Multivariate Chebyshev Inequality with Estimated Mean and Variance

TL;DR

This paper generalizes the Chebyshev inequality to multivariate cases with estimated mean and variance, demonstrating convergence to the theoretical bound as sample size increases.

Contribution

It introduces a multivariate Chebyshev inequality with estimated parameters, extending the scalar version and proving its asymptotic convergence.

Findings

The inequality applies to i.i.d. samples in multiple dimensions.

Convergence to the true Chebyshev bound as sample size grows.

Provides a practical tool for probabilistic bounds with estimated parameters.

Abstract

A variant of the well-known Chebyshev inequality for scalar random variables can be formulated in the case where the mean and variance are estimated from samples. In this paper we present a generalization of this result to multiple dimensions where the only requirement is that the samples are independent and identically distributed. Furthermore, we show that as the number of samples tends to infinity our inequality converges to the theoretical multi-dimensional Chebyshev bound.