Wavelet Linear Density Estimation for a GARCH Model under Various Dependence Structures

We consider n observations from the GARCH-type model: S = σ2Z, where σ2 and Z are independent random variables. We develop a new wavelet linear estimator of the unknown density of σ2 under four different dependence structures: the strong mixing case, the β- mixing case, the pairwise positive quadrant case and the ρ-mixing case. Its asymptotic mean integrated squared error properties are explored. In each case, we prove that it attains a fast rate of convergence.