Tifenbach, Ryan M.
(2011)
A Combinatorial Approach to Nearly Uncoupled Markov Chains.
PhD thesis, National University of Ireland Maynooth.
Abstract
A discretetime Markov chain on a state space S is a sequence of random variables
X = fx0; x1; : : :g that take on values in S. A Markov chain is a model of a system
which changes or evolves over time; the random variable xt is the state of the system
at time t.
A subset E S is referred to as an almost invariant aggregate if whenever xt 2 E,
then with high probability xt+1 2 E, as well. That is, if there is a small positive value
such that if xt 2 E then the probability that xt+1 =2 E is less than or equal to ,
then E is an almost invariant aggregate. If E is such an aggregate and xt 2 E, then
the probability that xt+1; : : : ; xt+s 2 E is at least (1E)s. A Markov chain tends to
remain within its almost invariant aggregates (if it possesses any) for long periods of
time.
We refer to the Markov chain X as nearly uncoupled (with respect to some positive
) if its associated state space contains two or more disjoint almost invariant
aggregates. Nearly uncoupled Markov chains are characterised by long periods of
relatively constant behaviour, punctuated by occasional drastic changes in state.
We present a series of algorithms intended to construct almost invariant aggregates
of a given Markov chain. These algorithms are iterative processes which utilise a
concept known as the stochastic complement. The stochastic complement is a method
by which a Markov chain on a state space S can be reduced to a random process on
a proper subset S0 S, while preserving many of the algebraic properties of the
original Markov chain.
We pay special attention to the reversible case. A Markov chain is reversible if
it is symmetric in time { by which we mean that if we were to reverse the order of
the variables x1; : : : ; xt, for some relatively large t, the resulting process would be
essentially indistinguishable from the original Markov chain.
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