Theory of Stochastic Signals/Markov Chains - Revision history

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	:$${\rm Pr}(A_ν\hspace{0.05cm}\vert\hspace{0.05cm} A_{ν–1}) = 0.2, \hspace{0.5cm} {\rm Pr}(B_ν\hspace{0.05cm} \vert\hspace{0.05cm} B_{ν–1}) = 0.4,$$		:$${\rm Pr}(A_ν\hspace{0.05cm}\vert\hspace{0.05cm} A_{ν–1}) = 0.2, \hspace{0.5cm} {\rm Pr}(B_ν\hspace{0.05cm} \vert\hspace{0.05cm} B_{ν–1}) = 0.4,$$
	the other two transition probabilities are also uniquely specified:		the other two transition probabilities are also uniquely specified:
	:$${\rm Pr}(B_ν\hspace{0.05cm} \vert\hspace{0.05cm} A_{ν–1}) = 1- 0.2 = 0.8, \hspace{0.5cm}		:$${\rm Pr}(B_ν\hspace{0.05cm} \vert\hspace{0.05cm} A_{ν–1}) = 1- 0.2 = 0.8, \hspace{0.5cm}{\rm Pr}(A_ν\hspace{0.05cm} \vert\hspace{0.05cm} B_{ν–1}) = 1- 0.4 = 0.6.$$}}
	{\rm Pr}(A_ν\hspace{0.05cm} \vert\hspace{0.05cm} B_{ν–1}) = 1- 0.4 = 0.6.$$}}

	==Homogeneous Markov chains==		==Homogeneous Markov chains==
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	==Stationary probabilities==		==Stationary probabilities==
	<br>		<br>
	Important properties of random processes are  [[Theory_of_Stochastic_Signals/Auto-~~Correlation_Function_(ACF)~~#Stationary_random_processes\|»$\text{stationarity}$«]]  and  [[Theory_of_Stochastic_Signals/Auto-~~Correlation_Function_(ACF)~~#Ergodic_random_processes\|»$\text{ergodicity}$«]].  Although these terms are not defined until the fourth main chapter  »Random Variables with Statistical Dependence«,  we apply them here already in advance to Markov chains.		Important properties of random processes are  [[Theory_of_Stochastic_Signals/Auto-Correlation_Function#Stationary_random_processes\|»$\text{stationarity}$«]]  and  [[Theory_of_Stochastic_Signals/Auto-Correlation_Function#Ergodic_random_processes\|»$\text{ergodicity}$«]].  Although these terms are not defined until the fourth main chapter  »Random Variables with Statistical Dependence«,  we apply them here already in advance to Markov chains.

	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=

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	{{Display}}		{{Display}}
			[[de:Stochastische_Signaltheorie/Markovketten]]

Guenter at 11:51, 6 February 2024

2024-02-06T11:51:59Z

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	:$${\mathbf{p}^{(\nu + 1)}} = {\mathbf{P}}^{\rm T} \cdot {\mathbf{p}^{(\nu )}} .$$		:$${\mathbf{p}^{(\nu + 1)}} = {\mathbf{P}}^{\rm T} \cdot {\mathbf{p}^{(\nu )}} .$$

	Here,  ${\mathbf{P}^{\rm T}}$  denotes the  "transposed matrix"  to  $\mathbf{P}$.  Thus,  after  $n$  steps,  the result is		Here,  ${\mathbf{P}^{\rm T}}$  denotes the  »transposed matrix»  to  $\mathbf{P}$.  Thus,  after  $n$  steps,  the result is
	:$${\mathbf{p}^{(\nu +{\it n})}} = \left( {\mathbf{P}}^{\rm T} \right)^n \cdot {\mathbf{p}^{(\nu )}} .$$		:$${\mathbf{p}^{(\nu +{\it n})}} = \left( {\mathbf{P}}^{\rm T} \right)^n \cdot {\mathbf{p}^{(\nu )}} .$$

