Let $f$ be a density on the real line and let $f_n$ be the kernel estimate of $f$ in which the smoothing factor is obtained by maximizing the cross-validated likelihood product according to the method of Duin and Habbema, Hermans and Vandenbroek. Under mild regularity conditions on the kernel and $f$, we show, among other things that $\int|f_n - f| \rightarrow 0$ almost surely if and only if the sample extremes of $f$ are strongly stable.
"On the Relationship Between Stability of Extreme Order Statistics and Convergence of the Maximum Likelihood Kernel Density Estimate." Ann. Statist. 17 (3) 1070 - 1086, September, 1989. https://doi.org/10.1214/aos/1176347256