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

Let {Zk}k ≥ 1 denote a sequence of independent Bernoulli random variables defined by P(Zk = 1) 1/k 1− 0) (k and put Tn ≔ ∑1 ≤ k n kZk. It is known that Tn/n convergesweakly to real variable D with density proportional the Dickman function, delay-differential equation uϱ ′ (u) + ϱ(u − 0 (u > initial condition ϱ(u) 1(0 u 1). Improving on earlier work, we propose asymptotic formulae remainders for corresponding local almost sure limit theorems: $$ \sum \limits_{m\ge 0}\left|\mathbf{P}\left({T}_n=m\right)-\frac{{\mathrm{e}}^{-\upgamma}}{n}\uprho \left(\frac{m}{n}\right)\right|=\frac{2\log n}{\pi^2n}\left\{1+O\left(\frac{1}{\log_2n}\right)\right\}\left(n\to \infty \right), \forall u>0,\kern1.25em \limits_{n\le N,{T}_n=\left\lfloor un\right\rfloor }1={\mathrm{e}}^{-\upgamma}\uprho (u)\log N+O\left({\left(\log N\right)}^{2/3+o(1)}\right)\ \mathrm{a}.\mathrm{s}\ \left(N\to where γ denotes Euler’s constant.