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	*At the start  $(ν =0)$  all events are equally probable.  For  $ν = 1$  then holds:		*At the start  $(ν =0)$  all events are equally probable.  For  $ν = 1$  then holds:
	:$${\mathbf{p}^{(1)} } = {\mathbf{P} }^{\rm T} \cdot {\mathbf{p}^{(0 )} }= \left[ \begin{array}{ccc} 1/2 & 1/2& 1/2 \\ 1/4 & 0 & 1/2 \\ 1/4& 1/2& 0 \end{array} \right] \left[ \begin{array}{c} 1/3 \\ 1/3 \\ 1/3 \end{array} \right] = \left[ \begin{array}{c} 1/2 \\ 1/4 \\ 1/4 \end{array} \right] .$$		:$${\mathbf{p}^{(1)} } = {\mathbf{P} }^{\rm T} \cdot {\mathbf{p}^{(0 )} }= \left[ \begin{array}{ccc} 1/2 & 1/2& 1/2 \\ 1/4 & 0 & 1/2 \\ 1/4& 1/2& 0 \end{array} \right] \left[ \begin{array}{c} 1/3 \\ 1/3 \\ 1/3 \end{array} \right] = \left[ \begin{array}{c} 1/2 \\ 1/4 \\ 1/4 \end{array} \right] .$$
	:From this it can be seen that we get from the matrix  $\mathbf{P}$  to the transposed matrix  ${\mathbf{P}^{\rm T} }$  by exchanging ~~the~~ rows and columns.		:From this it can be seen that we get from the matrix  $\mathbf{P}$  to the transposed matrix  ${\mathbf{P}^{\rm T} }$  by exchanging rows and columns.
	*At time  $ν =2$  (and also at all later times)  the same probabilities result:
			*At time  $ν =2$  $($and also at all later times$)$  the same probabilities result:
	:$${\mathbf{p}^{(2)} } = {\mathbf{P} }^{\rm T} \cdot {\mathbf{p}^{(1 )} }= \left[ \begin{array}{ccc} 1/2 & 1/2& 1/2 \\ 1/4 & 0 & 1/2 \\ 1/4& 1/2& 0 \end{array} \right] \left[ \begin{array}{c} 1/2 \\ 1/4 \\ 1/4 \end{array} \right] = \left[ \begin{array}{c} 1/2 \\ 1/4 \\ 1/4 \end{array} \right] .$$		:$${\mathbf{p}^{(2)} } = {\mathbf{P} }^{\rm T} \cdot {\mathbf{p}^{(1 )} }= \left[ \begin{array}{ccc} 1/2 & 1/2& 1/2 \\ 1/4 & 0 & 1/2 \\ 1/4& 1/2& 0 \end{array} \right] \left[ \begin{array}{c} 1/2 \\ 1/4 \\ 1/4 \end{array} \right] = \left[ \begin{array}{c} 1/2 \\ 1/4 \\ 1/4 \end{array} \right] .$$
	:This means:  The ergodic probabilities of the three events are  $1/2$  $($for  $E_1)$,  $1/4$  $($for  $E_2)$,  and $1/4$  $($for  $E_3)$.		:This means:  The ergodic probabilities of the three events are  $1/2$  $($for  $E_1)$,  $1/4$  $($for  $E_2)$,  and $1/4$  $($for  $E_3)$.

Guenter at 16:30, 5 February 2024

2024-02-05T16:30:15Z

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	==Considered scenario==		==Considered scenario==
	<br>		<br>
			Finally,  we consider the case where one conducts a random experiment continuously and that at each discrete time  $(ν = 1, 2, 3, \text{...})$  a new event  $E_ν$.  Here,  let hold:
	[[File:En_Sto_T_1_4_S1.png \|right\|frame\|Possible events and event sequence]]		[[File:En_Sto_T_1_4_S1.png \|right\|frame\|Possible events and event sequence]]
	Finally,  we consider the case where one conducts a random experiment continuously and that at each discrete time  $(ν = 1, 2, 3, \text{...})$  a new event  $E_ν$.  Here,  let hold:
	:$$E_\nu \in G = \{ E_{\rm 1}, E_{\rm 2}, \hspace{0.1cm}\text{...}\hspace{0.1cm}, E_\mu , \hspace{0.1cm}\text{...}\hspace{0.1cm}, E_M \}.$$		:$$E_\nu \in G = \{ E_{\rm 1}, E_{\rm 2}, \hspace{0.1cm}\text{...}\hspace{0.1cm}, E_\mu , \hspace{0.1cm}\text{...}\hspace{0.1cm}, E_M \}.$$

