On properties derived from different types of asymptotic distribution functions of ratio sequences

Let $X=\{x_1<x_2<\cdots\}$ be an infinite subset of positive integers and $X_n=\left(\frac{x_1}{x_n},\frac{x_2}{x_n},\dots,\frac{x_n}{x_n}\right)$, $n=1,2,\dots$. In this paper, we give new necessary and sufficient conditions for $X$ for that the sequence of blocks $X_n$ has an asymptotic distribution function.