The Lindeberg-Lévy theorem for martingales
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The central limit theorem of Lindeberg [7] and Levy [3] states that if {mi, m2, ■ ■ • } is an independent, identically distributed sequence of random variables with finite second moments, then the distribution of ra-1'2^^! uk approaches the normal distribution with mean 0 and variance £{m?} , assuming that £{mi} =0, which entails no loss of generality. In the following result, the assumption of indepence is weakened.
[1] J. Lindeberg. Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung , 1922 .
[2] J. Doob. Stochastic processes , 1953 .
[3] Patrick Billingsley,et al. Statistical inference for Markov processes , 1961 .