摘要
Ｓｌｕｔｓｋｙ定理指出：如果随机变量序列｛Ｘ１ｎ｝，｛Ｘ２ｎ｝，…，｛Ｘｍｎ｝分别依概率收敛到ｍ个有限常数ａ１，ａ２，…，ａｍ，那么任意一个有理函数Ｒ（Ｘ１ｎ，Ｘ２ｎ，…，Ｘｍｎ）也依概率收敛到常数Ｒ（ａ１，ａ２，…，ａｍ），只要Ｒ（ａ１，ａ２，…，ａｍ）有限．本文从两个方面推广了这一结果：第一，若上述随机变量序列分别依概率收敛到随机变量Ｘ１，Ｘ２，…，Ｘｍ，ｇ（ｘ１，ｘ２，…，ｘｍ）是ｍ维欧氏空间Ｒｍ上的连续函数，则ｇ（Ｘ１ｎ，Ｘ２ｎ，…，Ｘｍｎ）依概率收敛于ｇ（Ｘ１，Ｘ２，…，Ｘｍｎ）．第二，若上述随机变量序列分别收敛到ｍ个有限常数ａ１，ａ２，…，ａｍ，又Ｂｏｒｅｌ可测函数ｇ（ｘ１，ｘ２，…，ｘｍ）在点（ａ１，ａ２，…，ａｍ）处连续，则ｇ（Ｘ１ｎ，Ｘ２ｎ，…，Ｘｍｎ）依概率收敛到ｇ（ａ１，ａ２，…，ａｍ）．
In Slutsky′s theorem it is pointed out that if sequences of random variables {X1n},{X2n},…,{Xmn} converge to finite constants a1,a2,…,am in probability, respectively, then any rational function R(X1n,X2n,…,Xmn) also converges to constant R(a1,a2,…,am) as long as R(a1,a2,…,am) is finite We generalize Slutsky′s theorem in two aspects: (1) if the sequences of random variables mentioned above converge to random variables X1,X2,…,Xm in probability, and g(x1,x2,…,xm) is a continuous function on m-dimensional Eqclidean space, then g(X1n,X2n,…,Xmn) converges to g(X1,X2,…,Xmn) in probability, (2) if these sequences of random variables converge to finite constants a1,a2,…,am in probability, respectively, and Borel measurable function g(x1,x2,…,xm) is continuous at point (a1,a2,…,am) , then g(X1n,X2n,…,Xmn) converges to g(a1,a2,…,am) in probability
