# Convergence Rate for Multivariate Kernel Density Estimators

I know that univariate kernel density functions converge uniformly a.s. to the true distribution. Is this also true for multivariate kdf? Is there a theorem that gives the rate of convergence in the multivariate case? What about the univariate case?

Yes, convergence uniformly as. also applies to the multivariate KDE. The convergence rate for the univariate case is $\sqrt{nh}$, where $h$ is the bandwidth and $n$ is the sample size. For multivariate case, the convergence rate is $\sqrt{nh_1 \cdot h_q}$, where $h_1, \cdots, h_q$ are the bandwidth for each components of a random vector say $\boldsymbol{X}$, respectively. Here is a link to a note that have the proof for convergence rate. http://www.ssc.wisc.edu/~bhansen/718/NonParametrics1.pdf.