ON THE MAXIMUM DEVIATION OF THE SAMPLE DENSITY.

The maximum relative deviation of a sample density from the true one is shown to be relatively stable as the sample size tends to infinity provided both the sample and true densities are sufficiently regular. Conditions are given under which the sample density converges to the true density w.p. one uniformly on compact sets and uniformly on compact sets at a specified rate. The key to establishing this result is a theorem on the large deviations of independent, identically distributed bivariate random variables. Author