{"title":"Probability \u0026 Statistics","description":null,"products":[{"product_id":"simulation-and-the-monte-carlo-method","title":"Simulation and the Monte Carlo Method","description":"\u003cp\u003e\u003cb\u003eThis accessible new edition explores the major topics in Monte Carlo simulation that have arisen over the past 30 years and presents a sound foundation for problem solving\u003c\/b\u003e\u003cb\u003e \u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003ci\u003eSimulation and the Monte Carlo Method, Third Edition \u003c\/i\u003ereflects the latest developments in the field and presents a fully updated and comprehensive account of the state-of-the-art theory, methods and applications that have emerged in Monte Carlo simulation since the publication of the classic First Edition over more than a quarter of a century ago. While maintaining its accessible and intuitive approach, this revised edition features a wealth of up-to-date information that facilitates a deeper understanding of problem solving across a wide array of subject areas, such as engineering, statistics, computer science, mathematics, and the physical and life sciences. The book begins with a modernized introduction that addresses the basic concepts of probability, Markov processes, and convex optimization. Subsequent chapters discuss the dramatic changes that have occurred in the field of the Monte Carlo method, with coverage of many modern topics including: Markov Chain Monte Carlo, variance reduction techniques such as importance (re-)sampling, and the transform likelihood ratio method, the score function method for sensitivity analysis, the stochastic approximation method and the stochastic counter-part method for Monte Carlo optimization, the cross-entropy method for rare events estimation and combinatorial optimization, and application of Monte Carlo techniques for counting problems. An extensive range of exercises is provided at the end of each chapter, as well as a generous sampling of applied examples.\u003c\/p\u003e \u003cp\u003eThe \u003ci\u003eThird Edition \u003c\/i\u003efeatures a new chapter on the highly versatile splitting method, with applications to rare-event estimation, counting, sampling, and optimization. A second new chapter introduces the stochastic enumeration method, which is a new fast sequential Monte Carlo method for tree search. In addition, the \u003ci\u003eThird Edition \u003c\/i\u003efeatures new material on\u003ci\u003e:\u003c\/i\u003e\u003c\/p\u003e \u003cp\u003e• Random number generation, including multiple-recursive generators and the Mersenne Twister\u003c\/p\u003e \u003cp\u003e• Simulation of Gaussian processes, Brownian motion, and diffusion processes\u003c\/p\u003e \u003cp\u003e• Multilevel Monte Carlo method\u003c\/p\u003e \u003cp\u003e• New enhancements of the cross-entropy (CE) method, including the “improved” CE method, which uses sampling from the zero-variance distribution to find the optimal importance sampling parameters\u003c\/p\u003e \u003cp\u003e• Over 100 algorithms in modern pseudo code with flow control\u003c\/p\u003e \u003cp\u003e• Over 25 new exercises\u003c\/p\u003e \u003cp\u003e\u003ci\u003eSimulation and the Monte Carlo Method, Third Edition \u003c\/i\u003eis an excellent text for upper-undergraduate and beginning graduate courses in stochastic simulation and Monte Carlo techniques. 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Now re-issued in a new style and format, but with the reliable content that the third edition was revered for, this \u003ci\u003eAnniversary Edition\u003c\/i\u003e builds on its strong foundation of measure theory and probability with Billingsley's unique writing style. In recognition of 35 years of publication, impacting tens of thousands of readers, this Anniversary Edition has been completely redesigned in a new, open and user-friendly way in order to appeal to university-level students.\u003c\/p\u003e \u003cp\u003eThis book adds a new foreward by Steve Lally of the Statistics Department at The University of Chicago in order to underscore the many years of successful publication and world-wide popularity and emphasize the educational value of this book. The \u003ci\u003eAnniversary Edition\u003c\/i\u003e contains features including:\u003c\/p\u003e \u003cul\u003e \u003cli\u003eAn improved treatment of Brownian motion\u003c\/li\u003e \u003cli\u003eReplacement of queuing theory with ergodic theory\u003c\/li\u003e \u003cli\u003eTheory and applications used to illustrate real-life situations\u003c\/li\u003e \u003cli\u003eOver 300 problems with corresponding, intensive notes and solutions\u003c\/li\u003e \u003cli\u003eUpdated bibliography\u003c\/li\u003e \u003cli\u003eAn extensive supplement of additional notes on the problems and chapter commentaries\u003c\/li\u003e \u003c\/ul\u003e \u003cp\u003ePatrick Billingsley was a first-class, world-renowned authority in probability and measure theory at a leading U.S. institution of higher education. He continued to be an influential probability theorist until his unfortunate death in 2011. 