Convergence of functionals of sums of r.v.s to local times of fractional stable motions

Abstract

Consider a sequence X_k=\sum_{j=0}^{\infty}c_j\xi_{k-j}, k\geq 1, where c_j, j\geq 0, is a sequence of constants and \xi_j, -\infty <j<\infty, is a sequence of independent identically distributed (i.i.d.) random variables (r.v.s) belonging to the domain of attraction of a strictly stable law with index 0<\alpha \leq 2. Let S_k=\sum_{j=1}^kX_j. Under suitable conditions on the constants c_j it is known that for a suitable normalizing constant \gamma_n, the partial sum process \gamma_n^{-1}S_{[nt]} converges in distribution to a linear fractional stable motion (indexed by \alpha and H, 0<H<1). A fractional ARIMA process with possibly heavy tailed innovations is a special case of the process X_k. In this paper it is established that the process n^{-1}\beta_n\sum_{k=1}^{[nt]}f(\beta_n(\gamma_n^{-1}S_k+x)) converges in distribution to (\int_{-\infty}^{\infty}f(y) dy)L(t,-x), where L(t,x) is the local time of the linear fractional stable motion, for a wide class of functions f(y) that includes the indicator functions of bounded intervals of the real line. Here \beta_n\to \infty such that n^{-1}\beta_n\to 0. The only further condition that is assumed on the distribution of \xi_1 is that either it satisfies the Cram\'er's condition or has a nonzero absolutely continuous component. The results have motivation in large sample inference for certain nonlinear time series models.