SOLUTIONS

BUILTIN MATHEMATICA SYMBOL
MarkovProcessProperties
MarkovProcessProperties[mproc]
gives a summary of properties for the finite state Markov process mproc.
MarkovProcessProperties[mproc, "property"]
gives the specified for the process mproc.
DetailsDetails
 MarkovProcessProperties can be used for finite state Markov processes such as DiscreteMarkovProcess and ContinuousMarkovProcess.
 MarkovProcessProperties[mproc, "Properties"] gives a list of available properties.
 MarkovProcessProperties[mproc, "property", "Description"] gives a description of the property as a string.
 Basic properties include:

"InitialProbabilities" initial state probability vector "TransitionMatrix" conditional transition probabilities m "TransitionRateMatrix" conditional transition rates q "TransitionRateVector" state transition rates "HoldingTimeMean" mean holding time for a state "HoldingTimeVariance" variance of holding time for a state "SummaryTable" summary of properties  For a continuoustime Markov process gives the transition matrix of the embedded discretetime Markov process.
 The holding time is the time spent in each state before transitioning to a different state. This takes into account selfloops which may cause the process to transition to the same state several times.
 Structural properties include:

"CommunicatingClasses" sets of states accessible from each other "RecurrentClasses" communicating classes that cannot be left "TransientClasses" communicating classes that can be left "AbsorbingClasses" recurrent classes with a single element "PeriodicClasses" communicating classes with finite period greater than 1 "Periods" period for each of the periodic classes "Irreducible" whether the process has a single recurrent class "Aperiodic" whether all classes are aperiodic "Primitive" whether the process is irreducible and aperiodic  The states of a finite Markov process can be grouped into communicating classes where from each state in a class there is a path to every other state in the class.
 A communicating class can be transient when there is a path from the class to another class or recurrent when there isn't. A special type of recurrent class, called absorbing, consist of a single element.
 A state is periodic is if there is a nonzero probability that you return to the state after two or more steps. All the states in a class have the same period.
 Transient properties before the process enters a recurrent class:

"TransientVisitMean" mean number of visits to each transient state "TransientVisitVariance" variance of number of visits to each transient state "TransientTotalVisitMean" mean total number of transient states visited  A Markov process will eventually enter a recurrent class. The transient properties characterize how many times each transient state is visited or how many different transient states are visited.
 Limiting properties include:

"ReachabilityProbability" probability of ever reaching a state "LimitTransitionMatrix" Cesaro limit of the transition matrix "Reversible" whether the process is reversible  If a property is not available, this is indicated by Missing["reason"].
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