Abstract: Certain classical maximal probability inequalities for ordinary discrete-time submartingales (or supermartingales) are not (in general) true for discrete-time two dimensionally indexed submartingales (or supermartingales) and some others have their useful extensions but can not be used directly when dealing with the LIL convergency results for partial sums of two dimensionally indexed martingale difference. Maximal probability inequalities for two dimensionally indexed martingale difference in other forms and truncation properties for random fields was given to show the LIL convergency results for partial sums of two dimensional 1/4 martingale difference.

Key words: martingale difference, random fields, LIL

摘要： 一些一元上（下）鞅序列的极大概率不等式在多元的场合不再成立，其它的一些不等式在多元场合有类似的推广但不能直接用来证明随机场部分和的重对数律收敛性。本文给出一些特殊的随机场的极大概率不等式和一些较细的随机场的截断性质。利用这些性质证明了平稳遍历1/4鞅差随机场部分和的重对数律收敛性。

关键词: 鞅差, 随机场, 重对数律（LIL)