A Note on the Multivariate CLT and Convergence of Levy Processes at Long and Short Times

TL;DR

This paper establishes a natural condition based on truncated second moments for the convergence of sums of iid vectors and Levy processes to multivariate Gaussian distributions, including degenerate cases, at long and short times.

Contribution

It introduces a new necessary and sufficient condition involving slowly varying truncated second moments for multivariate CLT and Levy process convergence.

Findings

Truncated second moment matrix being slowly varying is key for convergence.

Conditions for Levy processes to converge to Gaussian at zero or infinity.

Allows for degenerate Gaussian limits under certain conditions.

Abstract

We show that a necessary and sufficient condition for the sum of iid random vectors to converge (under appropriate shifting and scaling) to a multivariate Gaussian distribution is that the truncated second moment matrix is slowly varying at infinity. This is more natural than the standard conditions, and allows for the possibility that the limiting Gaussian distribution is degenerate (so long as it is not concentrated at a point). We also give necessary and sufficient conditions for a d-dimensional Levy process to converge (under appropriate shifting and scaling) to a multivariate Gaussian distribution as time approaches zero or infinity.