ErgodicProbabilities
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Function Description
Returns the ergodic (in effect, the long run) probabilities
of a system characterised by a Markov chain being in its given states. The
Markov chain is characterised by a matrix defining the probability of
transitioning from a given state (first index) to a given state (second index).
Determination of the ergodic properties of a Markov chain is
in general a complicated task as a Markov chain may not actually have ergodic
properties (if e.g. it has two or more disjoint subchains and the system
always stays within a given subchain because it starts out in a state within a
specific subchain). If a Markov chain does have ergodic probabilities then
these can be found by iterating the Markov chain for long enough; the long run
probabilities of being in each state will then be the ergodic probabilities.
The algorithm used by the Nematrian website selects two
randomly chosen sets of starting probabilities for each state, projects forward
these two separate starting states up to IterationLimit number of times
and returns an error if the two sets of states have not by then converged to
the same apparent state probabilities, with convergence measured by the size of
the Tolerance parameter. To allow the algorithm to produce the same answer each
time it is run given the same inputs, it includes a random number seed used
merely in the selection of these two initial starting probabilities.
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Illustrative spreadsheet

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