Estimates for the moment-generating function for stationary random processes (Q1819814)

Let \(\{\xi\) (t), \(t\in T\}\) be an N-dimensional strictly stationary process. Let \(<.,.>\) be the scalar product. Define \(\xi =\xi (0)\). The moment generating function \(f(z)=E \exp (<z,\xi >)\) can be estimated by \[ f_{\mu,t}=\tau^{-1}\int^{\tau}_{0}\exp \{<z,\xi (t)>\}d\mu (t), \] where \(\mu\) is the counting measure if T is a set of all integers and \(\mu\) is the Lebesgue measure if \(T={\mathbb{R}}\). The author derives conditions, under which \(f_{\mu,\tau}\) is a strictly consistent estimate with an asymptotic normal distribution.