1st Edition

Martingale Methods in Statistics

By Yoichi Nishiyama Copyright 2022
    260 Pages 9 B/W Illustrations
    by Chapman & Hall

    260 Pages 9 B/W Illustrations
    by Chapman & Hall

    Martingale Methods in Statistics provides a unique introduction to statistics of stochastic processes written with the author’s strong desire to present what is not available in other textbooks. While the author chooses to omit the well-known proofs of some of fundamental theorems in martingale theory by making clear citations instead, the author does his best to describe some intuitive interpretations or concrete usages of such theorems. On the other hand, the exposition of relatively new theorems in asymptotic statistics is presented in a completely self-contained way. Some simple, easy-to-understand proofs of martingale central limit theorems are included.

    The potential readers include those who hope to build up mathematical bases to deal with high-frequency data in mathematical finance and those who hope to learn the theoretical background for Cox’s regression model in survival analysis. A highlight of the monograph is Chapters 8-10 dealing with Z-estimators and related topics, such as the asymptotic representation of Z-estimators, the theory of asymptotically optimal inference based on the LAN concept and the unified approach to the change point problems via "Z-process method". Some new inequalities for maxima of finitely many martingales are presented in the Appendix. Readers will find many tips for solving concrete problems in modern statistics of stochastic processes as well as in more fundamental models such as i.i.d. and Markov chain models.

    I Introduction

    1 Prologue

    Why Is the Martingale So Useful?

    Martingale as a tool to analyze time series data in real time

    Martingale as a tool to deal with censored data correctly

    Invitation to Statistical Modelling with Semimartingales

    From non-linear regression to diffusion process model

    Cox’s regression model as a semimartingale

    2 Preliminaries

    Remarks on Limit Operations in Measure Theory

    Limit operations for monotone sequence of measurable sets

    Limit theorems for Lebesgue integrals

    Conditional Expectation

    Understanding the definition of conditional expectation

    Properties of conditional expectation

    Stochastic Convergence

    3 A Short Introduction to Statistics of Stochastic Processes

    The "Core" of Statistics

    Two illustrations

    Filtration, martingale

    Motivation to Study Stochastic Integrals

    Intensity processes of counting processes

    Itˆo integrals and diffusion processes

    Square-Integrable Martingales

    Predictable quadratic variations

    Stochastic integrals

    Introduction to CLT for square-integrable martingales

    Asymptotic Normality of MLEs in Stochastic Process Models

    Counting process models

    Diffusion process models

    Summary of the approach


    Examples of counting process models

    Examples of diffusion process models

    II A User’s Guide to Martingale Methods

    4 Discrete-Time Martingales

    Basic Definitions, Prototype for Stochastic Integrals

    Stopping Times, Optional Sampling Theorem

    Inequalities for 1-Dimensional Martingales

    Lenglart’s inequality and its corollaries

    Bernstein’s inequality

    Burkholder’s inequalities

    5 Continuous-Time Martingales

    Basic Definitions, Fundamental Facts

    Discre-Time Stochastic Processes in Continuous-Time

    φ(M) Is a Submartingale

    "Predictable" and "Finite-Variation"

