Lindeberg's condition
In probability theory, Lindeberg's condition is a sufficient condition (and under certain conditions also a necessary condition) for the central limit theorem (CLT) to hold for a sequence of independent random variables. Unlike the classical CLT, which requires that the random variables in question have finite variance and be both independent and identically distributed, Lindeberg's CLT only requires that they have finite variance, satisfy Lindeberg's condition, and be independent. It is named after the Finnish mathematician Jarl Waldemar Lindeberg. Statement. Let formula_1 be a probability space, and formula_2, be "independent" random variables defined on  that space. Assume the expected values formula_3 and variances formula_4 exist and are finite. Also let formula_5 If this sequence of independent random variables formula_6 satisfies Lindeberg's condition: for all formula_8, where 1 is the indicator function, then the central limit theorem holds, i.e. the random variables converge in distribution to a standard normal random variable as formula_10 Lindeberg's condition is sufficient, but not in general necessary (i.e. the inverse implication does not hold in general). However, if the sequence of independent random variables in question satisfies then Lindeberg's condition is both sufficient and necessary, i.e. it  holds if and only if the result of central limit theorem holds. Remarks. Feller's theorem. Feller's theorem can be used as an alternative method to prove that Lindeberg's condition holds. Letting formula_12 and for simplicity formula_13, the theorem states This theorem can be used to disprove the central limit theorem holds for formula_6 by using proof by contradiction. This procedure involves proving that Lindeberg's condition fails for formula_6. Interpretation. Because the Lindeberg condition implies formula_21 as formula_22, it guarantees that the contribution of any individual random variable formula_6 (formula_24) to the variance formula_25 is arbitrarily small , for sufficiently large values of formula_26.