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

Abstract

Consider a sequence X k = ∞ j=0 cj ξ k−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 S k = k j=1 Xj . Under suitable conditions on the constants cj it is known that for a suitable normalizing constant γn, the partial sum process γ −1 n S [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 −1 βnis 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 −1 βn → 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.

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