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Levy's second arcsine law via the ballot theorem

Helmut H. Pitters

Abstract

We provide a new and elementary proof of Levy's second arcsine law for Brownian motion. The only tools required are basic properties of Brownian motion and Poisson processes, and the ballot theorem. Our proof is readily extended to Brownian motion with drift.

Levy's second arcsine law via the ballot theorem

Abstract

We provide a new and elementary proof of Levy's second arcsine law for Brownian motion. The only tools required are basic properties of Brownian motion and Poisson processes, and the ballot theorem. Our proof is readily extended to Brownian motion with drift.
Paper Structure (2 sections, 3 theorems, 14 equations)

This paper contains 2 sections, 3 theorems, 14 equations.

Table of Contents

  1. Introduction.
  2. New proof of Lévy's second arcsine law for Brownian motion.

Key Result

Theorem 1

For any fixed $t>0$ the occupation time def:occupation_time of Brownian motion has an arcsine distribution with support $(0, 1)$ and density

Theorems & Definitions (7)

  • Theorem 1: Lévy's second arcsine law for Brownian motion
  • Theorem 2: Bertrand Bertrand1888
  • Proposition 3: Sampling the occupation time
  • proof
  • Remark 4
  • Remark 5
  • Remark 6