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Sample averages of iid variables converge in probability to the mean.
Weak law of large numbers: Let be an iid sequence of random variables with expectation and finite variance . Define the sample mean
Then for every ,
This is a convergence-in-probability statement on the underlying probability space. A standard proof uses Chebyshev's inequality applied to , and the result is weaker than the strong law of large numbers, which upgrades the mode of convergence.
A sequence of i.i.d. random variables is a sequence of random variables on a common probability space such that the family is independent and the variables are identically distributed (equivalently, each has the same distribution law as ).
A random variable is a measurable function defined on a probability space , meaning that for every Borel set one has .
A variance of a random variable is the quantity
defined when (so in particular the expectation is finite).
A probability space is a triple where is a set, is a sigma-algebra on , and is a probability measure on .
Elements of are the events whose probabilities are evaluated by , and a random variable is a measurable function defined on .
Chebyshev's inequality: Let be a random variable with expectation and finite variance . Then for every ,
Strong law of large numbers: Let be an iid sequence of random variables such that , and let expectation . Define the sample mean
Then
i.e., almost surely.
This strengthens the weak law of large numbers by replacing convergence in probability with almost-sure convergence. The phrase “almost surely” is the probability-theory analogue of almost-everywhere convergence with respect to the probability measure.