Stable limits of empirical processes of moving averages with infinite variance.
Consider iid random variables \(\{Y_s\}\) with a distribution belonging to the domain of normal attraction of a symmetric \(\alpha \)-stable law, \(1<\alpha \leq 2\). Define \(X_t=\sum _{j=1}^{\infty } b_j \zeta _{t-j}\) where \(b_j\sim c_0 j^{-\beta }\) for \(\beta >1/\alpha \). Let \(F\) be the distribution function of \(X_t\) and \(F_N\) the empirical distribution function based on \(X_1,\dots ,X_N\). It is proved that \(N^{1- 1/\alpha \beta} (F_N(x)-F(x))\) weakly converges to the process \(c_x^+L^+ + c_x^-L^-\) where \(L^+\), \(L^-\) are independent totally skewed \(\alpha \beta \)-stable random variables and \(c_x^+\), \(c_x^-\) some deterministic functions. Further, let \(H\) be a bounded measurable function and \(S_{N,H}(t)=\sum _{s=1}^{[Nt]} (H(X_s)- EH(X_s))\), \(0\leq t\leq 1\). It is shown that the weak limit of \(N^{-1/\alpha \beta } S_{N,H}(t)\) is \(c_H^+L^+(t) + c_H^-L^-(t)\) where \(L^+(t)\), \(L^-(t)\) are independent copies of a totally skewed \(\alpha \beta \)-stable Lévy motion \(L(t)\) with independent and homogeneous increments.
scientific article; zbMATH DE number 2138927