WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 25, 2026
Radon-Nikodym Derivative Formula of a Shifted Gaussian Random Field under General Shift
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Abstract: Gaussian process considered in this paper is the Slepian fiel which plays a crucial role in testing for
constancy of parameters of a linear regression with spatial observations. The Slepian fiel appeared as the limit
process of the sequence of moving rectangle processes of recursive residuals. In this work we establish a sufficien
condition for the existence of Radon-Nikodym (R-N) derivative of a shifted two parameters Slepian field More
precisely we determine an infinit dimensional space of shift functions as well as the corresponding norm in which
the derivative of the shifted two parameters Slepian fiel exists. The exact formula of the derivative which is
commonly called Cameron-Martin formula is also derived. The result is obtained by means of the distributional
property of the Slepian random field.
Keywords:
Slepian random field, Wiener integral, Cameron-Martin formula, moving sums process, reproducing kernel, Gaussian random field, Radon-Nikodym derivative, model check, and limit process
Pages: 158-168
DOI: 10.37394/23206.2026.25.15