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Sasha Stoikov's paper provides an interesting finite state approach to modeling the mid. It makes good sense to me except one property.

On page 7 of the linked paper, $$G^1(x)=\left(\sum_s\mathbf{Q}^{s-1}\mathbf{R}\right)\mathbf{K}=(1-\mathbf{Q})^{-1}\mathbf{R}\mathbf{K},$$ where $\mathbf{Q}$ is the $nm\times nm$ transition matrix of the transient states and $\mathbf{R}$ the $nm\times 4$ transition matrix of the absorbing states. In my mind, the full transition matrix should look like the following:

$$\mathbf{T}=\left(\begin{array}{cc}\mathbf{Q}&\mathbf{R}\\\mathbf{0}&\mathbf{I}\\\end{array}\right),$$

which does agree with Sasha's own beamer.

Question: What is the property applied here so that $\sum_s\mathbf{Q}^{s-1}\mathbf{R}=(1-\mathbf{Q})^{-1}\mathbf{R}$?

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Found my answer on Wikipedia.

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