Generalized Stochastic Processes: Linear Relations to White Noise and Orthogonal Representations

Abstract

We present two linear relations between an arbitrary (real tempered second order) generalized stochastic process over

{\mathbb{R}}^d

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and White Noise processes over

{\mathbb{R}}^d

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. The first is that any generalized stochastic process can be obtained as a linear transformation of a White Noise. The second indicates that, under dimensional compatibility conditions, a generalized stochastic process can be linearly transformed into a White Noise. The arguments rely on the regularity theorem for tempered distributions, which is used to obtain a mean-square continuous stochastic process which is then expressed in a Karhunen–Loève expansion with respect to a convenient Hilbert space. The first linear relation obtained allows also to conclude that any generalized stochastic process has an orthogonal representation as a series expansion of deterministic tempered distributions weighted by uncorrelated random variables with summable variances. This representation is then used to conclude the second linear relation.

Keywords

Generalized stochastic process / White noise / Karhunen–Loève expansion / Orthogonal representations / 60H40 / 60G20 / 60G60 / 60B11 / 60G12 / 60H15

References

[1] Angulo J, Ruiz-Medina M. On the orthogonal representation of generalized random fields. Stat Probab Lett, 1997, 31(3): 145-153.

[2] Angulo J, Ruiz-Medina M, Anh V. Wavelet orthogonal approximation of fractional generalized random fields on bounded domains. Theory Probab Math Stat, 2006, 73: 1-17.

[3] Brislawn C. Traceable integral kernels on countably generated measure spaces. Pac. J. Math., 1991, 150(2): 229-240.

[4] Carrizo Vergara, R.: Development of geostatistical models using stochastic partial differential equations. PhD thesis, MINES ParisTech, PSL Research University (2018). https://hal.archives-ouvertes.fr/tel-02126057/file/CarrizoV

[5] Gelfand IM, Vilenkin NI. Generalized Functions, 1964, New York. Academic Press

[6] Hida T, Kuo H-H, Potthoff J, Streit L. White Noise: An Infinite Dimensional Calculus, 2013, Dordrecht. Springer253

[7] Holden H, Øksendal B, Ubøe J, Zhang T. Stochastic Partial Differential Equations: A Modeling. White Noise Functional Approach, 2009, New York. Springer

[8] Itô K. Stationary random distributions. Memoirs of the college of science, University of Kyoto. Ser. A: Math., 1954, 28(3): 209-223

[9] Lindgren F, Rue H, Lindström J. An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach. J R Stat Soc: Ser B (Stat Methodol), 2011, 73(4): 423-498.

[10] Loève, M.: Probability theory, Vol. II. Graduate texts in mathematics 46, 0–387 (1978)

[11] Matheron G. Les Variables Régionalisées et Leur Estimation: Une Application de la Théorie des Fonctions Aléatoires aux Sciences De la nature, 1965, Paris. Masson et CIE

[12] Medina JM, Frías BC. On the as convergence of certain random series to a fractional random field in D′\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ D^{\prime }$$\end{document}. Stat Probab Lett, 2005, 74(1): 39-49.

[13] Meidan R. Reproducing-kernel hilbert spaces of distributions and generalized stochastic processes. SIAM J. Math. Anal., 1979, 10(1): 62-70.

[14] Red-Horse JR, Ghanem RG. Elements of a function analytic approach to probability. Int. J. Numer. Meth. Eng., 2009, 80(6–7): 689-716.

[15] Reed M, Simon B. Methods of Modern Mathematical Analysis: Functional Analysis, 1980, Singapore. Academic Press

[16] Rozanov YA. Markov Random Fields, 1982, New York. Springer.

[17] Schwartz L. Théorie des Distributions, 1966, Paris. Hermann

[18] Trèves F. Topological Vector Spaces. Distributions and Kernels, 1967, New York. Academic Press

[19] Walsh, J.B.: An introduction to stochastic partial differential equations. In: École d’Été de Probabilités de Saint Flour XIV-1984, pp. 265–439. Springer, Berlin, Heidelberg (1986)

Funding

Swiss Ornithological Institute

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