Conditions on Normal Random Variable Let $X$ be a multivariate normal random variable with known mean and covariance, and $Y$ be another random variable. We know for sure $\|X−Y\|_2\le C$ where $C$ is a known constant. A few questions: Is it possible for $Y$ to be normally distributed? Can we say anything about its mean and/or covariance? If we know the mean of $Y$, can we say something about its covariance?
 A: To get a feeling for the question, consider the univariate case.  We can always choose an origin and unit of measurement to make $X$ a standard normal variate (with mean $0$ and unit variance).
The mean of $Y$ (call it $\nu$) can be anywhere between $-C$ and $C$ simply by setting $Y = X + \nu$.  Clearly $\|X-Y\| = |X-Y| = |\nu| \le C$.  This works in the other direction, though: when the mean of $Y$ exceeds $X+C$, then $E[Y-X] \gt C$ implies $\Pr(|X-Y| \gt C) \gt 0$, violating the assumption.  This gives definite, clean bounds on the mean.
We can maximize the variance of $Y$ by setting $Y=X+C$ when $X \gt 0$ and $Y=X-C$ when $X \lt 0$ (and choose whichever you want when $X=0$, because this zero-probability event does not contribute to the variance).  In other words, we spread $Y$ as far from the mean as we possibly can.  Doing the integral, we find this makes the variance as large as $1+C^2 + 2C\sqrt{\frac{2}{\pi}}$.
To minimize the variance of $Y$, I believe we should set $Y=0$ whenever $-C \lt X \lt C$ and otherwise set $Y=X-C$ for $X \ge C$ and $Y=X+C$ for $X \le -C$.  In other words, we push $Y$ towards the mean as much as we can.  By symmetry of construction, the mean of $Y$ is zero, whence its variance is the expectation of $Y^2$, equal to
$$\text{Var}(Y) = 2\int_C^\infty (x-C)^2 \phi(x)dx = \frac{-2C \exp(-C^2/2)}{\sqrt{2\pi}}+(1+C^2)\text{erfc}(\frac{C}{\sqrt{2}})$$
($\phi$ is the standard normal PDF).  A plot of this variance against $C$ is informative:

This makes intuitive sense: as $C$ grows above $0$, we may place more and more of the probability of $Y$ right on $0$, leaving all the contribution to $Y$'s variance from the tail of a normal distribution, which decreases rapidly.
Putting these results together yields a plot of the allowable interval of variances of $Y$ for small values of $C$:

The lower curve is the same as before; the upper curve is part of a parabola: it grows quadratically.
The analysis in the multivariate case will be the same for the mean--that is computed separately for the marginal distributions--but becomes much more complicated for the covariance, especially its off-diagonal elements.  A full, exact answer may be difficult even to write down.
