Local times for stochastic processes which are subordinate to Gaussian processes

Abstract

Let X and Y be random vectors of the same dimension such that Y has a normal distribution with mean vector O and covariance matrix R. Let g( x), x≥0, be a bounded nonincreasing function. X is said to be g-subordinate to Y if | Ee i u′X | ≤ g( u′Ru) for all real vectors u of the same dimension as X. This is used to define the g-subordination of a real stochastic process X( t), 0 ≤ t ≤ 1, to a Gaussian process Y( t), 0 ≤ t ≤ 1. It is shown that the basic local time properties of a given Gaussian process are shared by all the processes that age g-subordinate to it. It is shown in particular that certain random series, including some random Fourier series, are g-subordinate to Gaussian processes, and so have their local time properties.