On limiting distributions for moduli of sequential differences of independent variables

Let $(\xi,\eta)$ be a pair of independent equally distributed random variables, and $F(x)$ be their common distribution function. We define a sequence of pairs $(\xi_n,\eta_n)$ of independent equally distributed random variables with distribution functions $F_n(x)$:
$$ F_1(x)=\mathbf\{|\xi-\eta|<x\},\quad F_{n+1}(x)=\mathbf P\{|\xi_n-\eta_n|<x\}, $$
and prove two theorems concerning the limiting behaviour of $F_n(x)$.