Weak Law of Large Numbers

The weak law of large numbers (cf. the strong law of large numbers) is a result in probability theory also known as Bernoulli's theorem. Let , ..., be a sequence of independent and identically distributed random variables, each having a mean and standard deviation . Define a new variable

(1)

Then, as , the sample mean equals the population mean of each variable.

			(2)
			(3)
			(4)
			(5)

In addition,

			(6)
			(7)
			(8)
			(9)

Therefore, by the Chebyshev inequality, for all ,

(10)

As , it then follows that

(11)

(Khinchin 1929). Stated another way, the probability that the average for an arbitrary positive quantity approaches 1 as (Feller 1968, pp. 228-229).

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