Abstract
The notion of a stochastic (random) function is formalized and the classification of these functions is discussed. We present different ways to describe a stochastic process, in terms of a distribution function, a probability density function, and moment functions, and in particular the expectation, variance, covariance, and correlation functions. We consider a stationary stochastic process in the narrow and broad senses. We describe the Wiener–Khinchin transformation and generalized Wiener–Khinchin transformation. The spectral approach for describing a stochastic process is presented. The ergodic and fragmentary ergodic processes are considered.
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Notes
- 1.
This relates, in particular, to the N-dimensional probability density function:
$$ {f}_{\overrightarrow{x}}\left({x}_1,\dots, {x}_N;{t}_1,\dots, {t}_N\right)={f}_{\overrightarrow{x}}\left({x}_1,\dots, {x}_N;{t}_1+\tau, \dots, {t}_N+\tau \right), $$where Ï„ is an arbitrary number.
References
Gorban, I.I.: Teoriya Ymovirnostey i Matematychna Statystika dla Naukovykh Pratsivnykiv ta Inzheneriv (Probability Theory and Mathematical Statistics for Scientists and Engineers). IMMSP, NAS of Ukraine, Kiev (2003)
Gorban, I.I.: Sluchaynost i gipersluchaynost (Randomness and Hyper-randomness). Naukova Dumka, Kiev (2016)
Levin, B.R.: Teoreticheskie Osnovy Statisticheskoy Radiotekhniki (Theoretical Basis of Statistical Radio Engineering), vol. 1. Sovetskoe Radio, Moscow (1974)
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Gorban, I.I. (2018). Stochastic Functions. In: Randomness and Hyper-randomness. Mathematical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-60780-1_3
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DOI: https://doi.org/10.1007/978-3-319-60780-1_3
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