Are there any interesting results known about integer and fractional parts of $\xi \in \mathcal N(0, \sigma^2)$? In particular, I am interested in their expected values and covariance matrix.
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Are the integer and fractional parts of $-2.4$ equal to $-3$ and $+0.6$ or equal to $-2$ and $-0.4$?. If the former then expected values will be $-\frac12$ and $+\frac12$. If the latter, then the expected values will be $0$ and $0$. – Henry Feb 02 '16 at 14:13
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Partially answered here. – Lee David Chung Lin Feb 06 '20 at 06:56