The Lindeberg-Lévy theorem for martingales

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.