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.
Highlights
In some studies we have a two dimensional vector whose components belong to a Hilbert space H with the inner product H
For example when we have vector valued random variable whose components are members of L2(Ω, P), for some probability space (Ω, F, P). Another example is in the functional data analysis where we have a two dimensional vector whose elements are square integrable functions on the real line. In these cases if x1, y1, x2 and y2 belong to H z1 = (x1, y1) and z2 = (x2, y2) belong to the product space H × H and the ordinary inner product on the product space is defined by
< z1, z2 > = < x1, x2 >H + < y1, y2 >H. By this definition inner product in the product space always results in a scalar value and two vectors z1 = (x1, y1) and z2 = (x2, y2) are orthogonal if < x1, x2 >=< y1, y2 >= 0 or < x1, x2 >= − < y1, y2 >
Summary
In some studies we have a two dimensional vector whose components belong to a Hilbert space H with the inner product H. For B-valued random variables characteristic functional, mean and covariance operator are defined as follows. The set of B-valued random variables with finite mean is a Banach space.
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