k=1)=1/k=1-\mathbb{P}(Z_k=0)$ $(k⩾1)$ and put $T_n:={\sum }_{1⩽k⩽n}k{Z}_k$. It is then known that $T_n/n$ converges weakly to a real random variable $D$ with density proportional to the Dickman function, defined by the delay-differential equation $u{\varrho }^{\prime }(u)+\varrho (u-1)=0$ $(u>1)$ with initial condition $\varrho (u)=1$ $(0⩽u⩽1)$. Improving on earlier work, we propose asymptotic formulae with remainders for the corresponding local and almost sure limit theorems.