## 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

$R(t1,t2)=σ2ℛ(t1,t2)⁢(13.1)$

be observed at N equidistant discrete time instants ti, i = 1, 2,…, N in such a way that

$ti+1−tj=Δ=const.⁢(13.2)$

Then, at the measurer input we have a set of samples xi = x(ti). 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:

$p(x1,x2,...,xN|σ2)=1(2πσ2)N/2det‖ ℛij ‖exp{ −12σ2∑​i=1,j=1NxixjCij }⁢(13.3)$

where

• det ‖ℛij‖ is the determinant of matrix consisting of elements ...

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