Sample correlations of infinite variance time series models: An empirical and theoretical study (Q1271243)

Let \(\{X_n\}\) be a stationary sequence. Assume that \(var X_n=\infty\) and define the sample correlation function (SCR) \(\hat{\rho}_H(h)=\sum_{j=1}^{n-h} X_j X_{j+h} \bigl/ \sum_{j=1}^n X_j^2\bigr.\) (the index \(H\) indicates that it corresponds to a heavy tailed distribution). In certain cases SCR converges in probability to a constant, but not always. The authors simulated several heavy tailed stationary sequences to understand when SCR converges and when it does not. The simulated models include: (i) sum of two moving averages, (ii) mixed moving average \(S\alpha S\) process \(X_n=\int \int f(n-x,s) M(dx,ds)\) where \(M\) is an \(S\alpha S\) random measure, (iii) random walk, (iv) permutation process \(X_n=\sum _{i=0} ^{\infty} \psi_i^{(n)} Z_{n-i}\) arising from a linear process \(X_0=\sum_{j=0}^{\infty} \psi_j Z_{-j}\) in such a way that \(\{\psi_i^{(n)}\}\) is a permutation of \(\{\psi_j\}\). In the cases (i) and (iv) theoretical explanations for a random limit of SCR are derived.