On the Maximum Correlation Coefficient *
On the Maximum Correlation Coefficient *
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关于最大相关系数 *
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通讯作者：
And A Kagan
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作者：
W. Bryc;A. Dembo;And A Kagan
对于任意随机向量(X，Y)和独立随机变量Z，证明了X和Y+λZ之间的最大相关系数是λ的函数，在其达到最大值的地方是下半连续的，并且在零点连续.此外，如果Z是可自分解的随机变量，则最大相关系数对λ≥0是右连续的，不增加的，对λ≤0是左连续的，不减少。独立随机变量X和Z是高斯的当且仅当X和X+λZ之间的最大相关系数等于它们之间的线性相关。N个任意独立同分布随机变量之和与这些变量的第一个m<；n之和之间的最大相关系数等于m/n(以前仅证明了具有有限二阶矩的随机变量，其中它也等于线性相关)。所提供的例子揭示了对于更一般的Z和在Limitλ→∞中的最大相关系数的违反直觉的行为。
For an arbitrary random vector (X, Y) and an independent random variable Z it is shown that the maximum correlation coefficient between X and Y + λZ as a function of λ is lower semicontinuous everywhere and continuous at zero where it attains its maximum. If, moreover, Z is in the class of self-decomposable random variables, then the maximal correlation coefficient is right continuous, nonincreasing for λ ≥ 0 and left continuous, nondecreasing for λ ≤ 0. Independent random variables X and Z are Gaussian if and only if the maximum correlation coefficient between X and X +λZ equals the linear correlation between them. The maximum correlation coefficient between the sum of n arbitrary independent identically distributed random variables and the sum of the first m < n of these equals m/n (previously proved only for random variables with finite second moments, where it amounts also to the linear correlation). Examples provided reveal counterintuitive behavior of the maximum correlation coefficient for more general Z and in the limit λ → ∞.