Let $\xi_1,\xi_2,\ldots$ be i.i.d. positive random variables with infinite mean $\mathbb E[\xi_i]=\infty$, and consider the random walk $$T_n=\sum_{i=1}^n\xi_i,\qquad n\in\mathbb N.$$ Here's an example that illustrates why I'm interested in such questions.

Example.If $(S_n)_{n\in\mathbb N}$ is a simple symmetric random walk on $\mathbb Z$, then $\xi_1,\xi_2,\ldots$ could represent the time intervals between consecutive returns to zero of $S_n$.

**Q.** Does there exist general local limit theorems for the random walk $T_n$?

In the case of the example I have above, everything can be computed explicitly quite nicely, and it can be argued that $$n\cdot \mathbb P\{T_{\sqrt{n}x}=n\}\asymp xe^{-x^2/2}.$$ I'm interested in more general statements of this form.