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Ideas?

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- Thread starter Palindrom
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Ideas?

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mathman

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Ideas?

Naive(?) question. What is AC?

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Sorry- AC is absolutely continuous.

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mathman

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mathman

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be, I have a specific covariance function and I'm struggling to “take off”...

Thanks anyway though!

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be, I have a specific covariance function and I'm struggling to “take off”...

Thanks anyway though!

I'm not sure I understand your question regarding sampling times for stationary processes. In any case, for the variance-covariance matrix of two independent random sample vectors [tex]X_{i},X_{j}[/tex] having an equal number of comparable components, the matrix is symmetric, non-negative definite and, unless it is singular, invertible.r

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- #9

mathman

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Is this example supposed to be a 2-vector with both components sampled at the same time? This would be the exception to the general statement. I suspect that as long as the vector components are samples at different times you will have an invertible matrix.

be, I have a specific covariance function and I'm struggling to “take off”...

Thanks anyway though!

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- #11

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Covariance is defined for two random variables at the same mean times. A stationary process generally presumes a constant mean.

http://math.lbl.gov/~kourkina/math220/chapter4.pdf [Broken]

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How is that related to anything?

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How is that related to anything?

I'll ignore your attitude for the benefit of others reading this thread. Singular v-c matrices may arise with sampling from standardized Gaussian stationary distributions. They can be dealt with analytically.

http://www.riskglossary.com/link/positive_definite_matrix.htm

I don't see how random sample timing or frequency comes in here, in terms of individual samples.

EDIT: I'm assuming you know that singular matrices can be identified by simply obtaining the determinant.

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I really meant I don't see the connection between my question and what you wrote. I'm not talking about any random times, just about simple, basic non-degeneracy; for each n times t_1,...,t_n, the random vector X=(X_{t_1},...,X_{t_n}) has a density function. This can be otherwise stated as “the covariance matrix of X is invertible”.

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I really meant I don't see the connection between my question and what you wrote. I'm not talking about any random times, just about simple, basic non-degeneracy; for each n times t_1,...,t_n, the random vector X=(X_{t_1},...,X_{t_n}) has a density function. This can be otherwise stated as “the covariance matrix of X is invertible”.

Yes, this is almost always true for the variance-covariance matrix of any sample. However, given random sampling, it's possible that a singular (degenerate) matrix could be obtained. My second link shows how this can be handled analytically. If this is not your question, could you please rephrase it?

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As an example, if R(t,s)=1 for all s and t then the answer is “degenerate”, while for R(s,t)=min{s,t} the answer is “non-degenerate”.

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As an example, if R(t,s)=1 for all s and t then the answer is “degenerate”, while for R(s,t)=min{s,t} the answer is “non-degenerate”.

Well, it seems too simple, so I must be missing something. The determinant of the v-c matrix (or any matrix) of the components of a random vector will be zero if it is degenerate and non-zero if it's non-degenerate. If you're dealing with multiple matrices, the product of the determinants will be zero if at least one determinant is zero.

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