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Allthenewnotionsandstatementsrequiredforsolvingthepr- lemsaregiveneitherontheoreticalgroundsorintheformulationsoftheproblems vii viii Preface straightforwardly. However,sometimesanotionisusedinthetextbeforeitsformal de nition. Forinstance,theWienerandPoissonprocessesareprocesseswithin- pendentincrementsandthusareformallyintroducedinaTheoreticalgroundsfor Chapter5,buttheseprocessesareusedwidelyintheproblemsofChapters2to4. Theauthorsrecommendthatareaderwhocomestoanunknownnotionorobject usetheIndexinorderto ndthecorrespondingformalde nition. Thesamerec- mendationconcernssomestandardabbreviationsandsymbolslistedattheendofthe book. Someproblemsinthebookformcycles:solutionstooneofthemaregrounded onstatementsofothersoronauxiliaryconstructionsdescribedinsomepreceding solutions. Sometimes,onthecontrary,itisproposedtoprovethesamestatement withindifferentproblemsusingessentiallydifferenttechniques. Theauthorsrec- mendareaderpayspeci cattentiontothesefruitfulinternallinksbetweenvarious topicsofthetheoryofstochasticprocesses. Everypartofthebookwascomposedsubstantiallybyoneauthor. Chapters1-6, and16arecomposedbyA. Kulik,Chapters7,12-15,18,and19byYu. Mishura, Chapters 8-10 by A. Pilipenko, Chapter 17 by A. Kukush, and Chapter 20 by D. Gusak. Chapter11waspreparedjointlybyD. GusakandA. Pilipenko. Atthe sametime,everyauthorhasmadeacontributiontootherpartsofthebookbyprop- ingseparateproblemsorcyclesofproblems,improvingpreliminaryversionsoft- oreticalgrounds,andeditingthe naltext. The authors would like to express their deep gratitude to M. Portenko and A. Ivanovfortheircarefulreadingofapreliminaryversionofthebookandva- ablecommentsthatledtosigni cantimprovementofthetext. Theauthorsarealso gratefultoT. Yakovenko,G. Shevchenko,O. Soloveyko, Yu. Kartashov, Yu. K- menko,A. Malenko,andN. Ryabovafortheirassistanceintranslation,preparing lesandpictures,andcomposingthesubjectindexandreferences. Thetheoryofstochasticprocessesisanextendeddiscipline,andtheauthors- derstandthattheproblembookinitscurrentformmaycausecriticalremarksfrom readers,concerningeitherthestructureofthebookorthecontentofseparatech- ters. Whilepublishingtheproblembookinitscurrentform,theauthorsareopenfor remarks,comments,andpropositions,andexpressinadvancetheirgratitudetoall theircorrespondents. Kyiv DmytroGusak December2008 AlexanderKukush AlexeyKulik YuliyaMishura AndreyPilipenko Contents 1 De?nition of stochastic process. Cylinder?-algebra, ?nite-dimensional distributions, the Kolmogorov theorem. . . . . . . . . . 1 Theoreticalgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Hints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 AnswersandSolutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Characteristics of a stochastic process. Mean and covariance functions. Characteristic functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Theoreticalgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Hints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 AnswersandSolutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3 Trajectories. Modi?cations. Filtrations. . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Theoreticalgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Hints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 AnswersandSolutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4 Continuity. Differentiability. Integrability. . . . . . . . . . . . . . . . . . . . . . . . . 33 Theoreticalgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Hints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 AnswersandSolutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ix x Contents 5 Stochastic processes with independent increments. Wiener and Poisson processes. Poisson point measures. . . . . . . . . . . . . . . . . . . . .","brand":"None","offers":[{"title":"Paperback","offer_id":46155244568753,"sku":"9781461425069","price":72.95,"currency_code":"CAD","in_stock":true},{"title":"Hardcover","offer_id":46155244601521,"sku":"9780387878614","price":72.95,"currency_code":"CAD","in_stock":true},{"title":"Kobo eBook","offer_id":46155244634289,"sku":"57dec7eb-ff1c-304e-8b6d-14eadc6c4360","price":64.49,"currency_code":"CAD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0655\/8980\/5233\/files\/1_f4f9b7b4-39c9-43e6-af94-ee97382f0844.jpg?v=1763167008"},{"product_id":"elementary-statistical-analysis","title":"Elementary Statistical Analysis","description":"\u003cp\u003eA beginning text especially designed for those who probably will not go in to statistics professionally but who plan to go into the physical, biological, and social sciences. 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Traditional ergodic theorems focused on amenable groups, and relied on the existence of an asymptotically invariant sequence in the group, the resulting maximal inequalities based on covering arguments, and the transference principle. Here, Alexander Gorodnik and Amos Nevo develop a systematic general approach to the proof of ergodic theorems for a large class of non-amenable locally compact groups and their lattice subgroups. Simple general conditions on the spectral theory of the group and the regularity of the averaging sets are formulated, which suffice to guarantee convergence to the ergodic mean. In particular, this approach gives a complete solution to the problem of establishing mean and pointwise ergodic theorems for the natural averages on semisimple algebraic groups and on their discrete lattice subgroups. 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