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

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