On Hilbert C*-module-valued Random Variables

DOI

10.2991/jsta.2015.14.4.2How to use a DOI?

Keywords

Banach Valued random variable; central limit theorem, Covariance operator, Hilbert <i>C</i>*-modules

Abstract

In this paper random variables that take their values from a Hilbert C*-module are defined and three definitions for the mean, covariance operator, and Gaussian distribution of these random variables are given and it is shown that these definitions are equivalent. Furthermore, the concept of covariance of two real valued random variables and its properties are extended to two Hilbert C*-module valued random variables. These lead us to the generalization of Rao-Blackwell theorem for this type of random variables. Finally, in a special case, it is proved that the finiteness of second moment of the norm of such a random variable is a sufficient condition for the central limit theorem to be true.

Copyright

© 2017, the Authors. Published by Atlantis Press.

Open Access

This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - K. Shafie
PY  - 2015
DA  - 2015/12/01
TI  - On Hilbert C*-module-valued Random Variables
JO  - Journal of Statistical Theory and Applications
SP  - 359
EP  - 367
VL  - 14
IS  - 4
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.2015.14.4.2
DO  - 10.2991/jsta.2015.14.4.2
ID  - Shafie2015
ER  -