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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Definition:}$		$\text{Definition:}$
	*If,  in addition to the transition probabilities,  all event probabilities of a Markov chain are independent of ~~the~~ time  $ν$,  then the Markov chain is called  »'''stationary'''«.  One then omits the index  $ν$  and writes in the binary case:
			$(1)$  If,  in addition to the transition probabilities,  all event probabilities of a Markov chain are independent of time  $ν$,  then the Markov chain is called  »'''stationary'''«.

			$(2)$  One then omits the index  $ν$  and writes in the binary case:
	:$${\rm Pr}(A_\nu ) = {\rm Pr}(A ), \hspace{0.5 cm} {\rm Pr}(B_\nu ) = {\rm Pr}(B).$$		:$${\rm Pr}(A_\nu ) = {\rm Pr}(A ), \hspace{0.5 cm} {\rm Pr}(B_\nu ) = {\rm Pr}(B).$$
	*These quantities are also called ~~the~~  »'''ergodic probabilities'''«  of the Markov chain.}}		$(3)$  These quantities are also called  »'''ergodic probabilities'''«  of the Markov chain.}}


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	:$${\rm Pr}(A) = {\rm Pr}(A \hspace{0.05cm} \| \hspace{0.05cm}A) \cdot {\rm Pr}(A) \hspace{0.1cm} + \hspace{0.1cm} {\rm Pr}(A \hspace{0.05cm} \| \hspace{0.05cm}B) \cdot {\rm Pr}(B) ,$$		:$${\rm Pr}(A) = {\rm Pr}(A \hspace{0.05cm} \| \hspace{0.05cm}A) \cdot {\rm Pr}(A) \hspace{0.1cm} + \hspace{0.1cm} {\rm Pr}(A \hspace{0.05cm} \| \hspace{0.05cm}B) \cdot {\rm Pr}(B) ,$$
	:$${\rm Pr}(B) = {\rm Pr}(B \hspace{0.05cm} \| \hspace{0.05cm}A) \cdot {\rm Pr}(A) \hspace{0.1cm} + \hspace{0.1cm} {\rm Pr}(B \hspace{0.05cm} \| \hspace{0.05cm}B) \cdot {\rm Pr}(B) .$$		:$${\rm Pr}(B) = {\rm Pr}(B \hspace{0.05cm} \| \hspace{0.05cm}A) \cdot {\rm Pr}(A) \hspace{0.1cm} + \hspace{0.1cm} {\rm Pr}(B \hspace{0.05cm} \| \hspace{0.05cm}B) \cdot {\rm Pr}(B) .$$
	*Since these equations depend linearly on each other,  one may use only one of them.  ~~For example,  as~~ the second equation of determination one may use:		*Since these equations depend linearly on each other,  one may use only one of them.  As the second equation of determination one may use:
	:$${\rm Pr}(A) + {\rm Pr}(B) = 1.$$		:$${\rm Pr}(A) + {\rm Pr}(B) = 1.$$

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	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 4:}$  We consider a binary Markov chain with events  $A$  and $B$  and transition probabilities  ${\rm Pr}(A \hspace{0.05cm} \vert\hspace{0.05cm}A) = 0.4$  and  ${\rm Pr}(B \hspace{0.05cm} \vert \hspace{0.05cm}B) = 0.8$.		$\text{Example 4:}$  We consider a stationary and binary Markov chain with events  $A$  and $B$  and transition probabilities  ${\rm Pr}(A \hspace{0.05cm} \vert\hspace{0.05cm}A) = 0.4$  and  ${\rm Pr}(B \hspace{0.05cm} \vert \hspace{0.05cm}B) = 0.8$.
	[[File:P_ID642__Sto_T_1_4_S5_neu.png \|right\|frame\| Transient response of the event probabilities]]		[[File:P_ID642__Sto_T_1_4_S5_neu.png \|right\|frame\| Transient response of the event probabilities $($three decimal places$)$]]
	*Furthermore,  we assume that each realization ~~of this chain~~ starts with event  $A$  at ~~the~~ starting time  $ν = 0$ .
	*We then obtain the event probabilities listed ~~below  (with three decimal places)~~:		Furthermore,  we assume that each realization starts with event  $A$  at starting time  $ν = 0$.  We then obtain the event probabilities listed on the right:


