Convergence rates in the law of large numbers and new kinds of convergence of random variables

TL;DR

This paper explores convergence rates in the law of large numbers, introduces new types of convergence for random variables, and discusses their properties and relationships.

Contribution

It provides new convergence rate results in the law of large numbers and introduces novel convergence concepts for random variables.

Findings

Established strong $L^p$-convergence in the law of large numbers

Proved strongly almost sure convergence versions

Analyzed properties and relations of new convergence types

Abstract

In this paper, we first study convergence rates in the law of large numbers for independent and identically distributed random variables. We obtain a strong $L^p$-convergence version and a strongly almost sure convergence version of the law of large numbers. Second, we investigate several new kinds of convergence of random variables and discuss their relations and properties.