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The Markov property is unchanged. We can still
state that
P(Xn+1 = j | Xn = i,
Hn)
depends only on i, j and n,
and call it pn,ij.
These can be assembled into a transition matrix
Pn.
We can again use Chapman-Kolmogorov to show that the m-step transition probability P(Xn+m = j | Xn = i) is the (i, j)th element of a matrix obtained by matrix multiplication.
But the classification of states and the description of limiting behaviour no longer apply.
Marital status model:
A woman may be: never married, married for the first time,
divorced, widowed or remarried. We can draw a state space
diagram, but transition probabilities change with age.
Reversionary annuity:
While a husband is alive he pays into a scheme which will make regular payments
to his wife after he is dead. The four states: {H & W alive, H alive,
W alive, neither alive}, and transitions are clear, but again
probabilities are age-dependent.
It is possible to estimate a transition matrix for each year of age and use these to simulate the process. But estimating so many transition probabilities leads to inaccuracies.
If a number of young entrants arrive each year, the age distribution will converge to an equilibrium, in a statistical sense. The chain will then behave similarly to a time-homogeneous chain with transition matrix equal to the average of the Pn.
This approximation is often used and can be justified when the input level does not fluctuate very much.
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