No, you need more than that, for example, if $X_i$ are iid rvs taking values $\pm 1$ with prob 1/2 each and $Y_{2i} = X_i, Y_{2i+1} = -X_i$. If you avoid next door neighbors these are independent, so they have covariance that declines exponentially, but the partial sums are $\pm 1$
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