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Probability Spaces with an Infinite Number of Sample Points
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assume asymptotic binomial Borel field called central limit theorem Chapman-Kolmogorov equation Chapter characteristic function conditional probability Consider corresponding countable covariance function defined density function derived discrete discussion disjoint distributed random variables eigenfunctions eigenvalue elementary outcomes entropy equal ergodic theorem experiment finite number follows forward equations func given implies independent random variables inequality infinite interval invariant irreducible large numbers linear Markov chain Markov process Markovian mean square mean zero measurable function measurable with respect non-negative nondecreasing Notice obtained orthogonal prob probability distribution probability measure probability space probability vector problems process Xt random process real numbers representation sample points sequence sigma-field spectral density spectral distribution function stationary transition mechanism subsets Suppose tion transition probability function transition probability matrix uniformly weakly stationary process Wiener process Xk(w Xn(w Xt(w zero and variance