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Will the integral of the PSD of a down-sampled signal still equal the variance in all cases

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No. The aliased component will interfere with the non-aliased components and the interference can constructive or destructive.

Trivial example:

$$x[n] = \sin\left(\frac\pi2n\right)$$

If you down sample this to $y[n] = x[2n]$, you get all zeros.

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