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 formula_1 be an formula_2-dimensional random vector with expected value formula_3 and covariance matrix
 formula_4
If formula_5 is a positive-definite matrix, for any real number formula_6:
formula_7
Proof.
Since formula_5 is positive-definite, so is formula_9. Define the random variable
formula_10
Since formula_11 is positive, Markov's inequality holds:
formula_12
Finally,
formula_13
Infinite dimensions.
There is a straightforward extension of the vector version of Chebyshev's inequality to infinite dimensional settings[more refs. needed]. Let X be a random variable which takes values in a Fréchet space formula_14 (equipped with seminorms || ⋅ ||"α"). This includes most common settings of vector-valued random variables, e.g., when formula_14 is a Banach space (equipped with a single norm), a Hilbert space, or the finite-dimensional setting as described above.
Suppose that X is of "strong order two", meaning that
 formula_16
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.
Let formula_17 be the Pettis integral of X (i.e., the vector generalization of the mean), and let
formula_18
be the standard deviation with respect to the seminorm || ⋅ ||"α". In this setting we can state the following:
General version of Chebyshev's inequality. formula_19
Proof. 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.
If
formula_20
then ||"X" − "μ"||"α" ; 0, so we may safely divide by ||"X" − "μ"||"α". The crucial trick in Chebyshev's inequality is to recognize that formula_21.
The following calculations complete the proof:
formula_22
References.
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