Functional central limit theorems for

Functional central limit theorems for self-normalized partial sums of linear processes Alfredas Ra£kauskas and Charles Suquet July 10, 2010 Abstract We prove the invariance principle under self-normalization by blocks for linear processes with summable filters and i.i.d. innovations in the domain of attraction of the normal distribution. Keywords: Domain of attraction, invariance principle, linear processes, self-normalization, weak convergence. 1 Introduction and results We consider linear processes Xk = ∑ i?Z aik?i, k ? N, (1) where (ai, i ? N) is a square summable ( ∑ i?Z a 2 i <∞) sequence of real numbers and (i, i ? Z) are i.i.d. centered random variables in the domain of attraction of the normal law (writen 1 ? DAN). This implies in particular that E |i|p <∞ for each 0 < p < 2 and, consequently Xk is well defined (see, e.g., Brockwell and Davis [4]). Central limit theorem for partial sums Sn = X1 + · · · + Xn, n ? N, and functional limit theorems for processes build from partial sums (Sk, k ? N) has been extensively studied in the literature. We refer to the survey paper by Merlevède, Peligrad and Utev [15] for recent results on the central limit theorem and its weak invariance principle for stationary sequences under finite second moment assumption.

• dependent variable


• self-normalized partial


• convergence


• functional central


• n?∞


• banach space


• u?1n ?n


• central limit


• centered random