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Probability Spaces with an Infinite Number of Sample Points
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assumed Borel field bounded called central limit theorem Chapman-Kolmogorov equation Chapter characteristic function conditional probability Consider converges corresponding countable covariance function defined density function derived discrete discussion disjoint distributed random variables eigenfunctions eigenvalue elementary outcomes entropy ergodic theorem experiment finite number follows forward equations func given implies independent random variables inequality infinite interval invariant irreducible large numbers Let Xi linear Markov chain Markov process Markovian martingale mean square mean zero measurable function measurable with respect non-negative nondecreasing Notice obtained orthogonal P(Ai probability distribution probability measure probability space probability vector problem process Xt random process real numbers representation sample points satisfy sequence sigma-field sin2 spectral density spectral distribution function stationary process subset supermartingale Suppose tion transition probability matrix uniformly weakly stationary process Wiener process Xk(w Xn(w Xt(w zero and variance