Law of the iterated logarithm for stationary processes

By Ou Zhao, Michael Woodroofe. 2006

51 citations recorded by the source corpus.

There has been recent interest in the conditional central limit question for (strictly) stationary, ergodic processes X -1 , X 0 , X 1 ,... whose partial sums S n = X 1 +... + X n are of the form S n = M n + R n , where M n is a square integrable martingale with stationary increments and R n is a remainder term for which E(R n 2 ) = o(n). Here we explore the law of the iterated logarithm (LIL) for the same class of processes. Letting ∥ · ∥denote the norm in L 2 (P), a sufficient condition for the partial sums of a stationary process to have the form S n = M n + R n is that n -3/2 ∥ E(Sn|X 0 , X -1 ,...)∥ be summable. A sufficient condition for the LIL is only slightly stronger, requiring n -3/2 log 3/2 (n)∥E(Sn|X 0 ,X -1 ,...)|| to be summable. As a by-product of our main result, we obtain an improved statement of the conditional central limit theorem. Invariance principles are obtained as well.

Subfields: Finance, Management Science and Operations Research, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ad985fb0b3e5ebcfc733cb1e508ee05c019ac381

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Citations in the corpus

STRONG INVARIANCE PRINCIPLES FOR DEPENDENT RANDOM VARIABLES (2007, methodology): “Zhao and Woodroofe [72] obtained a LIL by letting l̃(i) = log i in (14).”