Local limit theorems and renewal theory with no moments

We study i.i.d. sums $τ_k$ of nonnegative variables with index $0$: this means $\mathbf{P}(τ_1=n) = φ(n) n^{-1}$, with $φ(\cdot)$ slowly varying, so that $\mathbf{E}(τ_1^\varepsilon)=\infty$ for all $\varepsilon>0$. We prove a local limit and local (upward) large deviation theorem, giving the asymptotics of $\mathbf{P}(τ_k=n)$ when $n$ is at least the typical length of $τ_k$. A recent renewal theorem by Nagaev [21] is an immediate consequence: $\mathbf{P}(n\inτ) \sim \mathbf{P}(τ_1=n)/\mathbf{P}(τ_1 > n)^2$ as $n\to\infty$. If instead we only assume regular variation of $\mathbf{P}(n\inτ)$ and slow variation of $U_n:= \sum_{k=0}^n \mathbf{P}(k\inτ)$, we obtain a similar equivalence but with $\mathbf{P}(τ_1=n)$ 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 $\mathbf{P}(τ_k \leq n)$ when $n$ is much smaller than the typical length of $τ_k$.

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