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

Abstract

Consider a sequence X_k=∑_j=0^∞c_jξ_k−j, k≥1, where c_j, j≥0, is a sequence of constants and ξ_j, −∞<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 S_k=∑_j=1^kX_j. Under suitable conditions on the constants c_j it is known that for a suitable normalizing constant γ_n, the partial sum process γ_n⁻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 X_k. In this paper it is established that the process n⁻¹β_n∑_k=1^[nt]f(β_n(γ_n⁻¹S_k+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 β_n→∞ such that n⁻¹β_n→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.