Occupation laws for some time-nonhomogeneous Markov chains
Zach Dietz, Tulane University
Sunder Sethuraman, Iowa State University
Abstract
We consider finite-state time-nonhomogeneous
Markov chains whose transition matrix at time n
is I+G/nz
where G is a ``generator'' matrix,
that is G(i,j)>0 for i,j distinct, and
G(i,i)= -∑
k≠i G(i,k), and z>0 is a strength parameter.
In these chains, as time grows, the positions are
less and less likely to change, and so form simple
models of age-dependent time-reinforcing schemes.
These chains, however, exhibit a trichotomy of
occupation behaviors depending on parameters.
We show that the average occupation or empirical distribution
vector up to time n, when variously
0<z<1, z>1 or
z=1,
converges in probability to a unique ``stationary'' vector
nG,
converges in law to a nontrivial mixture of point
measures, or converges in law to a distribution
mG
with no atoms and full support on a simplex respectively,
as n tends to infinity. This last type of limit can be
interpreted as a sort of ``spreading'' between the cases 0<z<1 and z>1.
In particular, when G is appropriately chosen, mG is a
Dirichlet distribution, reminiscent of results in Polya urns.
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