Limit theorems for decoupled renewal processes

The decoupled standard random walk is a sequence of independent random variables (S^n)n≥1, in which S^n has the same distribution as the position at time n of a standard random walk with nonnegative jumps. Denote by N^(t) the number of elements of the decoupled standard random walk which do not exceed t. The random process (N^(t))t≥0 is called decoupled renewal process. Under the assumption that t↦P{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 J1-topology for the decoupled renewal processes, properly scaled, centered and normalized. Also, under the assumption that t↦P{S^1>t} is regularly varying at infinity of index −α, α ∈ [0, 1) ∪ (1, 2) or the distribution of S^1 belongs to the domain of attraction of a normal distribution we prove a law of the iterated or single logarithm for 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.