Brownian Motion, the Fredholm Determinant, and Time Series Analysis

New insights on the distributional and statistical aspects of Brownian motion.

Katsuto Tanaka (Author)

9781009566995, Cambridge University Press

Hardback, published 2 January 2025

352 pages
23.5 x 16 x 2.5 cm, 0.627 kg

Brownian motion is an important topic in various applied fields where the analysis of random events is necessary. Introducing Brownian motion from a statistical viewpoint, this detailed text examines the distribution of quadratic plus linear or bilinear functionals of Brownian motion and demonstrates the utility of this approach for time series analysis. It also offers the first comprehensive guide on deriving the Fredholm determinant and the resolvent associated with such statistics. Presuming only a familiarity with standard statistical theory and the basics of stochastic processes, this book brings together a set of important statistical tools in one accessible resource for researchers and graduate students. Readers also benefit from online appendices, which provide probability density graphs and solutions to the chapter problems.

Part I. Theory: 1. Quadratic functionals of the Brownian motion
2. Integral equations and the Fredholm determinant
3. Integral equations and the resolvent
Part II. Applications: 4. Fredholm determinants for goodness of fit tests
5. Fredholm determinants in the state space model
6. Fredholm determinants in the moving average model
7. Fredholm determinants in the autoregressive model
8. Fredholm determinants for the fractional Brownian motion
References
Author index
Subject index.

Subject Areas: Probability & statistics [PBT]