Local limit theorems for positive random walks

Question

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.

Answer 1

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.