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).

REFERENCES:

Feller, W. "Laws of Large Numbers." Ch. 10 in An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, pp. 228-247, 1968.

Feller, W. "Law of Large Numbers for Identically Distributed Variables." §7.7 in An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, pp. 231-234, 1971.

Khinchin, A. "Sur la loi des grands nombres." Comptes rendus de l'Académie des Sciences 189, 477-479, 1929.

Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 69-71, 1984.