Liapunov Theorem

in probability theory a theorem that asserts certain extremely general sufficiency conditions for the convergence of the distribution of sums of independent random variables to a normal law. It was formulated and proved by A. M. Liapunov in 1901. The Liapunov theorem completed the investigations of P. L. Chebyshev, A. A. Markov (the elder), and Liapunov himself along what proved to be a fundamental line of research for probability theory as a whole.

A precise formulation of the Liapunov theorem is as follows. Let the independent random variables X₁,. .., X_n, … have finite mathematical expectations EX_k, dispersions DX_k, and, for δ > 0, absolute moments E|X_k-EX_k|^2+δ and let B_n = Σⁿ_k=1 be the dispersion of the sum X₁,… ,Xⁿ. We assert that if for some δ > 0

(the Liapunov condition), then the probability of the inequality

approaches, as n→∞, the limit

uniformly for all values of x₁ and x₂. Liapunov also provided an estimate of the rate of convergence in the Liapunov theorem. Subsequently, conditions were established that expanded the Liapunov condition and are not only sufficient but, in some sense, necessary.