Limit theorems for quadratic forms of linear processes (Q1079279)

Let \(\{\xi_ j\), \(j\in {\mathbb{Z}}\}\) be a sequence of iid random variables belonging to the domain of attraction of a stable law of index \(\alpha\), \(1<\alpha <2\); E \(\xi\) \({}_ 1=0\). Define \(X_ k=\sum_{j\leq k}(k- j)^{-\beta}H(k-j)\xi_ j\) where H(\(\cdot)\) is slowly varying. Further, for \(t\geq 0\), let \[ S_ N^{(1)}(t)=\sum^{[Nt]}_{k=0}X_ k,\quad S_ N^{(2)}(t)=\sum^{[Nt]}_{k=0}X^ 2_ k, \] \[ Q_ N(t)=\sum^{[Nt]}_{k=0}(X_{k+1}-X_ k)^ 2\quad and\quad R_ N(t)=S_ N^{(2)}(t)-\nu Q_ N(t). \] Results relating to asymptotic distributions (as \(N\to \infty)\) of suitably normed versions of the above processes as well as to the joint distribution of \((R_ N,Q_ N)\) have been provided.