ProbabilityandStochasticProcesses withApplications

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probability and Stochastic Processeswith ApplicationsOliver KnillContentsPreface31 What is probability theory? . . . . . . . . . . . . . . . . . . Some paradoxes in probability theory . . . . . . . . . . . . Some applications of probability theory . . . . . . . . . . . 182 Limit probability spaces, random variables, independence . . . . . Kolmogorov s 0 1 law, Borel-Cantelli lemma . . . . . . . . Integration, Expectation, Variance . . . . . . . . . . . . . . Results from real analysis . . . . . . . . . . . . . . . . . . . Some inequalities . . . . . . . . . . . . . . . . . . . . . . . . The weak law of large numbers . . . . . . . . . . . . . . . . The probability distribution function . . . . . . . . . . . . . Convergence of random variables.

random variables, for Poisson processes, see [49, 9]. For the geometry of numbers for Fourier series on fractals [45]. The book [114] contains examples which challenge the theory with counter examples. [33, 95, 71] are sources for problems with solutions. Probability theory can be developed using nonstandard analysis on finite probability ...

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