Summary
A random field over l p is a stochastic process X(t), where t is an element of l p .It is said to have homogeneous and isotropic increments if E(X(t) − X(s)) 2 is a function of ∥t-s∥. The subject of this work is the spectral theory of such processes. The main results are: a representation of the field as a series of filtered, orthogonal processes with a real time parameter; a representation as a white noise integral over l p ;limit theorems for the average of X over a sphere; and, finally, filtering of the orthogonal components.
In particular, we mention: (1) The averages over spheres of increasing dimension converge in quadratic mean for p=2 but not for 0<p<2. (2) The limiting distribution of a fixed coordinate of a point uniformly distributed over the l p -unit sphere in n-space, n → ∞, has the density [2γ(1/p+1)p 1/p]−1exp(-¦x¦p/P).
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Berman, S.M.: Some continuity properties of Brownian motion with the time parameter in Hilbert space. Trans. Amer. math. Soc. 131, 182–198 (1968).
Bretagnolle, J., D.Dacunha Castelle et J.L. Krivine: Lois stables et espaces L p.Ann. Inst. Henri Poincaré, n. Sér., Sect. B, 2, 231–259 (1966).
Lévy, P.: Processus stochastiques et mouvement brownien. Paris: Gauthier-Villars 1948; 2nd ed., 1966.
—: Le Mouvement Brownien, Mem. Sciences Mathématiques, CXXVI. Paris: Gauthier-Villars 1954.
Loéve, M.: Probability Theory. 3rd ed. Princeton: D. Van Nostrand 1963.
McKean, H.P.: Brownian motion with a several-dimensional time. Theor. Probab. Appl. 9, 357–368 (1963).
Schoenberg, I.J.: Metric spaces and completely monotone functions. Ann. of Math., II. Ser. 39, 811–841 (1938).
Strait, P.T.: Sample function regularity for Gaussian processes with the parameter in a Hilbert space. Pacific J. Math. 19, 159–173 (1966).
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This paper represents results obtained at the Courant Institute of Mathematical Sciences, New York University, under the sponsorship of the National Science Foundation Grants NSF-GP-7378 and NSF-GW-2049.
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Berman, S.M. Second order random fields over l p with homogeneous and isotropic increments. Z. Wahrscheinlichkeitstheorie verw Gebiete 12, 107–126 (1969). https://doi.org/10.1007/BF00531644
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DOI: https://doi.org/10.1007/BF00531644