Remarks on asymptotic independence

TL;DR

This paper explores various definitions of asymptotic independence for sequences of random elements, analyzing their properties, relationships, and providing counterexamples to clarify distinctions.

Contribution

It introduces multiple natural definitions of asymptotic independence and examines their properties and interrelations, especially for tight sequences.

Findings

Different definitions of asymptotic independence are characterized and compared.

Connections between asymptotic independence and weak dependence are established.

Counterexamples illustrate the distinctions between various definitions.

Abstract

In this paper we introduce several natural definitions of asymptotic independence of two sequences of random elements. We discuss their basic properties, some simple connections between them and connections with properties of weak dependence. In particular, the case of tight sequences is considered in detail. Finally, in order to clarify the relationships between different definitions, we provide some counterexamples.