Article Dans Une Revue Lithuanian Mathematical Journal Année : 2021

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

Résumé

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\geqslant 1) and put T_n:=Σ_{1≤ k≤ n}kZ_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 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, namely Σ_{m≥ 0} |P(T_n=m)-(\e^{-γ}/ n)ρ(m/n) |={(2log n)/ π^2 n} {1+O(1/ \og_2n) (n → oo), and $$(∀ u>0) Σ_{d≤N, T_n=[u_n]}1=\e^{-γ}ρ(u)log N+O( (\log N)^{2/3+o(1)}) a.s. (N → oo), where $γ$ denotes Euler's constant.

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Dates et versions

hal-03167177 , version 1 (11-03-2021)

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Régis de la Bretèche, Gérald Tenenbaum. On strong and almost sure local limit theorems for a probabilistic model of the Dickman distribution. Lithuanian Mathematical Journal, 2021, 61, pp.301-311. ⟨10.1007/s10986-021-09529-6⟩. ⟨hal-03167177⟩
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