Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes with infinite variance

We discuss the joint temporal and contemporaneous aggregation of N independent copies of random-coefficient AR(1) processes driven by independent and identically distributed innovations in the domain of normal attraction of an $\alpha$ -stable distribution, $0< \alpha \le 2$ , as both N and the time scale n tend to infinity, possibly at different rates. Assuming that the tail distribution function of the random autoregressive coefficient regularly varies at the unit root with exponent $\beta > 0$ , we show that, for $\beta < \max (\alpha, 1)$ , the joint aggregate displays a variety of stable and non-stable limit behaviors with stability index depending on $\alpha$ , $\beta$ and the mutual increase rate of N and n. The paper extends the results of Pilipauskaitė and Surgailis (2014) from $\alpha =2$ to $0 < \alpha < 2$ .

A local limit theorem for attractions under a stable law

Let {Xn} be a sequence of independent random variables each having a common d.f. V1. Suppose that V1 belongs to the domain of normal attraction of a stable d.f. V0 of index α 0 ≤ α ≤ 2. Here we prove that, if the c.f. of X1 is absolutely integrable in rth power for some integer r > 1, then for all large n the d.f. of the normalized sum Zn of X1, X2, …, Xn is absolutely continuous with a p.d.f. vn such thatas n → ∞, where v0 is the p.d.f. of Vo.