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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?

Thank you in advance.

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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.

I have been reading the book "Nonparameteric Econometrics" by Adrian Pagan and Aman Ullah. Section 2.4 and 2.5 provide a comprehensive proofs and conditions for the asymptotic unbiasedness and consistency, both in probability and uniformly. ISBN: 9780521586115. Link: http://www.cambridge.org/us/academic/subjects/economics/econometrics-statistics-and-mathematical-economics/nonparametric-econometrics.

This book mainly gives the theorem about consistency and asymptotic distribution of KDE in the univariate case, but the paper by Theophilos Cacoullos states the asymptotic properties of KDE in terms of multivariate case. Here is the link: http://www.ism.ac.jp/editsec/aism/pdf/018_2_0179.pdf.

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    $\begingroup$ Do you think you could provide a brief summary of the proof? In the long run, the link might go dead, which is why we like our answers to be self contained. $\endgroup$ – Silverfish Mar 14 '16 at 8:09

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