## On Almost Sure Convergence Rates for the Kernel Estimator of a Covariance Operator Under Negative Association

**Authors:** H. Jabbari1, M. Erfaniyan

Let fXn; n 1g be a strictly stationary sequence of negatively associated random
variables, with common continuous and bounded distribution function F. We consider
the estimation of the two-dimensional distribution function of (X1;Xk+1) based on kernel
type estimators as well as the estimation of the covariance function of the limit empirical
process induced by the sequence fXn; n 1g where k 2 IN0. Then, we derive uniform
strong convergence rates for the kernel estimator of two-dimensional distribution function
of (X1;Xk+1) which were not found already and do not need any conditions on the covari-
ance structure of the variables. Furthermore assuming a convenient decrease rate of the
covariances Cov(X1;Xn+1); n 1, we prove uniform strong convergence rate for covari-
ance function of the limit empirical process based on kernel type estimators. Finally, we
use a simulation study to compare the estimators of distribution function of (X1;Xk+1).

**Comments:** 10 Pages.

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### Submission history

[v1] 2015-12-14 09:37:41

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