Continuity Conditions for a Class of Gaussian Chaos Processes Related to Continuous Additive Functionals of Lévy Processes

Michael B. Marcus and Michel Talagrand
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Michael B. Marcus: City College of New York
Michel Talagrand: Université Paris VI

Journal of Theoretical Probability, 1998, vol. 11, issue 1, 157-179

Abstract: Abstract Let $$\left\{ {a_k } \right\}_{k \in Z^n } {\text{ and }}\left\{ {b_k } \right\}_{k \in Z^n }$$ be sequences of real numbers which are symmetric in k. Let $$\left\{ {g_k } \right\}_{k \in Z^n } {\text{ and }}\left\{ {g\prime _k } \right\}_{k \in Z^n }$$ be independent sequences of independent normal random variables with mean zero and variance one. For each fixed choice of $$\left\{ {a_k } \right\}_{k \in Z^n } {\text{ and }}\left\{ {b_k } \right\}_{k \in Z^n }$$ we consider $$Q\left( x \right) = \sum\limits_{j,k \in Z^n } {a_k a_j g_k g_j \prime b_{k - j} e^{i\left( {k - j} \right)x} } { }x \in \left[ {0,2{\pi }} \right]^n$$ Let $$d_2 \left( {x,y} \right) = \left( {E\left| {Q\left( x \right) - Q\left( y \right)} \right|^2 } \right)^{1/2}$$ Several examples are given in which the condition $$\int_0^\infty {\left( {\log N_{d_2 } \left( {\left[ {0,2{\pi }} \right]^n ,\varepsilon } \right)} \right)^{1/2} d} \varepsilon

Keywords: Gaussian processes; Lévy processes; additive functionals (search for similar items in EconPapers)
Date: 1998
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Citations: View citations in EconPapers (1)

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DOI: 10.1023/A:1021699009373

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