Probability Theory: An Analytic View
This second edition of Daniel W. Stroock's text is suitable for first-year graduate students with a good grasp of introductory, undergraduate probability theory and a sound grounding in analysis. It is intended to provide readers with an introduction to probability theory and the analytic ideas and tools on which the modern theory relies. It includes more than 750 exercises. Much of the content has undergone significant revision. In particular, the treatment of Levy processes has been rewritten, and a detailed account of Gaussian measures on a Banach space is given.
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abstract Wiener space addition apply assertion assume Banach space Borel measurable bounded Brownian motion Brownian paths CC G choose compact subsets complete the proof conclude continuous Corollary countable cr-algebra define denote distribution Doob's dTegG EP[X ergodic Exercise exists Finally finite ﬁrst Fourier transform G l,oo G Z+ Gaussian measure given harmonic function Hence Hilbert space Hint Inequality integrable Lemma Levy process lim^oo linear M(dy Markov property martingale mean value measurable function metric Moreover non-decreasing observe open subset orthogonal orthonormal basis P-almost surely P-independent P-integrable particular Poisson process preceding probability measure probability space progressively measurable prove R-valued random variables result right-continuous satisfies separable Banach space sequence submartingale subspace suppose symmetric topology uniformly on compacts unique