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You want to work directly with $\overline{X}$, and not some other r.v. with the same distribution, since equivalence in distribution doesn't imply that correlation remains the same. For ease of notation, I'll assume that $\mu = 0$ and $\sigma = 1$. I claim that $$\text{cov}\left(\overline{X},X \right) = \frac{1}{t} \int_0^t s \ ds.$$ Note that this is ...