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

Triacca, Umberto, and Volodin, Andrei. "On a characterization of orthogonality with respect to particular sequences of random variables in $L^2$." Applications of Mathematics 55.4 (2010): 329-335. <http://eudml.org/doc/37851>.

@article{Triacca2010,
abstract = {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.},
author = {Triacca, Umberto, Volodin, Andrei},
journal = {Applications of Mathematics},
keywords = {Hilbert space; orthogonality; ergodic theorem},
language = {eng},
number = {4},
pages = {329-335},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On a characterization of orthogonality with respect to particular sequences of random variables in $L^2$},
url = {http://eudml.org/doc/37851},
volume = {55},
year = {2010},
}

TY - JOUR
AU - Triacca, Umberto
AU - Volodin, Andrei
TI - On a characterization of orthogonality with respect to particular sequences of random variables in $L^2$
JO - Applications of Mathematics
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 55
IS - 4
SP - 329
EP - 335
AB - 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.
LA - eng
KW - Hilbert space; orthogonality; ergodic theorem
UR - http://eudml.org/doc/37851
ER -