Statistics Papers

Document Type

Journal Article

Date of this Version

1966

Publication Source

Annals of Mathematical Statistics

Volume

37

Issue

2

Start Page

321

Last Page

354

DOI

10.1214/aoms/1177699516

Abstract

We give simple necessary and sufficient conditions on the mean and covariance for a Gaussian measure to be equivalent to Wiener measure. This was formerly an unsolved problem [26].

An unsolved problem is to obtain the Radom-Nikodym derivative dμ/dν where μ and ν are equivalent Gaussian measure [28]. We solve this problem for many cases of μ and ν, by writing dμ/dν in terms of Fredholm determinants and resolvents. The problem is thereby reduced to the calculation of these classical quantities, and explicit formulas can often be given.

Our method uses Wiener measure μw as a catalyst; that is, we compute derivatives with respect to μw and then use the chain rule: dμ/dν = (dμ/dμw) / (dν/dμw). Wiener measure is singled out because it has a simple distinctive property--the Wiener process has a random Fourier-type expansion in the integrals of any complete orthonormal system.

We show that any process equivalent to the Wiener process W can be realized by a linear transformation of W. This transformation necessarily involves stochastic integration and generalizes earlier nodulation transformations studied by Legal [21] and others [4], [27].

New variants of the Wiener process are introduced, both conditioned Wiener processes and free n-fold integrated Wiener processes. We given necessary and sufficient conditions for a Gaussian process to be equivalent to any one of the variants and also give the corresponding Radon-Niels (R-N) derivative.

Last, some novel uses of R-N derivatives are given. We calculate explicitly: (i) the probability that W cross a slanted line in a finite time, (ii) the first passage probability for the process W (T + 1) − W(t), and (iii) a class of function space integrals. Using (iii) we prove a zero-one law for convergence of certain integrals on Wiener paths.

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Date Posted: 27 November 2017