Limiting behavior of U-statistics for stationary, absolutely regular processes
Limiting behavior of U-statistics for stationary, absolutely regular processes
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DOI：
10.1007/bf00532676
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发表时间：
1976-09
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影响因子：
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通讯作者：
K. Yoshihara
中科院分区：
文献类型：
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作者：
K. Yoshihara
令{~ i}为严格平稳、绝对正则过程，即满足条件fl (n)= E {sup [P {AlJC/~(n-* oo))的过程，其中~/Y/~(a< b)是由~ a,...,~ b生成的事件的a代数。如果合适的过程是根据 W. Hoeffding 的 [Ann.数学。统计 19, 293-325 (1947; 本 Zbl. 32, 41)] 绝对正则过程的 U 统计量，然后弱收敛到布朗运动过程和斯特拉森版本的迭代对数定律 [Z. Wahrscheinlichkeitstheorie verw。 Gebiete 3, 211-226 (1964，本 Zbl. 132, 129)] 被确立。结果是 Sen 的结果的扩展 [同上。 25, 71-82 (1972；本 Zbl. 238, 6097)]。由类似于 Sen 的广义 U 统计构建的过程的弱收敛性 [Ann.很可能。 2, 90-102 (1974)]和几乎确定不变性原理和积分检验[Sen;2, 90-102 (1974)]安.统计 2, 387-395 (1974)，Jain 等人；安.很可能。 3, 119-145,(1975)] 考虑由一些 q% 混合序列定义的 U 统计量。 R. von Mises 的类似问题[Ann。数学。统计 18, 309-348 (1947: 本 Zbl. 37, 84)] 还处理可微统计泛函。
Let {~ i} be a strictly stationary, absolutely regular processes, ie, the process satisfying the condition fl (n)= E {sup [P {AlJC/~(n-* oo) where~/Y/~(a< b) is the a-algebra of events generated by~ a,...,~ b. If the suitable processes are constructed from the sequence of W. Hoeffding's [Ann. Math. Statistics 19, 293-325 (1947; this Zbl. 32, 41)] U-statistics for the absolutely regular processes, then weak convergence to Brownian motion processes and the Strassen's version of the law of the iterated logarithm [Z. Wahrscheinlichkeitstheorie verw. Gebiete 3, 211-226 (1964, this Zbl. 132, 129)] are established. The results are extensions of Sen's ones [ibid. 25, 71-82 (1972; this Zbl. 238, 6097)]. Weak convergence of the processes constructed by generalized U-statistics analogous to Sen's [Ann. Probab. 2, 90-102 (1974)] and almost sure invariance principle and integral tests [Sen; Ann. Statistics 2, 387-395 (1974), Jain etal; Ann. Probab. 3, 119-145,(1975)] for U-statistics defined by some q% mixing sequences are considered. Analogous problems for R. von Mises'[Ann. Math. Statistics 18, 309-348 (1947: this Zbl. 37, 84)] differentiable statistical functionals are also treated.