THE STRONG UNIFORM CONSISTENCY OF KERNEL DENSITY ESTIMATES FOR φ—MIXING SAMPLE

Let X₁, X₂, … be a φ-mixing Sample with density function f（x）, f_n（x₁, K_n）=\frac1n h_n \sum_i=1^n K_n\left(\fracx-X_ih_n\right), be an estimate of the f（x）, under proper conditions, we obtain he best convergence rate \sup _\epsilon \in J\left|f_n\left(\infty, K_n\right)-f(x)\right|=O\left((n / \log n)^-1 / 3\right), a.s..