Abstract:
Let $\xi_1,\dots$ be a sequence of independent identically distributed non-negative random variables. If the distribution function of $\xi_1$ has an absolutely continuous component, $\mathbf M\xi_1^{\alpha}<\infty$, $\alpha\ge 1$, then
$$
\biggl|H-\frac{1}{a}L-\frac{1}{a}F_2\biggr|([t,t+y))=
\begin{cases}
o(t^{-2(\alpha-1)}), &1\le\alpha<2,
\\
o(t^{-\alpha}), &2\le\alpha,
\end{cases}
$$
as $t\to\infty$ for $y>0$. Here: for a Borel set $A$,
$$
H(A)+\sum_{n=0}^{\infty}\mathbf P(S_n\in A),\qquad S_n=\sum_{k=1}^n\xi_k,\qquad S_0=0;
$$ $L$ is the Lebesgue measure; $a=\mathbf M\xi_1$;
$$
F_2(A)=\int_A\biggl(\int_x^{\infty}\mathbf P(\xi_1>u)\,du\biggr)\,dx;
$$ $|\mu|(A)$ stands for the total variation of a measure $\mu$ on a set $A$.