Role of correlations in the maximum distribution of strongly correlated stationary Markovian processes

We are interested in numerically investigating the statistical properties of extreme values for strongly correlated variables. The main motivation for this study is to understand how the strong-correlation properties of power-law distributed processes affect the possibility of exploring the whole domain of a stochastic process when performing time-average numerical simulations. This problem is relevant when investigating the convergence properties in the numerical evaluation of the autocorrelation function of a stochastic process. In fact, by performing extensive numerical simulations we observe that for power-law correlated variables whose probability distribution function decays like a power-law $1/x^\alpha$, the maximum distribution has a tail compatible with a decay, $1/Z^{\alpha+2}$ while for i.i.d. variables we expect a $1/Z\alpha$ decay. As a consequence, we also show that the numerically estimated autocorrelation function converges to its theoretical prediction according to a factor that depends on the length of the simulated time-series n according to a power-law: $1/n^{\alpha^\delta}$ with $\delta<1$. This accounts for a very slow convergence rate.