## 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 μ_{w} as a catalyst; that is, we compute derivatives with respect to μ_{w} and then use the chain rule: dμ/dν = (dμ/dμ_{w}) / (dν/dμ_{w}). 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.

#### Recommended Citation

Shepp, L. A.
(1966).
Radon-Nikodym Derivatives of Gaussian Measures.
*Annals of Mathematical Statistics,*
*37*
(2),
321-354.
http://dx.doi.org/10.1214/aoms/1177699516

**Date Posted:** 27 November 2017