BERRY-ESSEEN BOUNDS OF RECURSIVE KERNEL ESTIMATOR OF DENSITY UNDER STRONG MIXING ASSUMPTIONS

Let {X-i} be a sequence of stationary alpha-mixing random variables with probability density function f (x). The recursive kernel estimators of f (x) are defined by (f) over cap (n)(x) = 1/n root bn Sigma(n)(j=1)b(j)(-1/2) K(x-X-j/b(j)) and (f) over tilde (n)(x) = 1/n Sigma(n)(j=1)1/b(j)K (x-X-j/b(j)), where 0 < b(n) -> 0 is bandwith and K is some kernel function. Under appropriate conditions, we establish the Berry-Esseen bounds for these estimators of f (x), which show the convergence rates of asymptotic normality of the estimators.