Almost sure convergence of classification procedures using Hermite series density estimates

A multidimensional classification procedure is examined derived from the multiple Hermite series estimate of probability density functions. Conditions for the almost sure convergence of the integrated square error for the estimate are presented and the rate of the convergence is studied. The probability of misclassification, conditioned on a learning sequence of length n, is shown to converge to the Bayes risk almost surely as rapidly as $\text{O}\text{(n}{}^{\mathrm{<mtext>?1</mtext><mtext>2+\delta </mtext>}}\text{)}$, δ positive.