BUILT-IN 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.

Details Details

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 continuous-time Markov process gives the transition matrix of the embedded discrete-time Markov process.
The holding time is the time spent in each state before transitioning to a different state. This takes into account self-loops 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 non-zero 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