Multidimensional Chebyshev's inequality

In probability theory, the multidimensional Chebyshev's inequality is a generalization of Chebyshev's inequality, which puts a bound on the probability of the event that a random variable differs from its expected value by more than a specified amount.
Let be an -dimensional random vector with expected value and covariance matrix
If is a positive-definite matrix, for any real number :

Proof

Since is positive-definite, so is. Define the random variable
Since is positive, Markov's inequality holds:
Finally,

Infinite dimensions

There is a straightforward extension of the vector version of Chebyshev's inequality to infinite dimensional settings. Let be a random variable which takes values in a Fréchet space . This includes most common settings of vector-valued random variables, e.g., when is a Banach space, a Hilbert space, or the finite-dimensional setting as described above.
Suppose that is of "strong order two", meaning that
for every seminorm. This is a generalization of the requirement that have finite variance, and is necessary for this strong form of Chebyshev's inequality in infinite dimensions. The terminology "strong order two" is due to Vakhania.
Let be the Pettis integral of, and let
be the standard deviation with respect to the seminorm. In this setting we can state the following:
Proof. The proof is straightforward, and essentially the same as the finitary version. If, then is constant almost surely, so the inequality is trivial.
If
then, so we may safely divide by. The crucial trick in Chebyshev's inequality is to recognize that.
The following calculations complete the proof: