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3080 results in Probability theory and stochastic processes

Page 7 of 154

Brownian Motion, the Fredholm Determinant, and Time Series Analysis
  • Katsuto Tanaka
  • Published online:
    19 December 2024
    Print publication:
    02 January 2025
  • 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.

  • 6 - Some Extensions and Applications of Martingale Theory

  • Daniel W. Stroock, Massachusetts Institute of Technology
  • Book:
    Probability Theory, An Analytic View
    Published online:
    07 November 2024
    Print publication:
    21 November 2024, pp 200-234

  • Summary

    Chapter 6 opens with extensions of martingale theory in two directions: to σ-finite measures and to random variables with values in a Banach space. In §6.2 I prove Burkholder’s Inequality for martingales with values in a Hilbert space. The derivation that I give is essentially the same as Burkholder’s second proof, the one that gives optimal constants. Finally, the results in §6.1 are used in §6.3 to derive Birkhoff’s Individual Ergodic Theorem and a couple of its applications.

  • 7 - Continuous Parameter Martingales

  • Daniel W. Stroock, Massachusetts Institute of Technology
  • Book:
    Probability Theory, An Analytic View
    Published online:
    07 November 2024
    Print publication:
    21 November 2024, pp 235-263

  • Summary

    Section 7.1 provides a brief introduction to the theory of martingales with a continuous parameter. As anyone at all familiar with the topic knows, anything approaching a full account of this theory requires much more space than a book like this can provide. Thus, I deal with only its most rudimentary aspects, which, fortunately, are sufficient for the applications to Brownian motion that I have in mind. Namely, in §7.2 I first discuss the intimate relationship between continuous martingales and Brownian motion (Lévy’s martingale characterization of Brownian motion), then derive the simplest (and perhaps most widely applied) case of the Doob–Meyer Decomposition Theory, and finally show what Burkholder’s Inequality looks like for continuous martingales. In the concluding section, §7.3, the results in §7.1 and §7.2 are applied to derive the Reflection Principle for Brownian motion.

  • 3 - Infinitely Divisible Laws

  • Daniel W. Stroock, Massachusetts Institute of Technology
  • Book:
    Probability Theory, An Analytic View
    Published online:
    07 November 2024
    Print publication:
    21 November 2024, pp 101-130

  • Summary

    This chapter is devoted to the study of infinitely divisible laws. It begins in §3.1 with a few refinements (especially the Lévy Continuity Theorem) of the Fourier techniques introduced in §2.3. These play a role in §3.2, where the Lévy–Khinchine formula is first derived and then applied to the analysis of stable laws.

Page 7 of 154