I am studying the Parzen Windows technique.
Although I could find and understand an outline of the proof (for the convergence in mean square), e.g. that provided in Duda's book or sketched at http://www.ehu.eus/ccwintco/uploads/8/89/Borja-Parzen-windows.pdf (which is essentialy Duda's materials that have been condensed), I do not immediately see where the following condition is needed ($\varphi$ denotes the window function):
$lim_{||u|| \rightarrow+\infty}{\varphi(u)}\prod_{i=1}^{d}u_{i} = 0$
I understand this gives some nice asymptotic smoothness to the window function but fail to see why this is necessary for convergence in mean square...