Multivariate records based on dominance

TL;DR

This paper studies three types of multivariate records in high-dimensional spaces, deriving their statistical properties and establishing central limit theorems, with practical methods for precise variance computation.

Contribution

It introduces new theoretical results on the mean, variance, and distributional limits of multivariate records in hypercubes and simplices, including numerical procedures.

Findings

Derived mean and variance formulas for multivariate records.

Established central limit theorems with convergence rates.

Provided numerical methods for variance constant computation.

Abstract

We consider three types of multivariate records in this paper and derive the mean and the variance of their numbers for independent and uniform random samples from two prototype regions: hypercubes $[0,1]^d$ and $d$-dimensional simplex. Central limit theorems with convergence rates are established when the variance tends to infinity. Effective numerical procedures are also provided for computing the variance constants to high degree of precision.