A direct construction of the Standard Brownian Motion

TL;DR

This paper presents a detailed method for constructing a standard Brownian motion from a Brownian motion defined on non-negative dyadic numbers, combining previous approaches and including a proof of Etemadi's inequality.

Contribution

It offers a new, self-contained construction of standard Brownian motion by integrating existing methods and adding a proof of Etemadi's inequality for clarity.

Findings

Provides a detailed, sequential construction method

Combines approaches of Billingsley and Csörgo & Révész

Includes a proof of Etemadi's inequality

Abstract

In this note, we combine the two approaches of Billingsley (1998) and Cs\H{o}rg\H{o} and R\'ev\'esz (1980), to provide a detailed sequential and descriptive for creating s standard Brownian motion, from a Brownian motion whose time space is the class of non-negative dyadic numbers. By adding the proof of Etemadi's inequality to text, it becomes self-readable and serves as an independent source for researches and professors.