  We analyze how the choice of the sampling weight affects the efficiency of
the Monte Carlo evaluation of classical time autocorrelation functions.
Assuming uncorrelated sampling or sampling with constant correlation length, we
propose a sampling weight for which the number of trajectories needed for
convergence is independent of the correlated quantity, dimensionality,
dynamics, and phase-space density. In contrast, it is shown that the
computational cost of the "standard" intuitive algorithm which samples directly
from the phase-space density may scale exponentially with the number of degrees
of freedom. Yet, for the stationary Gaussian distribution of harmonic systems
and for the autocorrelation function of a linear function of phase-space
coordinates, the computational cost of this standard algorithm is also
independent of dimensionality.
