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Suppose we have a sequence $x_i, \ldots, x_n$ of correlated random variables with pairwise covariances $\Sigma_{ij}$, (the $(i,j)$-entry of the covariance matrix $\Sigma$). Consider $\hat{x}=\sum_{j}c_jx_j$, where $c_j=\frac{\sum_i K_{ij}}{\sum_{i,m} K_{im}}$ (here $K_{ij}$ denotes the $(i,j)$-entry of the inverse of the covariance matrix $K = \Sigma^{-1}$). If each $x_j$ has mean $\mu$, then one can easily see that $\mathbb{E}[\hat{x}]=\mu$. However, I'm having trouble proving that $Var[\hat{x}]=Cov[\hat{x},\hat{x}]=(\sum_{i.j}K_{ij})^{-1}$. Can anyone shed some light on how to obtain this value for the variance? My instinct was to use $Var[X]=E(X^2)-[E(X)]^2$, but this only creates more trouble when we get to the $[E(X)]^2$ term. Any help is much appreciated.

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Substituting in the formula for the variance the definition of the variable $\hat{x}$ and the fact that $E[\hat{x}]=\mu$:

$$Var[\hat{x}]=E(\hat{x}^2)-E(\hat{x})^2=\sum_{jk}c_jc_kE(x_jx_k)-\mu^2$$

We know that $$E(x_jx_k)=\text{Cov}[x_j,x_k]+E[x_j]E[x_k]=\Sigma_{jk}+\mu^2$$

and now we would like to try and compute the sum in the first expression. First we notice that $\sum_{i}c_i=1$. Then we substitute the 2nd equation into the sum and decompose:

$$\sum_{jk}c_jc_kE(x_jx_k)=\sum_{jk}{c_jc_k\Sigma_{jk}}+\mu^2\sum_{j}c_j\sum_{k}c_k=\sum_{jk}{c_jc_k\Sigma_{jk}}+\mu^2$$

so the mean of the variable cancels and we are left with the evaluation of sum which can be performed as follows:

$$\sum_{jk}{c_jc_k\Sigma_{jk}}=\frac{1}{(\sum_{im}K_{im})^2}\sum_{i_1i_2}\sum_{jk}K_{i_1j}\Sigma_{jk}K^{T}_{ki_2}\\=\frac{1}{(\sum_{im}K_{im})^2}\sum_{i_1i_2}(K\Sigma K^{T})_{i_1i_2}\\=\frac{1}{(\sum_{im}K_{im})^2}\sum_{i_1i_2}(K^{T})_{i_1i_2}\\=\frac{1}{(\sum_{im}K_{im})^2}\sum_{i_1i_2}K_{i_1i_2}=(\sum_{ij}K_{ij})^{-1}\\$$

and finally we conclude that:

$$Var[\hat{x}]=\sum_{jk}{c_jc_k\Sigma_{jk}}+\mu^2-\mu^2=(\sum_{ij}K_{ij})^{-1}$$

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