  We are interested in investigating the statistical properties of extreme
values for strongly correlated variables. The starting motivation 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 (the real axis in most cases) when performing time-average numerical
simulations and how this relates to the numerical evaluation of the
autocorrelation function.
  We show that correlations decrease the heterogeneity of the maximum values.
Specifically, through numerical simulations we observe that for strongly
correlated variables whose probability distribution function decays like a
power-law $1/x^\alpha$, the maximum distribution has a tail compatible with a
$1/x^{\alpha+2}$ decay, while for i.i.d. variables we expect a $1/x^\alpha$
decay. As a consequence, we also show that the numerically estimated
autocorrelation function converges to the 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.
