Integral criteria for Strong Renewal Theorems with infinite mean

Let $F$ be a probability measure on $\mathbb{R}$ in the domain of attraction of a stable law with exponent $\alpha\in (0, 1)$. We establish integral criteria on $F$ that significantly expand the probabilistic approach to Strong Renewal Theorems (SRTs). The criterion for $\alpha \in (0,1/2]$ is much weaker than currently available ones and in some cases provides sufficient and necessary conditions for the SRT. The criterion for $\alpha \in (1/2, 1)$ establishes the SRT in full generality and in a unified way, barring the Limit Local Theorems employed. As an application, for infinitely divisible $F$, an integral criterion on its L\'evy measure is established for the SRT. As another application, for $F$ in the domain of attraction of a stable law without centering, an integral criterion on $F$ is established for the SRT for the ladder height process of a random walk with step distribution $F$.