Law of the Iterated Logarithm for Weighted Sums of Functionals of Long-Memory Gaussian Sequences

Abstract

We establish laws of the iterated logarithm for weighted sums of linear and nonlinear functionals of long-memory stationary Gaussian sequences. The leading Wiener chaos determines the scale of the almost-sure fluctuations; all higher chaoses are negligible on it and the exponent of the iterated-logarithm factor is half the Hermite rank. In the linear case, we obtain the exact LIL constant, and it is the classical one. In the nonlinear case, we identify a functional cluster set and give a variational formula for the LIL constant. We also present explicit upper and lower bounds for this constant. The lower bound is expressed through beta functions and depends on the memory parameter, the weight exponent, and the Hermite rank; numerically it matches the constant to within a fraction of one percent, and can therefore be used in its place. Thus, the weights affect the normalization, the geometry of the limiting cluster set, and the nonlinear LIL constant itself.