| Summary: | In this paper random variables that take their values from a Hilbert <i>C</i>*-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 <i>C</i>*-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.
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