Central limit theorem by polynomial dependence coefficients

We study the asymptotic behaviour of normalized sums of n random variables ∑n = n−12(X1 + X2 + … + Xn). We obtain a necessary and sufficient condition in order that the moments of ∑n converge to the moments of a normal distribution. This condition is expressed by means of polynomial dependence coefficients ϱj1,j2, … ,jn. Then, one can obtain central limit theorems with minimal assumptions.