an:03844381
Zbl 0532.46011
Bajdak, G. I.; Braverman, M. Sh.; Petunin, Yu. I.
Additivity of the variance is a characteristic property of the Hilbert space \(L_ 2(\Omega,{\mathfrak A},\mu)\)
EN
Funct. Anal. Appl. 17, 218-220 (1983); translation from Funkts. Anal. Prilozh. 17, No. 3, 66-68 (1983).
00148837
1983
j
46E30 60A10
rearrangement-invariant Banach function space; random variable; dispersion
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.
A.V.Bukhvalov