## 13 Estimation of Stochastic Process Variance

### 13.1 Optimal Variance Estimate of Gaussian Stochastic Process

Let the stationary Gaussian stochastic process *ξ(t)* with the correlation function

$$R({t}_{1},{t}_{2})={\sigma}^{2}\mathcal{R}({t}_{1},{t}_{2})\left(13.1\right)$$

be observed at *N* equidistant discrete time instants *t*_{i}, *i* = 1, 2,…, *N* in such a way that

$${t}_{i+1}-{t}_{j}=\Delta =\text{const}.\left(13.2\right)$$

Then, at the measurer input we have a set of samples *x*_{i} = *x*(*t*_{i}). Furthermore, we assume that the mathematical expectation of observed stochastic process is zero. Then, the conditional *N*-dimensional pdf of Gaussian stochastic process can be presented in the following form:

$$p({x}_{1},{x}_{2},\mathrm{...},{x}_{N}|{\sigma}^{2})=\frac{1}{{\left(2\pi {\sigma}^{2}\right)}^{N/2}\sqrt{\mathrm{det}\Vert {\mathcal{R}}_{ij}\Vert}}\mathrm{exp}\left\{-\frac{1}{2{\sigma}^{2}}\underset{i=1,j=1}{\overset{N}{{{\displaystyle \sum}}^{\text{}}}}{x}_{i}{x}_{j}{C}_{ij}\right\}\left(13.3\right)$$

where

det ‖ℛ

_{ij}‖ is the determinant of matrix consisting of elements ...

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