In probability theory, the multidimensional Chebyshev's inequality[1] 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.
is positive, Markov's inequality holds: Finally, There is a straightforward extension of the vector version of Chebyshev's inequality to infinite dimensional settings[more refs.
[3] Let X be a random variable which takes values in a Fréchet space
This includes most common settings of vector-valued random variables, e.g., when
Suppose that X is of "strong order two", meaning that for every seminorm || ⋅ ||α.
This is a generalization of the requirement that X 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 X (i.e., the vector generalization of the mean), and let be the standard deviation with respect to the seminorm || ⋅ ||α.
The proof is straightforward, and essentially the same as the finitary version[source needed].
If σα = 0, then X is constant (and equal to μ) almost surely, so the inequality is trivial.
The crucial trick in Chebyshev's inequality is to recognize that