    Predictable and optional processes

    Processes with finite-variation

    A role of the two properties

    Stopping Times, First Hitting Times

    Localizing Procedure

    Integrability of Martigales, Optional Sampling Theorem

    Doob-Meyer Decomposition Theorem

    Doob’s inequality

    Doob-Meyer decomposition theorem

    Predictable Quadratic Co-Variations

    Decompositions of Local Martingales

    6 Tools of Semimartingales


    Stochastic Integrals

    Starting point of constructing stochastic integrals

    Stochastic integral w.r.t. locally square-integrable martingale

    Stochastic integral w.r.t. semimartingale

    Formula for the Integration by Parts

    Itˆo’s Formula

    Likelihood Ratio Processes

    Likelihood ratio process and martingale

    Girsanov’s theorem

    Example: Diffusion processes

    Example: Counting processes

    Inequalities for 1-Dimensional Martingales

    Lenglart’s inequality and its corollaries

    Bernstein’s inequality

    Burkholder-Davis-Gundy’s inequalities

    III Asymptotic Statistics with Martingale Methods

    7 Tools for Asymptotic Statistics

    Martingale Central Limit Theorems

    Discrete-time martingales

    Continuous local martingales

    Stochastic integrals w.r.t. counting processes

    Local martingales

    Functional Martingale Central Limit Theorems


    The functional CLT for local martingales

    Special cases

    Uniform Convergence of Random Fields

    Uniform law of large numbers for ergodic random fields

    Uniform convergence of smooth random fields

    Tools for Discrete Sampling of Diffusion Processe

    8. Parametric Z-Estimators

    Illustrations with MLEs in I.I.D. Models

    Intuitive arguments for consistency of MLEs

    Intuitive arguments for asymptotic normality of MLEs

    General Theory for Z-estimators

    Consistency of Z-estimators, I

    Asymptotic representation of Z-estimators, I

    Examples, I-1 (Fundamental Models)

    Rigorous arguments for MLEs in i.i.d. models

    MLEs in Markov chain models

    Interim Summary for Approach Overview


    Asymptotic normality

    Examples, I-2 (Advanced Topics)

    Method of moment estimatorsQuasi-likelihood for drifts in ergodic diffusion models

    Quasi-likelihood for volatilities in ergodic diffusion modelsPartial-likelihood for Cox’s regression models

    More General Theory for Z-estimators

    Consistency of Z-estimators, II

    Asymptotic representation of Z-estimators, II

    Example, II (More Advanced Topic: Different Rates of Convergence)Quasi-likelihood for ergodic diffusion models

    9 Optimal Inference in Finite-Dimensional LAN Models

    Local Asymptotic Normality

    Asymptotic Efficiency

    How to Apply

    10 Z-Process Method for Change Point Problems

    Illustrations with Independent Random Sequences

    Z-Process Method: General Theorem


    Rigorous arguments for independent random sequences

    Markov chain models

    Final exercises: three models of ergodic diffusions

    A Appendices

    A1 Supplements

    A1.1 A Stochastic Maximal Inequality and Its Applications

    A1.1.1 Continuous-time case

    A1.1.2 Discrete-time case

    A1.2 Supplementary Tools for the Main Parts

    A2 Notes

    A3 Solutions/Hints to Exercises



    Yoichi Nishiyama is a professor in mathematical statistics and probability at the School of International Liberal Studies of Waseda University; he is also engaged in the education of master’s and doctoral students at the Department of Pure and Applied Mathematics at the same university. Prior to his assignment to Waseda University, he worked at the Institute of Statistical Mathematics, Tokyo, from 1994 to 2015. He was the Editor-in-Chief of Journal of the Japan Statistical Society and a Co-Editor of Annals of the Institute of Statistical Mathematics and he received the JSS Ogawa Award from the Japan Statistical Society in 2009.

    "This book is expected to be an excellent reference for researchers who need to perform statistical analysis based on the martingale theory. It is very rare to find a book that systematically and rigorously summarizes martingale theory from the point of view of its application to statistics. This textbook harmonically organizes the mathematical facts related to martingales and their statistical applications, and by doing so, it helps researchers to establish a theoretically concrete foundation. Therefore, this book can be evaluated as an excellent textbook where mathematics and statistics meet together."
    -Insuk Seo, in Journal of the American Statistical Association, November 2023

    "The martingale theory is an important topic in probability theory and related tools have been widely applied in statistical analysis, such as financial data or survival analysis. ...This book well summarizes useful tools in martingale and provides rigorous theorems. ... In summary, this book is a nice reference because of rich and comprehensive materials in martingale
    theory. This book is suitable to researchers who are working on related research topics."
    - Li-Pang Chen, in Journal of the Royal Statistical Society: Series A, April 2022