Epsilon Entropy of Gaussian Processes

Abstract

This paper shows that the epsilon entropy of any mean-continuous Gaussian process on L2[ 0, 1 ] is finite for all positive ε. The epsilon entropy of such a process is defined as the infimum of the entropies of all partitions of L2[ 0, 1 ] by measurable sets of diameter at most ε, where the probability measure on L2 is the one induced by the process. Fairly tight upper and lower bounds are found as ε → 0 for the epsilon entropy in terms of the eigenvalues of the process.