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I think so, ie if the sample covariance converges to the real covariance, then the sample mean also converges to the expected value. But I couldn't get a nice proof. Also, I'm not sure if it holds in the almost sure convergence, only in L2 convergence.

Thanks.

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The convergence of the covariances and the convergence of the means are quite unrelated.

Consider $X_t=x(t)+\xi_t$ and $Y_t=y(t)+\eta_t$ for some deterministic $x(t)$ and $y(t)$ and some centered random variables $\xi_t$ and $\eta_t$, then $\mathbb E(X_t)=x(t)$, $\mathbb E(Y_t)=y(t)$ and $\mathrm{Cov}(X_t,Y_t)=\mathbb E(\xi_t\eta_t)$, and the quantities $(x(t),y(t))$ and $\mathbb E(\xi_t\eta_t)$ can be adjusted at will.

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    ?? Please read carefully my answer, you are in for some surprises...2012-12-06