*Article* **Second Order Expansions for High-Dimension Low-Sample-Size Data Statistics in Random Setting**

#### **Gerd Christoph 1,2,† and Vladimir V. Ulyanov 2,3,\*,†**


Received: 12 May 2020; Accepted: 9 July 2020; Published: 14 July 2020

**Abstract:** We consider high-dimension low-sample-size data taken from the standard multivariate normal distribution under assumption that dimension is a random variable. The second order Chebyshev–Edgeworth expansions for distributions of an angle between two sample observations and corresponding sample correlation coefficient are constructed with error bounds. Depending on the type of normalization, we ge<sup>t</sup> three different limit distributions: Normal, Student's *t*-, or Laplace distributions. The paper continues studies of the authors on approximation of statistics for random size samples.

**Keywords:** second order expansions; high-dimensional; low sample size; random sample size; Laplace distribution; Student's *t*-distribution

**MSC:** 62E17 (Primary) 62H10; 60E05 (Secondary)
