Algorithmically random Fourier series and Brownian motion

TL;DR

This paper explores the construction of algorithmically random Fourier series and demonstrates how certain series converge to an algorithmically random Brownian motion, linking computability and stochastic processes.

Contribution

It introduces a method to generate algorithmically random Brownian motion via Fourier series with computable coefficients, extending previous work on Rademacher series.

Findings

Fourier series with computable coefficients can converge to algorithmically random Brownian motion

Extension of Rademacher series to Fourier series on the circle

Establishment of a link between computability and stochastic processes

Abstract

We consider some random series parametrised by complex binary strings. The simplest case is that of Rademacher series, independent of a time parameter. This is then extended to the case of Fourier series on the circle with Rademacher coefficients. Finally, a specific Fourier series which has coefficients determined by a computable function is shown to converge to an algorithmically random Brownian motion.