Abstract: Let X=（x₁, x₂,…, x_m）, be a m-dimensional random vector, X₁, X₂,…, X subsamples from the populatton X, the m-dimensional random vector Z have the N_m（0, I_m） distribution and B_m＞0 be a sequence of real numbers. Let \hatF_n^\ell / B m(x)=\frac1n \#\left\i: Z^\prime X_4 / B_i X be multinomial distribution\mathscrB\left(k_;, p_1, p_2, \cdots, p_m\right), with each p_i≥0, m=\sum_i=1^m p_i^2, there be a sequence of real numbers B_m＞0, k/B_m²→σ, k*d_m→0, as m→∞ then \hatF_n^Z / B_m(x) \xrightarrowP N\left(0, \sigma^2\right) as n→∞, m→∞. （2） Let X^~N_m（u,V）,V＞0, there be a sequence of real numbers B_m^-2 T_r(V) \rightarrow \sigma^2, B_m^-2\|u\|^2 \rightarrow 0, \left(T_r(V)+2 u^\prime V u\right) / B_m^4 \rightarrow 0,as m→∞,then \hatF_n^\mathrmZ / B m(x) \xrightarrowP N\left(0, \sigma^2\right) as n→∞, m→∞.