Chebyshev's Inequality and the Weak Law of Large Numbers for iid Two-Vectors

Chebyshev's inequality states that if are independent, identically distributed random variables (an iid sample) with common mean and common standard deviation and is the average of these random variables, then An immediate consequence is the weak law of large numbers, which states that as . These results are usually stated for real-valued random variables but also hold for random vectors, provided you interpret all absolute values as Euclidean distances and the variance as . The blue dots in the image are the means of 100 different iid samples from a bivariate normal distribution with mean and standard deviation specified by the locators on the left—is the square of the magnitude of this standard deviation. The orange dot is the common mean, , and the circle shown is centered at with radius . The fraction of blue dots outside the circle will usually be smaller than the theoretical upper bound given in Chebyshev's inequality—in many instances this bound is very crude.

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It should be noted that Chebyshev's inequality and the weak law hold for any underlying distribution—the bivariate normal is used for convenience.
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