We obtain a strong renewal theorem with infinite mean beyond regular variation, when the underlying distribution belongs to the domain of geometric partial attraction of a semistable law with index \(\alpha \in (1/2,1]\). In the process we obtain local limit theorems for both finite and infinite mean, that is, for the whole range \(\alpha \in (0,2)\). We also derive the asymptotics of the renewal function for \(\alpha \in (0,1]\).

中文翻译：

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在没有规则变化的情况下的强更新定理和局部极限定理

当基础分布属于索引为\（\ alpha \ in（1 / 2,1] \）的半稳定定律的几何部分吸引域时，我们获得了具有除正则变化之外的无穷均值的强更新定理。我们获得有限均值和无限均值的局部极限定理，即整个范围\（\ alpha \ in（0,2）\）。我们还导出了\（\ alpha \ in（ 0,1] \）。