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Limiting distribution and eigenvalues

+2 votes
asked Mar 24, 2016 in STA 3041F - Time Series and Markov Chains by Dean_Bunce (1,160 points)

Does anyone have a clear explanation for why a CTMC on a finite state space has a limiting dbn iff Q has exactly one 0 eigenvalue?

commented Mar 31, 2016 by simon_rigby (4,220 points)

Where did you get this fact? :) I'm trying to picture why this would be so, but haven't done any working out.

commented Apr 3, 2016 by Dumie Nyathi (260 points)

It was part of one of the exercises we had to do in class...

1 Answer

+2 votes
answered Apr 4, 2016 by Andrew_Soane (480 points)

If \(Q\) has exactly one zero eigenvalue then it means there exists a vector \(\mathbf{x}\) such that \(Q\mathbf{x}=0\), which implies that \(\mathbf{x}\) has to be equal to your stationary distribution