A general method of density estimation for associated random variables

Let {X-n,n greater than or equal to 1} be a sequence of stationary associat ed random variables having a common marginal density function f(x). Let phi (n)(x,y), n = 1,2,..., be a sequence of Borel-measurable functions defined on R-2. Let f(n)(x)= 1/n Sigma(k=1)(n) phi(n)(x,X-k) be the empirical densi ty function. Here we study a set of sufficient conditions under which the p robability Pr(sup(a+delta less than or equal to x less than or equal to b-d elta)\f(n)(x) - f(x)\ > epsilon) --> 0 at an exponential rate as n --> infi nity where the rate possibly depends on epsilon, delta and f and [a, b] is a finite or an infinite interval.