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.



Local limit theorems for positive random walks
Yes, there is a general result, see Chapter 9 of Gnedenko-Kolmogorov book. 


The theorem says that if $\xi\_i$ are i.i.d with values in $\mathbb{Z}$ such that $$\text{gcd}\{s-s':\mathbb{P}(\xi\_1=s)>0,\;\mathbb{P}(\xi\_1=s')>0\}=1,$$ and $b\_n^{-1}(T\_n-a\_n)$ converges in distribution to a stable law, then also the local limit theorem holds, namely, 
$$
b\_n\mathbb{P}(T\_n=k)-g(b\_n^{-1}(k-a\_n))\to 0
$$uniformly in $k$, where $g$ is the density of the stable law.


There is also a necessary and sufficient condition for the existence of $b\_n$ and $a\_n$ such that $b\_n^{-1}(T\_n-a\_n)$ converges in distribution, see e. g. Feller's or Durrett's book.


