The Estimation of the Fourth-Order Cumulant for Dependent Data: Consistency and Asymptotic Normality

Let {X
_i
} be a stationary dependent random process with finite eight-order moments. For broad classes of processes (¿-mixing and strongly mixing), we obtain the convergence in probability, with sharp rates, of the estimate of the fourth-order cumulant from
n
observations {X
_i
}
_i=1

ⁿ
. We also establish the asymptotic distribution of the estimation error. The asymptotic expression of the variance is explicitly specified.