Let $\{X_n, -\ity < n < \ity \}$ be a
stationary $\phi$-mixing process with the one-dimensional marginal
distribution function $F$ and the density function $f.$ Let $F_n(x)$
be the empirical distribution function based on the observations
$\{X_i, 1 \leq i \leq n\}$ and $W_n^*= \sup_{-\ity < x < \ity}\sqrt
n|F_n(x)-F(x)|.$ We obtain upper bounds for $E(W_n^*)$. We give an
application to get bounds on the expectation of the supremum of the
deviation of a kernel density estimator $\hat f_n(x)$ from true
density function $f(x)$ . Similar results were obtained for a kernel
type estimator $\hat F_n(x)$ for the true distribution function
$F(x).$