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I am currently considering a set of random variables, [tex] \vec{x} = [x_1,x_2,...x_N][/tex] which are know to follow a multivariate normal distribution,

[tex] P(\vec{x}) \propto \mathrm{exp}(-\frac{1}{2}(\vec{x}-\vec{\mu})^\mathrm{T}\Sigma^{-1}(\vec{x}-\vec{\mu}))[/tex]

The covariance matrix Σ and the vector of mean values μ are constructed numerically from quantum optics, and I the purpose is to calculate the Fisher information of estimating a parameter θ,

[tex] \mathcal{I}(\theta) = \frac{\partial \vec{\mu}^\mathrm{T}}{\partial \theta}\Sigma^{-1} \frac{\partial \vec{\mu}}{\partial \theta}[/tex]

Now, I wish to take the limit as the set of variables approach a continuous distribution,

[tex] x_i \rightarrow x(t), \quad \quad \Sigma_{i,j} \rightarrow \Sigma(t,t') [/tex]

In this case Σ^{-1}is ill-defined and numerically the calculations break down because Σ^{-1}becomes singular when close columns become (nearly) identical in this limit.

I would be very grateful for any suggestions to solve a problem like this.

Is there perhaps a "standard" way to take this continuum limit of a multivariate normal distribution or maybe just of the inverse matrix?

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# Continous limit of a multivariate normal distribution

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