Invariant Random Fields on Spaces with a Group Action by Anatoliy Malyarenko

Author:Anatoliy Malyarenko
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

(4.19)

Luschgy and Pagès (2009) call a sequence of elements {f k ∈B:k≥1} admissible for X if the expansion (4.19) holds true. By part a) of Corollary 1 of Luschgy and Pagès (2009), if for every s, t∈T,

(4.20)

then the sequence {f j ∈E:j≥1} is admissible for X.

The right-hand side of the series (4.17) is the sum of products of independent standard normal random variables by continuous functions. Equation (4.20) follows from Karhunen’s theorem. Therefore, the expansion (4.17) converges uniformly in the centred closed ball of radius R in the space ℝ N .

By the Riesz representation theorem, the dual space, C ∗(T), coincides with the space of finite signed Borel measures on T. The covariance operator of X is defined as a linear operator acting from B ∗ to B by

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