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

Let { Z k } k ≥ 1 denote a sequence of independent Bernoulli random variables defined by P ( Z k = 1) = 1 /k = 1 − P ( Z k = 0) ( k ≥ 1) and put T n ≔ ∑ 1 ≤ k ≤ n kZ k . It is known that T n /n convergesweakly to a real random variable D with density proportional to the Dickman function, defined by the delay-differential equation u ϱ ′ ( u ) + ϱ( u − 1) = 0 ( u > 1) with initial condition ϱ( u ) = 1(0 ≤ u ≤ 1). Improving on earlier work, we propose asymptotic formulae with remainders for the corresponding local and almost sure limit theorems: ∑ m ≥ 0 P T n = m − e − γ n ϱ m n = 2 log n π 2 n 1 + O 1 log 2 n n → ∞ , and ∀ u > 0 , ∑ n ≤ N , T n = un 1 = e − γ ϱ u log N + O log N 2 / 3 + o 1 a . s N → ∞ , where γ denotes Euler’s constant.