Asymptotic distributions of basic statistics in geometric representation for high-dimensional data and their error bounds

In geometric representation of n observations on p variables, it is necessary to examine asymptotic behaviors of the three statistics; the length of a p-dimensional observation vector, the distance between two independent observation vectors, and the angle between these observation vectors. Hall et al. (2005) found the asymptotic values of these three statistics in high-dimensional framework when the dimension p tends to infinity, while the sample size n is fixed. In this paper, their results are extended by deriving asymptotic expansions of the distributions of the three statistics. Further, computable error bounds for the limiting distributions of the length and the distance were obtained. These results will be useful to obtain statistical insights in middle — as well as in high-dimensional data sets.