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.

-dimensional random vector with expected value and covariance matrix

is a positive-definite matrix, for any real number:

Proof

. Define the random variable

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[more refs. needed]. Let be a random variable which takes values in a Fréchet space (equipped with seminorms is a Banach space (equipped with a single norm), 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.

be the Pettis integral of (i.e., the vector generalization of the mean), and let

be the standard deviation with respect to the seminorm . In this setting we can state the following:

1. General version of Chebyshev's inequality.

Proof. The proof is straightforward, and essentially the same as the finitary version[source needed]. If , then is constant (and equal to ) almost surely, so the inequality is trivial.

If

then , so we may safely divide by .

The following calculations complete the proof:

References