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 \boldsymbolR, 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)\leftP\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) .