Functional Analysis for Probability and Stochastic Processes: An IntroductionCambridge University Press, 11.08.2005 - 393 Seiten This text is designed both for students of probability and stochastic processes, and for students of functional analysis. It presents some chosen parts of functional analysis that can help understand ideas from probability and stochastic processes. The subjects range from basic Hilbert and Banach spaces, through weak topologies and Banach algebras, to the theory of semigroups of bounded linear operators. Numerous standard and non-standard examples and exercises make the book suitable as a course textbook as well as for self-study. |
Inhalt
| 37 | |
| 80 | |
Brownian motion and Hilbert spaces | 121 |
Dual spaces and convergence of probability measures | 147 |
The Gelfand transform and its applications | 201 |
Semigroups of operators and Lévy processes | 234 |
Markov processes and semigroups of operators | 294 |
Appendixes | 363 |
References | 385 |
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Functional Analysis for Probability and Stochastic Processes: An Introduction Adam Bobrowski Eingeschränkte Leseprobe - 2005 |
Häufige Begriffe und Wortgruppen
algebraic subspace Banach algebra Banach space BM(R Borel measures bounded linear operator Brownian motion BUC(R C(R+ Co(R Co(S continuous functions Convergence Theorem converges weakly convolution semigroup Corollary defined definition denote dense density distribution e-at elements equals equation Example Let Exercise Let Exercise Prove Exercise Show exists follows formula functional F H₁ Hence Hilbert space Hint to Exercise implies independent inequality integrable isometrically isomorphic L¹(R+ Lebesgue measure Lemma Lévy processes limit limn linear functional linear space martingale matrix measurable functions measure space Moreover multiplicative functionals non-negative norm o-algebra parameter particular Poisson probability measure probability space Proof random variables satisfies semigroup strongly continuous semigroup subset suffices to show Suppose t₁ topological space topology vector zero μη Ξη
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Seite 187 - S is said to be tight if for every e > 0 there exists a compact set K such that P(A") > 1 - e for all P in this family. 5.7.10 Urysohn's Theorem The universal space where the supports of limit measures "live" is the Hubert cube H = [0, 1]N = H^j S¡ where all Si — [0, 1].
Seite 24 - This follows from the fact that the probability distribution of the sum of two independent random variables is the convolution of their probability distributions.
Seite 297 - I = 0, 1, 2, . . . } or a continuous parameter stochastic process {X(t), t > 0} is said to be a Markov process if, for any set of n time points...
Seite 75 - Y, and suppose that there exists a constant C > 0 such that for all x € Xo, Hazily < CHxHx, where || • ||x and || • ||y are norms in X and Y, respectively.
Seite 49 - Completeness of this algebra follows from the fact that the uniform limit of a sequence of continuous functions is continuous.
Seite 121 - This is a good topic to center a discussion around because Brownian motion is in the intersection of many fundamental classes of processes. It is a continuous martingale, a Gaussian process, a Markov process or more specifically a process with independent increments...
Seite 3 - N. A sequence is said to be convergent if it converges to some element x° of the space.
Seite 25 - Use convolution to show that if X and Y are independent exponential random variables with parameters a and...