Asymptotic of spectral covariance for linear random fields with infinite variance

Abstract

In the paper we continue to investigate measures of dependence for random variables with infinite variance. The asymptotic of spectral covariance $\rho (X_{(0,0)}, X_{(k_1,k_2)})$ for linear random field $X_{k,l}=\sum_{i,j=0}^\infty c_{i,j}\epsilon_{k-i, l-j}, \ (k, l)\in \mathbb{Z}^2,$ with special form of filter $\{c_{i,j}\}$ and with innovations $\{ \epsilon_{i, j}\}$ having infinite second moment is investigated. Different behavior of $\rho (X_{(0,0)}, X_{(k_1,k_2)})$ is obtained in the cases $n\to \infty, \ m\to \infty$ and $n\to \infty, \ m\to -\infty$, the latter case being much more complicated.