Measure Theory, Probability, and Stochastic Processes
Le Gall, Jean-François
- 出版商: Springer
- 出版日期: 2023-10-31
- 售價: $2,170
- 貴賓價: 9.5 折 $2,062
- 語言: 英文
- 頁數: 406
- 裝訂: Quality Paper - also called trade paper
- ISBN: 3031142071
- ISBN-13: 9783031142079
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商品描述
This textbook introduces readers to the fundamental notions of modern probability theory. The only prerequisite is a working knowledge in real analysis. Highlighting the connections between martingales and Markov chains on one hand, and Brownian motion and harmonic functions on the other, this book provides an introduction to the rich interplay between probability and other areas of analysis.
Arranged into three parts, the book begins with a rigorous treatment of measure theory, with applications to probability in mind. The second part of the book focuses on the basic concepts of probability theory such as random variables, independence, conditional expectation, and the different types of convergence of random variables. In the third part, in which all chapters can be read independently, the reader will encounter three important classes of stochastic processes: discrete-time martingales, countable state-space Markov chains, and Brownian motion. Each chapter ends with a selectionof illuminating exercises of varying difficulty. Some basic facts from functional analysis, in particular on Hilbert and Banach spaces, are included in the appendix.
Measure Theory, Probability, and Stochastic Processes is an ideal text for readers seeking a thorough understanding of basic probability theory. Students interested in learning more about Brownian motion, and other continuous-time stochastic processes, may continue reading the author's more advanced textbook in the same series (GTM 274).
商品描述(中文翻譯)
這本教科書向讀者介紹了現代概率論的基本概念。唯一的先備知識是實分析的基本知識。本書強調了鞅與馬可夫鏈之間的聯繫，以及布朗運動與調和函數之間的關係，提供了概率與其他分析領域之間豐富互動的入門介紹。
本書分為三個部分，首先以嚴謹的方式處理測度論，並考慮其在概率中的應用。第二部分專注於概率論的基本概念，如隨機變數、獨立性、條件期望以及隨機變數的不同收斂類型。在第三部分中，所有章節均可獨立閱讀，讀者將接觸到三個重要的隨機過程類別：離散時間鞅、可數狀態空間的馬可夫鏈以及布朗運動。每章結尾都有一系列不同難度的啟發性練習題。附錄中包含了一些來自泛函分析的基本事實，特別是關於希爾伯特空間和巴拿赫空間的內容。
《測度論、概率與隨機過程》是希望深入理解基本概率論的讀者的理想教材。對於有興趣進一步了解布朗運動及其他連續時間隨機過程的學生，可以繼續閱讀作者在同一系列中的更高級教材（GTM 274）。
作者簡介
Jean-François Le Gall is Professor of Mathematics at the University of Paris-Saclay in France. As one of the leading experts in probability theory, he has done extensive research on stochastic processes, including Brownian motion, random trees, random planar maps, and other related objects. His research accomplishments have been recognized with various awards, most recently the Wolf prize. He is the author of two successful textbooks on Brownian Motion, Martingales, and Stochastic Calculus (2016) in the Graduate Texts in Mathematics series and Spatial Branching Processes, Random Snakes and Partial Differential Equations (1999) in the Lectures in Mathematics, ETH Zürich series.
作者簡介(中文翻譯)
Jean-François Le Gall 是法國巴黎薩克雷大學的數學教授。作為概率論的領軍專家之一，他在隨機過程方面進行了廣泛的研究，包括布朗運動、隨機樹、隨機平面地圖及其他相關對象。他的研究成就獲得了多項獎項的認可，最近獲得了沃爾夫獎。他是兩本成功教科書的作者，分別是《布朗運動、馬丁蓋爾與隨機微積分》（2016年，發表於數學研究生教材系列）和《空間分支過程、隨機蛇與偏微分方程》（1999年，發表於蘇黎世聯邦理工學院數學講座系列）。
