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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?

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    $\begingroup$ Trivial observation to address the first question: when $Y=X$ the condition is satisfied for any non-negative $C$ and obviously $Y$ is normal. Suggestion: draw pictures of the situation when $X$ is univariate. They will make it easy to see that the answers to the questions are "Yes," "A little," and "A little" (in the sense of some bounds). $\endgroup$ – whuber Mar 26 '12 at 20:36
  • $\begingroup$ Thanks, how can I compute these bounds? What should I be looking at? $\endgroup$ – sks Mar 26 '12 at 20:41
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    $\begingroup$ Write $Y$ as $X+U$ where $U$ is distributed over the ball of radius $C$. This should tell you something about the mean of $Y$. Same thing about the covariance. $\endgroup$ – Xi'an Mar 26 '12 at 20:52
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    $\begingroup$ Nice idea, X'ian. Note, however, that this approach will not give the full range of possibilities for the covariance: for that to occur, you need $U$ to depend on $X$. $\endgroup$ – whuber Mar 26 '12 at 21:15
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    $\begingroup$ @whuber: ah! Strictly speaking, : "$U$ distributed over the ball of radius $C$" does not exclude $U$ dependent on $X$! Even though in truth I was not considering this case... $\endgroup$ – Xi'an Mar 28 '12 at 9:23
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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:

Figure 1

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 $C$:

Figure 2

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.

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  • $\begingroup$ Wow, this is very detailed, thanks a lot! I think this is exactly what I was looking for. Now I just have to generalize this to the multivariate case :) the good thing is that there are no off-diagonal elements - individual components are all independent. $\endgroup$ – sks Mar 26 '12 at 21:16
  • $\begingroup$ I suspect you can get best bounds on the diagonal elements of the covariance using the same approach. But to bound the off-diagonal elements will be much more difficult. The full bounds will be connected sets of symmetric positive-definite matrices potentially having complicated shapes. $\endgroup$ – whuber Mar 26 '12 at 21:18

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