Central Limit Theorem by Higher Order Correlation Coefficients

René Blacher (1)

1 IPS - Inférence Processus Stochastiques

The higher order correlation coefficients are able to detect any dependence. So, in a previous paper, we obtained conditions about these coefficients equivalent to the convergence of moments. We have deduced a central limit theorem with minimal assumptions. However, it was assumed that all random variables have the same distribution. In this report, we remove this condition. This allows us to reduce the assumptions necessary for the convergence of moments for martingales and even to replace this assumption by a weaker hypothesis. On the other hand, we shall prove that these assumptions can be simplified when the random variables are bounded. On the other hand, we will compare the different assumptions of asymptotic independence between them, in particular, strong mixing condition, weak dependence and condition $ H_ {mI} $ which we introduced in a previous paper We understand that it is this condition $ H_ {mI}$ which is closest to the minimum conditions to ensure asymptotic normality. Finally, we see that, if one has a process whose moments converge, moments converge also for almost all processes which has only the same multilinear correlation coefficients that the first process.

Mots clés: Central Limit Theorem; moments; strongly mixing sequence; weak dependence; martingale; dependence density; higher order correlation coefficients

Domaines: Probabilités [math.PR]

René Blacher. Central Limit Theorem by Higher Order Correlation Coefficients. [Research Report] LJK. 2011. ⟨hal-00600639⟩