Multidimensional Chebyshev’s inequality
Let be an N-dimensional random variable with mean and covariance matrix
.
If is invertible (i.e., strictly positive), for any :
Proof: is positive, so is.Define the random variable
is positive, then Markov’s inequality holds:
Since is symmetric, a rotation
(i.e., ) and a diagonal matrix
(i.e., ) exist such that
Since is positive .Besides
clearly .
Define .
The following identities hold:
and
then