Asymptotics of renewal functions

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 $A$.