Necessary and sufficient conditions for a second-order Wiener-Itô integral process to be mixing

Let (Xs,s[set membership, variant]) be a stationary Gaussian process with spectral measure [sigma], time-shift operator U, and the associated pth order multiple Wiener-Itô integrals,Ip,p = 1,2,..., defined on their domains L2([sigma]p,sym). Let f[set membership, variant]L2([sigma]p,sym). We give a necessary and sufficient spectral condition for the stationary process (Us(Ipf),s[set membership, variant]) to be mixing in the case p = 2; a simplified sufficient condition is given for f of the form f = g1[circle times operator]h1+g2[circle times operator]h2+...+gn[circle times operator]hn, where gi, hi[set membership, variant]L2([sigma]1,sym). Similar results are obtained in the case p = 4. A necessary and sufficient spectral condition is given for (Us(Ip(h[circle times operator]...[circle times operator]h)),sisin;) to be mixing, for any p[greater-or-equal, slanted]1 and h[set membership, variant]L2([sigma]1,sym). An example of a non-mixing stationary Gaussian process with a mixing factor process is given.