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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.
List of contents
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.
About the author
Katsuto Tanaka is Professor Emeritus at Hitotsubashi University, Tokyo. He is a recipient of the Tjalling C. Koopmans Econometric Theory Prize (1996), the Japan Statistical Society Prize (1998), and the Econometric Theory Award (1999). He previously authored 'Time Series Analysis' (First Edition 1996, Second Edition 2017) and six statistics and econometrics books in Japanese.
Summary
Introducing Brownian motion from a statistical viewpoint, this text gathers many important statistical tools in one accessible resource for researchers and graduate students. In particular, it explores how to derive the distribution of various statistics arising in time series analysis that are the quadratic functionals of Brownian motion.
