Local limit theorems and renewal theory with no moments

We study i.i.d. sums $\tau_k$ of nonnegative variables with index $0$: this means $\mathbf{P}(\tau_1=n) = \varphi(n) n^{-1}$, with $\varphi(\cdot)$ slowly varying, so that $\mathbf{E}(\tau_1^\varepsilon)=\infty$ for all $\varepsilon>0$. We prove a local limit and local (upward) large deviation theorem, giving the asymptotics of $\mathbf{P}(\tau_k=n)$ when $n$ is at least the typical length of $\tau_k$. A recent renewal theorem by Nagaev [21] is an immediate consequence: $\mathbf{P}(n\in\tau) \sim \mathbf{P}(\tau_1=n)/\mathbf{P}(\tau_1 > n)^2$ as $n\to\infty$. If instead we only assume regular variation of $\mathbf{P}(n\in\tau)$ and slow variation of $U_n:= \sum_{k=0}^n \mathbf{P}(k\in\tau)$, we obtain a similar equivalence but with $\mathbf{P}(\tau_1=n)$ replaced by its average over a short interval. We give an application to the local asymptotics of the distribution of the first intersection of two independent renewals. We further derive downward moderate and large deviations estimates, that is, the asymptotics of $\mathbf{P}(\tau_k \leq n)$ when $n$ is much smaller than the typical length of $\tau_k$.