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图形投影寻踪的渐近理论
Mathematical tools are developed for describing low-dimensional projections of high-dimensional data. Theorems are given to show that under suitable conditions, most projections are approximately Gaussian.
报告显著性水平的非随机解释
Tests of significance are often made in situations where the standard assumptions underlying the probability calculations do not hold. As a result, the reported significance levels become difficult to interpret. This article sketches an alternative interpretation of a reported significance level, valid in considerable generality. This level locates the given data set within the spectrum of other data sets derived from the given one by an appropriate class of transformations.…
傅里叶系数的经验分布
Suppose $X_1, X_2, \cdots$ are independent, identically distributed complex-valued $L^2$ random variables with $EX_1 = 0$ and $E(|X_1|^2) = 1$. Let $Y_{nk}$ be the $k$th Fourier coefficient of $X_1, \cdots, X_n$: $Y_{nk} = \sum^n_{j=1} X_j \exp \big(\frac{2\pi(-1)^{1/2}kj}{n}\big).$ Let $\mu_n$ be the empirical distribution of $\{n^{-1/2}Y_{nk}: k = 1, \cdots, n\}$. Then $\mu_n$ converges to the distribution of $U + iV$, where $U$ and $V$ are independent normal variables…