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**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

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**

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

Preliminaries

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

Consistency

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

Examples

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

** **

### Biography

**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, inJournal 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, inJournal of the Royal Statistical Society: Series A, April 2022