  The momentum or velocity autocorrelation function C(t) for a tagged
oscillator in a finite harmonic system decays like that of an infinite system
for short times, but exhibits erratic behavior at longer time scales. We
introduce the autocorrelation function of the long-time noisy tail of C(t) ("a
correlation of the correlation"), which characterizes the distribution of
recurrence times. Remarkably, for harmonic systems with same-mass particles
this secondary correlation may coincide with the primary correlation C(t) (when
both functions are normalized) either exactly, or over a significant initial
time interval. When the tagged particle is heavier than the rest, the equality
does not hold, correlations shows non-random long-time scale pattern, and
higher order correlations converge to the lowest normal mode.
