The theory of stochastic processes has developed so much in the last twenty years that the need for a systematic account of the subject has been felt, particularly by students and instructors of probability. This book fills that need. While even elementary definitions and theorems are stated in detail, this is not recommended as a first text in probability and there has been no compromise with the mathematics of probability. Since readers complained that omission of certain mathematical detail increased the obscurity of the subject, the text contains various mathematical points that might otherwise seem extraneous. A supplement includes a treatment of the various aspects of measure theory. A chapter on the specialized problem of prediction theory has also been included and references to the literature and historical remarks have been collected in the Appendix.
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INTRODUCTION AND PROBABILITY BACKGROUND
DEFINITION OF A STOCHASTIC PROCESSPRINCIPAL
PROCESSES WITH MUTUALLY INDEPENDENT RAN
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absolutely continuous according to Theorem Baire function Borel field Borel set characteristic function closed linear manifold conditional expectation conditional probability continuous parameter converges with probability corresponding defined definition discrete parameter discussion enumerable equation ergodic ergodic set example exists fact finite number Fourier function measurable Gaussian given Hence hypothesis implies independent increments independent random variables inequality infinite integrand large numbers law of large Lebesgue measure Lemma limit Markov process martingale matrix measurable with respect metrically transitive monotone non-decreasing mutually independent random non-negative obtained orthogonal orthogonal increments parameter set points of discontinuity probability measure proof prove sample functions sample space satisfied semi-martingale sequence set of probability spectral distribution function stationary processes stationary wide sense step functions stochastic integral stochastic process strictly stationary subset suppose Theorem 2.1 transition probabilities true uniformly integrable uniquely determined values vanishes xn process