【摘要】 <正>本文给出了独立随机变量组列所产生的部分和过程弱收敛于Brown运动过程的充要条件的一个直接证明,并运用这一结果对m相依情形给出了一个充分条件。前者推广了Billingsley及Loeve提到的Donsker及Lecam的定理,后者改进了Orey有关m相依情形的结果。更多还原

【Abstract】 In this paper, we give a necessary and sufficient condition for partial sum process generated by array of independent random variables converges to Brown motion process weakly: i. e.,Theorem 1 Let kn(t) be an integral valued, increasing function on [0.1], right continuous in t, and be an array of independent random variables, be an array of constants. PutThen in order that and the summand is infinitesimal if and only if that the following conditions are satisfied:(i) (ii) For some t>0, every ,here is the distribution function ofWe improve a result of Orey[4] by using theorem 1, and prove the following: Theorem 2 Let be an array of m-dependent random variables. If forsome t>0, every ε>0, the following conditions are satisfied:(i) (ii)(iii) (iv) then converges to W(t) weakly, for kn(t) =g(i)m, here g(i) is anonnegative integral valued function.更多还原