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Lyapunov Condition

The Lyapunov condition, sometimes known as Lyapunov's central limit theorem, states that if the (2+epsilon)th moment (with epsilon>0) exists for a statistical distribution of independent random variates x_i (which need not necessarily be from same distribution), the means mu_i and variances sigma_i^2 are finite, and

r_n^(2+epsilon)=sum_(i=1)^n<|x_i-mu_i|^(2+epsilon)>,
(1)

then if

lim_(n->infty)(r_n)/(s_n)=0,
(2)

where

s_n^2=sum_(i=1)^nsigma_i^2,
(3)

the central limit theorem holds.

See also

Central Limit Theorem

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References

Ash, R. B. and Doléans-Dade, C. A. Probability & Measure Theory, 2nd ed. New York: Academic Press, p. 307, 1999.Billingsley, P. Probability and Measure, 2nd ed. New York: p. 371, Wiley, 1986.Resnik, S. A Probability Path. Boston, MA: Birkhäuser, p. 319, 1999.

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Lyapunov Condition

Cite this as:

Weisstein, Eric W. "Lyapunov Condition." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LyapunovCondition.html

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