Law of the iterated logarithm for stationary processes

Abstract

There has been recent interest in the conditional central limit question for (strictly) stationary, ergodic processes …, X₋₁, X₀, X₁, … whose partial sums S_n=X₁+⋯+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_n²)=o(n). Here we explore the law of the iterated logarithm (LIL) for the same class of processes. Letting ‖⋅‖ denote the norm in L²(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(S_n|X₀, X₋₁, …)‖ be summable. A sufficient condition for the LIL is only slightly stronger, requiring n^−3/2log^3/2(n)‖E(S_n|X₀, 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.