论文标题
关于迪克曼分布的概率模型的强，几乎确定的局部限制定理
On strong and almost sure local limit theorems for a probabilistic model of the Dickman distribution
论文作者
论文摘要
令$ \ {z_k \} _ {k \ geqslant 1} $表示由$ {\ mathbb p}定义的独立bernoulli随机变量序列$ t_n：= \ sum_ {1 \ leqslant k \ leqslant n} kz_k $。然后知道，$ t_n/n $薄弱地收敛到真正的随机变量$ d $，密度与迪克曼函数成正比，由延迟差异方程$ \ u \ varrho'（u）+\ varrho（u-varrho（u-1）= 0 $（u->）= 0 $（u-> 1）$，其初始条件$ \ varrho（u） 1）$。为了改善早期工作，我们提出了与相应局部且几乎肯定限制定理的剩余渐近公式。
Let $\{Z_k\}_{k\geqslant 1}$ denote a sequence of independent Bernoulli random variables defined by ${\mathbb P}(Z_k=1)=1/k=1-{\mathbb P}(Z_k=0)$ $(k\geqslant 1)$ and put $T_n:=\sum_{1\leqslant k\leqslant n}kZ_k$. 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 $u\varrho'(u)+\varrho(u-1)=0$ $(u>1)$ with initial condition $\varrho(u)=1$ $(0\leqslant u\leqslant 1)$. Improving on earlier work, we propose asymptotic formulae with remainders for the corresponding local and almost sure limit theorems.
