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Additivity of the variance is a characteristic property of the Hilbert space \(L_ 2(\Omega,{\mathfrak A},\mu)\). (English. Russian original) Zbl 0532.46011
Funct. Anal. Appl. 17, 218-220 (1983); translation from Funkts. Anal. Prilozh. 17, No. 3, 66-68 (1983).
Let (\(\Omega\),\({\mathfrak A},\mu)\) be a measure space with finitely continuous measure \(\mu\), E be a rearrangement-invariant Banach function space on \(\Omega\). If x is a random variable and \(m(x)=\int x(\omega)d\mu(\omega)\) its mean value, we define \(\delta(x)=\| x- m(x)\|_ E\). For \(E=L_ 2(\Omega,{\mathfrak A},\mu)\) we have \(\delta^ 2(x)=D(x)\), where D(x) is the usual dispersion of x. In this case \(D(x+y)=D(x)+D(y)\) iff the random variables x and y are uncorrelated. The aim of this work is to show, that the validity of equality \(\delta^ 2(x+y)=\delta^ 2(x)+\delta^ 2(y)\) for all independent x,\(y\in E\) is a characteristic property of \(L^ 2(\Omega,{\mathfrak A},\mu)\) in some class of rearrangement-invariant spaces E.
Reviewer: A.V.Bukhvalov
MSC:
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
60A10 Probabilistic measure theory
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References:
[1] S. G. Krein, Yu. I. Petunin, and E. M. Semenov, Interpolation of Linear Operators [in Russian], Nauka, Moscow (1978).
[2] A. A. Ershov, Avtom. Telemekh., No. 8, 66-100 (1978).
[3] E. M. Krasnenker, Avtom. Telemekh., No. 5, 65-88 (1980).
[4] B. L. Van der Waerden, Mathematical Statistics [Russian translation], IL, Moscow (1960). · Zbl 0184.21701
[5] J. Neveu, Mathematical Foundations of Probability Theory [Russian translation], Mir, Moscow (1969).
[6] Yu. I. Kuritsyn, Yu. I. Petunin, and E. M. Semenov, Mat. Zametki,10, No. 2, 195-205 (1971).
[7] S. N. Bernstein, Collected Works [in Russian], Vol. 4, Nauka, Moscow (1964).
[8] Yu. I. Petunin, Dokl. Akad. Nauk SSSR,170, No. 3, 516-519 (1966).
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