Random ProcessesThis text has as its object an introduction to elements of the theory of random processes. Strictly speaking, only a good background in the topics usually associated with a course in Advanced Calculus (see, for example, the text of Apostol [1]) and the elements of matrix algebra is required although additional background is always helpful. N onethe less a strong effort has been made to keep the required background on the level specified above. This means that a course based on this book would be appropriate for a beginning graduate student or an advanced undergraduate. Previous knowledge of probability theory is not required since the discussion starts with the basic notions of probability theory. Chapters II and III are concerned with discrete probability spaces and elements of the theory of Markov chains respectively. These two chapters thus deal with probability theory for finite or countable models. The object is to present some of the basic ideas and problems of the theory in a discrete context where difficulties of heavy technique and detailed measure theoretic discussions do not obscure the ideas and problems. |
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Contents
Introduction | 3 |
Markov Chains | 36 |
Probability Spaces with an Infinite Number of Sample Points | 68 |
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A₁ assumed asymptotic Borel field called central limit theorem Chapman-Kolmogorov equation Chapter characteristic function conditional probability Consider continuous converges corresponding countable covariance function defined density function derived dF(x discrete discussion disjoint eigenfunctions eigenvalue elementary outcomes experiment finite number follows forward equations given implies independent random variables inequality infinite interval invariant irreducible jump Kolmogorov large numbers linear Markov chain Markov process Markovian martingale mean square mean zero measurable with respect non-negative nondecreasing Notice obtained orthogonal parameter Poisson positive probability distribution probability measure probability space probability vector problem random process random variable X(w real numbers representation sample points satisfy sequence sigma-field sin2 spectral density stationary process stationary transition mechanism subset supermartingale tion transition probability function transition probability matrix uniformly variables X1 weakly stationary process X₁ Xn(w zero and variance