stochastic process

plural stochastic processes

1. mathematicsA function that maps elements of an index set to elements of a collection of random variables; historically, a collection of random variables indexed by a set of numbers usually regarded as points in time.

   A unified treatment of all the aspects of Fourier theory relevant to the Fourier analysis of stochastic processes is not only unavoidable, but also intellectually satisfying, and in fact time-saving.

   2014, Pierre Brémaud, Fourier Analysis and Stochastic Processes, Springer, page xi

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   A stochastic process #92;#123;#92;zeta#95;t#92;#125; is a family of random variables that depends on a parameter t, usually called time.[…]Two important properties stochastic processes may have are given in the following definition.

   1994, Andrzej Lasota, Michael C. Mackey, Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics, 2nd edition, Springer, page 254

   Hence even Bernoulli processes, often regarded as the most random stochastic processes, are simulated by deterministic systems in science when observed with specific observation functions. But then any attempt to separate such deterministic systems from the deterministic systems needed to simulate the stochastic processes in science must fail. The philosophical implication of this result is that the above guess is wrong, viz. the deterministic systems which simulate the stochastic processes in science include deterministic systems in science. This indicates that from a predictive viewpoint, stochastic processes and deterministic systems are very similar.

   2009, Charlotte Werndl, “Deterministic Versus Indeterministic Descriptios: Not That Different After All”, in Alexander Hieke, Hannes Leitgeb, editors, Reduction - Abstraction - Analysis, page 71

   Synonyms: probabilistic process, probability process, random process