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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:
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