On a property of sums of independent random variables

Let $\xi_i$, $j=1,2,\dots$ be independent identically distributed random variables with $\mathbf M\xi_1=0$, $\mathbf D\xi_1=1$. Put $P_n(x)=\mathbf P\{\xi_1+\dots+\xi_n\ge x\}$. In the paper, a class of distributions $P_1(x)$ is described having the following property: for $x\ge x_n$, $n\to\infty$
$$ P_n(x)=nP_1(x)(1+o(1)). $$
The dependence of the sequence $\{x_n\}$ on properties of $P_1(x)$ is also analyzed.