关于一类随机适应序列的强收敛定理

Strong convergence theorem for arbitrary adapted stochastic sequence

引入了随机变量的截尾和停时的概念,在定理条件及其证明过程中适当地定义随机变量的截尾和停时,并通过定义鞅差序列和利用鞅差序列的收敛定理,得到了一类关于任意随机适应序列的强收敛定理.通过利用截尾方法和条件三级数定理,得到了在更高阶条件下的任意随机适应序列的强收敛定理.作为推论,得到了相应的关于独立随机变量序列的强收敛定理以及一些关于任意随机序列的已有的结果,进一步发展和完善了关于任意随机适应序列的强收敛性的研究.

A class of strong convergence theorem for an arbitrarily adapted stochastic sequence without the non-decrease condition was established by using the truncation method of stochastic variables and the stop time. Another strong convergence theorem is obtained by using the truncation method and conditional three series theorem. As corollaries, the strong laws of large numbers for the sequence of independent stochastic variables are obtained.