Re: Do time-varying transition matrices exist?

On Jan 9, 10:13 pm, y...@xxxxxxxxxxxxxxxxxxx ( ) wrote:
Suppose that a transition matrix P of nxn contains the transition
probabilities between n states, we know that the probability for state i
to state j after time t is P^t. By doing so, we assume that the
transition matrix remains the same. In reality, different time (months,
for example) should have different transition matrices.
Is it possible that P1 is the transition matrix at t=1, P2 is the
transition matrix at t=2 and so on such that the probability for state i
to state j after t=5 is P1*P2*P3*P4*P5?
Please let me know if any books or articles discuss this subject.
Thank you very much for your help.
Lee

I would say, why not? No principles of probability are violated with
time-varying transition matrices.

Time-varying transition matrices are common in physics, in particular
in the path integral formulation of quantum physics. One book on the
topic is:
http://www.amazon.com/Techniques-Applications-Integration-Dover-Physics/dp/0486445283
With a path integral, you transition between one state and any of a
variety of other states based on something like a probability of
transition (it includes a probability amplitude and a phase, which
makes things interesting). If the Hamiltonian for the system changes
in time (or space, as the wave function propagates), then the matrix
also changes in time (or space).

I imagine the same would be true in any other application of
transition matrices--I've done it myself, in an informal application,
for land-use transitions--but I don't know of books or articles.

Eric
.

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