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

Abstract

Consider a sequence Xk=∑j=0∞cjξk−j, k≥1, where cj, j≥0, is a sequence of constants and ξj, −∞<j<∞, 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<α≤2. Let Sk=∑j=1kXj. Under suitable conditions on the constants cj it is known that for a suitable normalizing constant γn, the partial sum process γn−1S[nt] converges in distribution to a linear fractional stable motion (indexed by α and H, 0<H<1). A fractional ARIMA process with possibly heavy tailed innovations is a special case of the process Xk. In this paper it is established that the process n−1βn∑k=1[nt]f(βn(γn−1Sk+x)) converges in distribution to ( ∫−∞∞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 βn→∞ such that n−1βn→0. The only further condition that is assumed on the distribution of ξ1 is that either it satisfies the Cramér’s condition or has a nonzero absolutely continuous component. The results have motivation in large sample inference for certain nonlinear time series models.