取值于有限域上的独立随机变量和的极限分布定理

Limit Distribution Theorem for a Sum of Independent Random Variables Whose Values Belong to a Finite Field

利用文献［１］中所定义的取值于有限域上随机变量的特征函数，首先得到了在一定条件下取值于有限域上的独立随机变量和的极限分布为均匀分布的充分必要条件，再利用无穷乘积的有关性质，又给出了取值于有限域上的独立随机变量和的极限分布为均匀分布的若干充分条件，特别地，当独立随机变量取值于素域时，所得的充分条件更易于验证且不难满足．从所得结论看，取值于有限域上独立随机变量和的极限分布收敛于均匀分布的条件并不难满足，这与相互独立的实值随机变量和的分布在许多情况下收敛于正态分布是类似的．

With the characteristic function defined in document ,this paper gives a sufficient and necessary condition for the distribution of a sum of independent random variables whose values belong to a finite field to converge to uniform distribution .In addition,we have given the sufficient conditions that the limit distribution of a sum of independent random variables whose values belong to a finite field is uniform. Particularly,when the independent ramdon variables' values belong to a prime field,the sufficient condition is easy to be satified.It isn't difficult to satisfy these conditions.Therefore,it is easy for the distribution of a sum of independent random variables whose values belong to a finite field to converge to uniform distribution.which is similar to that the asymptotic distribution of a sum of independent real valued random variables is usually normal.