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Mathematics - Probability Theory and Stochastic Processes | The Generic Chaining - Upper and Lower Bounds of Stochastic Processes

The Generic Chaining

Upper and Lower Bounds of Stochastic Processes

Talagrand, Michel

2005, VIII, 222 p.

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What is the maximum level a certain river is likely to reach over the next 25 years? (Having experienced three times a few feet of water in my house, I feel a keen personal interest in this question. ) There are many questions of the same nature: what is the likely magnitude of the strongest earthquake to occur during the life of a planned building, or the speed ofthe strongestwind a suspension bridge will have to stand? All these situations can be modeled inthesamemanner. Thevalue X of the quantity of interest (be it water t level or speed of wind) at time t is a random variable. What can be said about the maximum value of X over a certain range of t? t A collection of random variables (X ), where t belongs to a certain index t set T, is called a stochastic process, and the topic of this book is the study of the supremum of certain stochastic processes, and more precisely to ?nd upper and lower bounds for the quantity EsupX . (0. 1) t t?T Since T might be uncountable, some care has to be taken to de?ne this quantity. For any reasonable de?nition of Esup X we have t t?T EsupX =sup{EsupX ; F?T,F ?nite} , (0. 2) t t t?T t?F an equality that we will take as the de?nition of the quantity Esup X . t t?T Thus, the crucial case for the estimation of the quantity (0.

Content Level » Research

Keywords » Gaussian process - Gaussian processes - Stochastic processes - infinitely divisible processes - matchings - stochastic process

Related subjects » Analysis - Physical & Information Science - Probability Theory and Stochastic Processes

Table of contents / Errata 

Overview and Basic Facts.- Gaussian Processes and Related Structures.- Matching Theorems.- The Bernoulli Conjecture.- Families of distances.- Applications to Banach Space Theory.

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