[ This is a repost of the following article: ]
[ From: Law Hiu Chung <[EMAIL PROTECTED]> ]
[ Subject: Question on Gaussian distribution ]
[ Newsgroups: sci.stat.math ]
[ Message-ID: <9ond41$sk6$[EMAIL PROTECTED]> ]
We define a function f(x) as a Gaussian process if
for any n, and for any x1, ... xn, f(x1), f(x2), ... f(xn)
follows a Gaussian distribution.
Can I interpret this definition intuitively as
Given a f(x) in a set X of functions (satisfying some conditions),
the "projection" of f(x) to a finite set of basis
{ delta(x1), delta(x2), ... delta(xn) }
must be Gaussian irrespect of the number of xi's and their values.
Then f(x) "follows a Gaussian distribution".
(The above is meaningless without defining the inner product, but I would
like to know if my intuition is correct or not.)
Can I generalize the above to:
Given an inner product space X (with possibly infinite dimension), I can
define "a Gaussian distribution" (or other appropriate term)
on X such that
For x \in X, if we project it to a finite set of orthonormal vectors
( phi_1, phi_2, ..., phi_n) and get the projection (a1, a2, ... an),
the tuple follows an n-dimensional Gaussian distribution.
This should hold for all values of n and all set of orthonormal vectors.
Is this definition legal?
I guess "X being an inner product space" may not be enough. If that is
the case, what other conditions are needed?
If this looks like a text book question to you, can you point me to
some good introductory books on this topic? I have tried to read some books
on Gaussian measures, but
they are too technical for me -- an engineering person without a strong
background in measure theory.
Thank you for your help.
--
Martin Law
Computer Science Department
Hong Kong University of Science and Technology
--
Martin Law
Computer Science Department
Hong Kong University of Science and Technology
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