Weak-stationarity conditions for wavelet processes (Q5942199)

Let \(\{X(t)\}\) be a zero mean process with a finite second moment. We call it a wavelet process if \(X(t)=\int |a|^{-1/2} \psi[(t-b)/a] dZ(a,b)\) where \(\psi\) is a measurable function (``mother wavelet'') and \(Z\) is an orthogonal random measure. It is assumed that the \(\sigma\)-finite measure \(\mu\) corresponding to \(Z\) through the formula \(E Z(S)\overline{Z(T)}=\mu(S\cap T)\) is a product measure \(\mu=\mu_1\times\mu_2\) where \(\mu_2\) is allowed to have only continuous and discrete parts. More precisely, \(\mu_2\) is characterized by \(\mu_2(S)=\int_S g(b) db +\sum_{kT\in S} g_k\) where \(T>0\) is a constant independent of \(S\). Define NEWLINE\[NEWLINEK_v(b) = \int |a|^{-1}\psi[(b+v)/a]\overline{\psi[(b-v)/a]} d\mu_1(a).NEWLINE\]NEWLINE Then the covariance function \(r(s,t)=E X(s)\overline{X(t)}\) can be written in the form NEWLINE\[NEWLINEr(u+v,u-v) = \int K_v(u-b) g(b) db +\sum_k g_k K_v(u-kT).NEWLINE\]NEWLINE The author presents criteria for weak stationarity of \(\{X(t)\}\), i.e., for the case when \(r(u+v,u-v)\) does not depend on \(u\).