Probability density estimates are generated by a kernel or weight function. Limit theorems are obtained for the maximum of the normalized deviation of the estimate from its expected value. The results are, in part, an extension to the $k$-dimensional case $(k > 1)$ of those obtained by P. Bickel and M. Rosenblatt (Ann. Statist. 1973, 1071-1095) one dimensionally.
"On the Maximal Deviation of $k$-Dimensional Density Estimates." Ann. Probab. 4 (6) 1009 - 1015, December, 1976. https://doi.org/10.1214/aop/1176995945