Abstract:
Let $(\theta, X)$ be a random vector with $\theta \in\{1, \cdots, M\}, X \in R^d$, and $\left(\theta_i, X_i\right), i=1, \cdots$, $n$, i.i.d. random samples of $(\theta, X)$. For a distance function $\rho(\cdot, \cdot)$ given on $R^u$, denote $$ k_n=\max \left\{j: \rho\left(x_y, x\right)=\min _{1< j< n} \rho\left(x_j, x\right), j \leqslant n\right\}, \quad \theta_n^{(2)}=\theta_{k_n}, $$ Where $\theta_n^{(2)}$ is called NN discrimination in decreasing order of affixes of $\theta$. In this paper we prove that if $P(\theta-j \mid X=x), j=1, \cdots, M$, are $Q K$-continuous then the following statements are equivalent: ( I ) $L_n^{(2)} \rightarrow \boldsymbol{R}$, a.s. as $n \rightarrow \infty, \cdots$ (II) For every atom $a_i$ of $X, 1 \leqslant j< k< M$ $$ P\left(\theta=j, X=a_i\right) P\left(\theta=k, X=a_i\right)\left[P\left(\theta=j, X=a_i\right)-P\left(\theta=k, X=a_i\right)\right]^2=0 $$ If (II) is not true, then $L_n^{(2)}$ is divergent, a.s., where $$ L_n^{(2)}=P\left(\theta_n^{(2)} \neq \theta \mid Z^n\right), Z^n=\left(\left(\theta_i, X_i\right), i=1, \cdots, n\right) . $$
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