Aggregation of a random-coefficient ar(1) process with infinite variance and idiosyncratic innovations

Abstract

Contemporaneous aggregation ofNindependent copies of a random-coefficient AR(1) process with random coefficienta∈ (−1, 1) and independent and identically distributed innovations belonging to the domain of attraction of an α-stable law (0 < α < 2) is discussed. We show that, under the normalizationN1/α, the limit aggregate exists, in the sense of weak convergence of finite-dimensional distributions, and is a mixed stable moving average as studied in Surgailis, Rosiński, Mandrekar and Cambanis (1993). We focus on the case where the slope coefficientahas probability density vanishing regularly ata= 1 with exponentb∈ (0, α − 1) for α ∈ (1, 2). We show that in this case, the limit aggregate {X̅t} exhibits long memory. In particular, for {X̅t}, we investigate the decay of the codifference, the limit of partial sums, and the long-range dependence (sample Allen variance) property of Heyde and Yang (1997).