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MathGroup Archive 2005

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General 3-state stochastic matrix

  • To: mathgroup at smc.vnet.net
  • Subject: [mg58164] General 3-state stochastic matrix
  • From: Virgil Stokes <virgil.stokes at it.uu.se>
  • Date: Tue, 21 Jun 2005 06:02:37 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

I have tried to find the limit (as n, the power of the matrix, goes to 
infinity) for the general 3-state stochastic matrix using the following 
code:

Clear[α2, α3, β1, β3, γ1, γ2]
T = {{1 - α2 - α3, α2, α3}, {β1, 1 - β1 - β3, β3}, {γ1, γ2, 1 - γ1 - γ2}};
MatrixForm[T]
xx = MatrixPower[T, n];
TimeUsed[]
zz = Limit[xx, n -> â??, Assumptions -> 0 < α2 < 1 && 0 <
α3 < 1 &&
0 < α2 + α3 < 1 && 0 < β1 < 1 && 0 <
β3 < 1 && 0 < β1 + β3 < 1 &&
0 < γ1 < 1 && 0 < γ2 < 1 && 0 < γ1 + γ2 < 1] // FullSimplify;
TimeUsed[]
MatrixForm[zz]

However, it does not find a symbolic solution. I would appreciate it 
greatly if someone else could look at this and see if they are able to 
get a symbolic solution. Warning! this can take considerable CPU time.

Note, for a general 2-state stochastic matrix, the above approach works 
fine.

--Thanks, V. Stokes


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