In 1972 Delange [9] observed in analogy of the classical Erdős-Wintner theorem that q-additive functions $f(n)$ has a distribution function if and only if the two series $\sum f(d{q}^j)$, $\sum f{(d{q}^j)}^2$ converge. The purpose of this paper is to provide quantitative versions of this theorem as well as generalizations to other kinds of digital expansions. In addition to the q-ary and Cantor case we focus on the Zeckendorf expansion that is based on the Fibonacci sequence, where we provide a sufficient and necessary condition for the existence of a distribution function, namely that the two series $\sum f(F_j)$, $\sum f{(F_j)}^2$ converge (previously only a sufficient condition was known [2]).

1972 年，Delange [9] 用经典 Erdős-Wintner 定理的类比观察到q -additive 函数$F(n)$ 有分布函数当且仅当两个系列 $\sum F(d{q}^j)$, $\sum F{(d{q}^j)}^2$收敛。本文的目的是提供该定理的定量版本以及对其他类型的数字扩展的推广。除了q- ary 和 Cantor 的情况，我们关注基于斐波那契数列的 Zeckendorf 展开，我们提供了分布函数存在的充分必要条件，即两个级数$\sum F(F_j)$, $\sum F{(F_j)}^2$ 收敛（以前只知道一个充分条件 [2]）。