English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Distance covariance is a popular dependence measure for two random vectors$X$ and $Y$ of possibly different dimensions and types. Recent years havewitnessed concentrated efforts in the literature to understand thedistributional properties of the sample distance covariance in ahigh-dimensional setting, with an exclusive emphasis on the null case that $X$and $Y$ are independent. This paper derives the first non-null central limittheorem for the sample distance covariance, and the more general sample(Hilbert-Schmidt) kernel distance covariance in high dimensions, primarily inthe Gaussian case. The new non-null central limit theorem yields anasymptotically exact first-order power formula for the widely used generalizedkernel distance correlation test of independence between $X$ and $Y$. The powerformula in particular unveils an interesting universality phenomenon: the powerof the generalized kernel distance correlation test is completely determined by$n\cdot \text{dcor}^2(X,Y)/\sqrt{2}$ in the high dimensional limit, regardlessof a wide range of choices of the kernels and bandwidth parameters.Furthermore, this separation rate is also shown to be optimal in a minimaxsense. The key step in the proof of the non-null central limit theorem is aprecise expansion of the mean and variance of the sample distance covariance inhigh dimensions, which shows, among other things, that the non-null Gaussianapproximation of the sample distance covariance involves a rather subtleinterplay between the dimension-to-sample ratio and the dependence between $X$and $Y$.

Generalized kernel distance covariance in high dimensions: non-null CLTs and power universality