@ACuriousMind From our brief discussion yesterday...In Sakurai he gives the correlation amplitude as $$C(t) = \int dE |g(E)|^2 \rho(E) \text{exp}\big( \frac{-iEt}{\hbar} \big)$$
What is the effect on the correlation amplitude $C(t)$ for large $t$? I see how large $t$ implies that the integrand oscillates more as a function of $E$ but how does this imply that $C(t)$ decreases for large $t$, which is what I assume it does?