Renewal theorems in $R^d$

Let $\zeta_n$ be the sum of i. i. d. random vectors. The renewal measure $\displaystyle H(E)=\sum_{n=1}^{\infty}\mathbf P\{\zeta_n\in E\}$ with $E$ being a bounded Borel set is considered. Some results concerning the asymptotic behaviour of $H(E+x)$ as $|x|\to\infty$ are obtained.