Limiting distributions for individual Mahalanobis distances

The problem of estimating individual Mahalanobis distances in situations where the dimension of the parent variable increases proportionally with the sample size is discussed. In particular, central limit theorems are derived for two frequently used estimators. Although they behave similarly in finite dimensions, they are shown to have different convergence rates and are centred at two different points in high-dimensional settings. The limiting distributions are valid under some general moment conditions and hence available in a wide range of applications.