2阶标准U-统计量的LIL
The LIL for canonical U-statistics of order 2
摘要：
设X，X_1，X_2，.是独立同分布的随机变量，设h（x，y）=h（y，x）是二元可测函数。证明了有界重对数律$\limsup_n (n\log\log n)^{-1}|\sum_{1<= i< j<= n}h(X_i,X_j)|<\infty$ a.s.，当且仅当满足以下三个条件时成立：h对于X的定律是正则的（即对于几乎y，Eh（X，y）=0），并且存在$C<\infty$使得，对于所有大u，$E\min(h^2(X_1,X_2),u)<C\log\log u$和$sup\{Eh(X_1,X_2)f(X_1)g(X_2):|f(X)|_2<1,\|g(X)\|_2<1, \|f\|_\infty<\infty, \|g\|_\infty<\infty\}< C$都是。
英文摘要：
Let X,X_1,X_2,... be independent identically distributed random variables and let h(x,y)=h(y,x) be a measurable function of two variables. It is shown that the bounded law of the iterated logarithm, $\limsup_n (n\log\log n)^{-1}|\sum_{1<= i< j<= n}h(X_i,X_j)|<\infty$ a.s., holds if and only if the following three conditions are satisfied: h is canonical for the law of X (that is Eh(X,y)=0 for almost y) and there exists $C<\infty$ such that, both, $E\min(h^2(X_1,X_2),u)<C\log\log u$ for all large u and $sup\{Eh(X_1,X_2)f(X_1)g(X_2):|f(X)|_2<1,\|g(X)\|_2<1, \|f\|_\infty<\infty, \|g\|_\infty<\infty\}< C$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
