Growth rates of sample covariances of stationary symmetric \(\alpha \)-stable processes associated with null recurrent Markov chains (Q1613597)

A canonical class of stochastic processes with infinite variance consists of symmetric \(\alpha \)-stable (S\(\alpha \)S) processes of the form \[ \int _{\Omega }{f_t(\omega)M(\omega)},\;\;\;\;\;\;t=1,2,\dots , \] where \(M\) is an (S\(\alpha \)S) random measure on \(\Omega \) with a \(\sigma \)-finite control measure \(m\), and \(0<\int _{\Omega }{|f_t(\omega)|^{\alpha }m(d\omega)}<\infty \) for all \(t\) and \(0<\alpha <2\). Such a process is associated with a null recurrent Markov chain. The authors show that it are the properties of the Markov chain that most noticeably determine the properties of the resulting stationary (S\(\alpha \)S) process. The resulting process exhibits such strong dependence that its sample covariance grows at a surprising rate which is slower than one would expect based on the fatness of the marginal distribution tails. An additional feature of the process is that the sample autocorrelations converge to non-random limits.