Probability and Random Processes
In this chapter we develop basic mathematical models of discrete time random processes. Such processes are also called discrete time stochastic processes, information sources, and time series. Physically a random process is something that produces a succession of symbols called “outputs” in a random or nondeterministic manner. The symbols produced may be real numbers such as produced by voltage measurements from a transducer, binary numbers as in computer data, two-dimensional intensity fields as in a sequence of images, continuous or discontinuous waveforms, and so on. The space containing all of the possible output symbols is called the alphabet of the random process, and a random process is essentially an assignment of a probability measure to events consisting of sets of sequences of symbols from the alphabet. It is useful, however, to treat the notion of time explicitly as a transformation of sequences produced by the random process. Thus in addition to the common random process model we shall also consider modeling random processes by dynamical systems as considered in ergodic theory.
KeywordsProbability Measure Random Process Probability Space Measurable Space Event Space
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- 1.R. B. Ash, Real Analysis and Probability, Academic Press, New York, 1972.Google Scholar
- 7.M. Denker, C. Grillenberger, and K. Sigmund, Ergodic Theory on Compact Spaces, 57, Lecture Notes in Mathematics, Springer-Verlag, New York, 1970.Google Scholar
- 13.A. N. Kolmogorov, Foundations of the Theory of Probability, Chelsea, New York, 1950.Google Scholar
- 14.U. Krengel, Ergodic Theorems, De Gruyter Series in Mathematics, De Gruyter, New York, 1985.Google Scholar
- 17.D. Ornstein, Ergodic Theory, Randomness, and Dynamical Systems, Yale University Press, New Haven, 1975.Google Scholar