Consider a sequence $(X_n)_{n \ge 1}$ of i.i.d. Pareto($2$) random variables, which means $$ \mathbb{P}( X_1 > x) = \begin{cases} 1/x \qquad &\text{for } x \ge 1 \\ 1 \qquad & \text{else}. \end{cases} $$ Consider the stochastic process $(S_n)_{n\ge 0}$ taking the partial sums $$ S_n=X_1+\cdots +X_n, \qquad \text{with} \qquad S_0=0 $$ and define the renewal process $(N_t)_{t \in [0,+\infty)}$ as $$ N_t = \sup\{n: S_n\le t\}. $$ Question. Is the renewal process $(N_t)_{t \in [0,+\infty)}$ studied anywhere in literature?
Comment. Renewal theory is a classic in probability and it is treated in the book by Feller (1968-Volume 2, Chapter XI). Feller mainly focuses on the case where the inter-arrival times have mean and variance, stating the Central Limit Theorem for $N_t$ in Chapter XI, Section 5. Such variant of CLT states that, in the notation established above, when $X_n$ are i.i.d random variables with $\mathbb{E}(X_1)=\mu$ and $\mathrm{Var}(X_1)=\sigma^2$, then $N_t \approx \mathscr{N}(t/\mu ,t\sigma^2/\mu^3 )$.
What about analogs of the CLT for heavy tailed $X_n$ in terms of stable laws? In the case when $X_n$ have Pareto distribution with coefficient $2$ it is known that $\mathbb{E}(N_t) \approx t/\log(t)$ and quick heuristic calculations suggest that $\mathrm{Var}(N_t) \approx t^2 /\log(t)^3$. Given how simple and natural this model is I suspect it's treated somewhere in literature, but I cannot find where.
