Aggregation of random-coefficient AR(1) process with infinite variance and common innovations

We discuss aggregation of random-coefficient AR(1) processes X _i,t = a _i X _i,t?1 + ε _t , i = 1,…,N, with i.i.d. coefficients a _i ∈ (?1, 1) and common i.i.d. innovations {ε _t } belonging to the domain of attraction of an α-stable law (0 < α ≤ 2). Particular attention is given to the case of slope coefficient having probability density growing regularly to infinity at points a = 1 and a = ?1. We obtain conditions under which the limit aggregate \( {\bar X_t} = {\lim_{N \to \infty }}{N^{ - 1}}\sum\nolimits_{i = 1}^N {{X_{i,t}}} \) exists and exhibits long memory in a certain sense. In particularly, we show that suitably normalized partial sums of the \( {\bar X_t} \)’s tend to a fractional α-stable motion and that \( \left\{ {{{\bar X}_t}} \right\} \) satisfies the long-range dependence (sample Allen variance) property of Heyde and Yang. We also extend some results of Zaffaroni from the finite variance case to the infinite variance case.