Limit theorems for decoupled renewal processes

The decoupled standard random walk is a sequence of independent random variables $(\hat S_n)_{n\geq 1}$, in which $\hat S_n$ has the same distribution as the position at time $n$ of a standard random walk with nonnegative jumps. Denote by $\hat N(t)$ the number of elements of the decoupled standard random walk which do not exceed $t$. The random process $(\hat N(t))_{t\geq 0}$ is called decoupled renewal process. Under the assumption that $t\mapsto \mathbb{P}\{\hat S_1>t\}$ is regularly varying at infinity of nonpositive index larger than $-1$ we prove a functional central limit theorem in the Skorokhod space equipped with the $J_1$-topology for the decoupled renewal processes, properly scaled, centered and normalized. Also, under the assumption that $t\mapsto \mathbb{P}\{\hat S_1>t\}$ is regularly varying at infinity of index $-\alpha$, $\alpha\in [0,1)\cup (1,2)$ or the distribution of $\hat S_1$ belongs to the domain of attraction of a normal distribution we prove a law of the iterated or single logarithm for $\hat N(t)$, again properly normalized and centered. As an application, we obtain a law of the single logarithm for the number of atoms of a determinantal point process with the Mittag-Leffler kernel, which lie in expanding discs.