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 "X" be an "N"-dimensional random vector with expected value formula_1 and covariance matrix If formula_3 is a positive-definite matrix, for any real number formula_4: Proof. Since formula_3 is positive-definite, so is formula_7. Define the random variable Since formula_9 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 formula_12 (equipped with seminorms ). This includes most common settings of vector-valued random variables, e.g., when formula_12 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. Let formula_15 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: Proof. The proof is straightforward, and essentially the same as the finitary version. If , then is constant (and equal to ) 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 formula_19. The following calculations complete the proof: