| Abstract [eng] |
Aggregation of random coefficient AR(1) processes $X_{i,t} = a_i X_{i,t-1} + \vep_t, \ i=1,\cdots, N$ with i.i.d. coefficients $a_i \in (-1,1)$ and common i.i.d. innovations $\{\vep_t\}$ belonging to the domain of attraction of $\alpha-$stable law $(0< \alpha \le 2)$ is discussed. Particular attention is given to the case of slope coefficient having probability density growing regularly to infinity at points $a = 1 $ and $a=-1$. Conditions are obtained under which the limit aggregate $\bar X_t = \lim_{N \to \infty} N^{-1}\sum_{i=1}^N X_{i,t} $ exists and exhibits long memory, in certain sense. In particularly, I show that suitably normalized partial sums of the $\bar X_t$'s tend to fractional $\alpha-$stable motion, and that $\{\bar X_t\}$ satisfies the long-range dependence (sample Allen variance) property and distributional long memory. The present paper also extends some results of P. Zaffaroni from finite variance case to infinite variance case. |
