On a characterization of orthogonality with respect to particular sequences of random variables in $L^2$

summary:This note deals with the orthogonality between sequences of random variables. The main idea of the note is to apply the results on equidistant systems of points in a Hilbert space to the case of the space $L^2(\Omega ,\mathcal F,\mathbb P)$ of real square integrable random variables. The main result gives a necessary and sufficient condition for a particular sequence of random variables (elements of which are taken from sets of equidistant elements of $L^2(\Omega ,\mathcal F,\mathbb P)$) to be orthogonal to some other sequence in $L^2(\Omega ,\mathcal F,\mathbb P)$. The result obtained is interesting from the point of view of the time series analysis, since it can be applied to a class of sequences random variables that exhibit a monotonically increasing variance. An application to ergodic theorem is also provided