Аннотация:A first investigation of high-dimensional low-sample-size (HDLSS) asymptotics, Hall, Marron and Neeman (2005) discovered a surprisingly rigid geometric structure. A sample of size k taken from the standard m-dimensional normal distribution is for large m close to the vertices of the k-dimensional simplex in m-dimensional vector space. It follows from the analysis of three geometric statistics: the length of an observation, the distance between any two independent observations and the angle between these vectors. We generalize and refine the results constructing the second order Chebyshev-Edgeworth expansions under assumption that the data dimension is random and different scaling factors are chosen.