Let $X_i$ ($i=1, \dots$) be an orthonormal basis for $L^2(\Omega, \mathbb P)$. In particular, it holds that $$\mathbb E[X_iX_j] = \delta_{ij}.$$ Now take $Z\in L^2(\Omega, \mathbb P)$ and define $\tilde X_i:=ZX_i$. Let $\Sigma_m$ be the covariance matrix of $\tilde X_1, \dots, \tilde X_m$. Are there conditions on $Z$ ensuring that the rank of $\Sigma_m$ is bounded in $m$?
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4$\begingroup$ In general, $ZX_i$ are only in $L^1(\Omega,P)$ so that the covariance is not necessarily defined. $Z\in L^\infty$ would be okay. $\endgroup$– Jochen WengenrothCommented Nov 5, 2014 at 8:11
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1 Answer
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The rank of $\Sigma_m$ is the dimension of the vector spaces spanned by $\tilde{X}_1,\dotsc,\tilde{X}_m$. In particular, if $\newcommand{\bP}{\mathbb{P}}$ $\bP(Z=0)=0$, then the rank of $\Sigma_m$ is $m$ because the random variables $ZX_1,\dotsc, ZX_m$ are linearly independent in this case.
answered Nov 4, 2014 at 19:39