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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 X_1, ..., X_n be a sequence of independent and identically distributed random variables, each having a mean <X_i>=mu and standard deviation sigma. Define a new variable

X=(X_1+...+X_n)/n.
(1)

Then, as n->infty, the sample mean <x> equals the population mean mu of each variable.

<X>=<(X_1+...+X_n)/n>
(2)
=1/n(<X_1>+...+<X_n>)
(3)
=(nmu)/n
(4)
=mu.
(5)

In addition,

var(X)=var((X_1+...+X_n)/n)
(6)
=var((X_1)/n)+...+var((X_n)/n)
(7)
=(sigma^2)/(n^2)+...+(sigma^2)/(n^2)
(8)
=(sigma^2)/n.
(9)

Therefore, by the Chebyshev inequality, for all epsilon>0,

P(|X-mu|>=epsilon)<=(var(X))/(epsilon^2)=(sigma^2)/(nepsilon^2).
(10)

As n->infty, it then follows that

lim_(n->infty)P(|X-mu|>=epsilon)=0.
(11)

(Khinchin 1929). Stated another way, the probability that the average |(X_1+...+X_n)/n-mu|<epsilon for epsilon an arbitrary positive quantity approaches 1 as n->infty (Feller 1968, pp. 228-229).

See also

Asymptotic Equipartition Property, Central Limit Theorem, Chebyshev Inequality, Frivolous Theorem of Arithmetic, Law of Truly Large Numbers, Strong Law of Large Numbers

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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.

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Weak Law of Large Numbers

Cite this as:

Weisstein, Eric W. "Weak Law of Large Numbers." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WeakLawofLargeNumbers.html

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