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Strong Law of Large Numbers


The sequence of variates X_i with corresponding means mu_i obeys the strong law of large numbers if, to every pair epsilon,delta>0, there corresponds an N such that there is probability 1-delta or better that for every r>0, all r+1 inequalities

 (|S_n-m_n|)/n<epsilon
(1)

for n=N, N+1, ..., N+r will be satisfied, where

S_n=sum_(i=1)^(n)X_n
(2)
m_n=<S_n>=mu_1+...+mu_n
(3)

(Feller 1968). Kolmogorov established that the convergence of the sequence

 sum(sigma_k^2)/(k^2),
(4)

sometimes called the Kolmogorov criterion, is a sufficient condition for the strong law of large numbers to apply to the sequence of mutually independent random variables X_k with variances sigma_k (Feller 1968).


See also

Frivolous Theorem of Arithmetic, Law of Large Numbers, Law of Truly Large Numbers, Strong Law of Small Numbers

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References

Feller, W. "The Strong Law of Large Numbers." §10.7 in An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, pp. 243-245, 1968.Feller, W. "Strong Laws for Martingales." §7.8 in An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, pp. 234-238, 1971.

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Strong Law of Large Numbers

Cite this as:

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

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