	In a strict sense,  this is a non-stationary Markov chain, but it is  (nearly)  settled after a short time:		In a strict sense,  this is a non-stationary Markov chain,  but it is  $($nearly$)$  settled after a short time:
	*At later times  $(ν > 5),$  the event probabilities  ${\rm Pr}(A_ν) \approx 1/4$,  ${\rm Pr}(B_ν) \approx 3/4$  are no longer seriously changed.		#At later times  $(ν > 5),$  the event probabilities  ${\rm Pr}(A_ν) \approx 1/4$,  ${\rm Pr}(B_ν) \approx 3/4$  are no longer seriously changed.
	*However,  from  ${\rm Pr}(A_{ν=5}) = 0.250$  and  ${\rm Pr}(B_{ν=5}) = 0.750$  it should not be concluded that the Markov chain is already perfectly steady at time   $ν = 5$.		#However,  from  ${\rm Pr}(A_{ν=5}) = 0.250$  and  ${\rm Pr}(B_{ν=5}) = 0.750$  it should not be concluded that the Markov chain is already perfectly steady at time   $ν = 5$.
	*Indeed,  the exact values are  ${\rm Pr}(A_{ν=5}) = 0.25024~~$  and  $~~{\rm Pr}(B_{ν=5}) = 0.74976$.}}		#Indeed,  the exact values are:
			::$${\rm Pr}(A_{ν=5}) = 0.25024,\hspace{0.5cm}{\rm Pr}(B_{ν=5}) = 0.74976.$$}}


	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Conclusion:}$		$\text{Conclusion:}$  For a stationary Markov chain, the ergodic probabilities always occur with good approximation very long after the chain is switched on   $(ν → ∞)$ , regardless of the initial conditions  ${\rm Pr}(A_0)$  and  ${\rm Pr}(B_0) = 1 - {\rm Pr}(A_0)$.

	~~'''~~For a stationary Markov chain, the ergodic probabilities always occur with good approximation very long after the chain is switched on~~'''~~   $(ν → ∞)$ , regardless of the initial conditions  ${\rm Pr}(A_0)$  and  ${\rm Pr}(B_0) = 1 - {\rm Pr}(A_0)$.
	}}		}}

Guenter at 16:07, 5 February 2024

2024-02-05T16:07:40Z

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Guenter at 14:17, 13 December 2022

2022-12-13T14:17:27Z

← Older revision		Revision as of 16:17, 13 December 2022
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	:$$0 < {\rm Pr}(B \hspace{0.05cm} \| \hspace{0.05cm}A) + {\rm Pr}(B \hspace{0.05cm} \| \hspace{0.05cm}B)< 2.$$		:$$0 < {\rm Pr}(B \hspace{0.05cm} \| \hspace{0.05cm}A) + {\rm Pr}(B \hspace{0.05cm} \| \hspace{0.05cm}B)< 2.$$

	You can calculate and display the transient behavior of the event probabilities of such a binary Markov chain with the  (German language)  interactive SWF applet		⇒   You can calculate and display the transient behavior of the event probabilities of such a binary Markov chain with the  (German language)  interactive SWF applet
	: [[Applets:Markovketten\|"Ereigniswahrscheinlichkeiten einer Markov-Kette erster Ordnung"]]   ⇒   "Event Probabilities of a First Order Markov Chain".		: [[Applets:Markovketten\|"Ereigniswahrscheinlichkeiten einer Markov-Kette erster Ordnung"]]   ⇒   "Event Probabilities of a First Order Markov Chain".

Hwang at 21:28, 11 December 2022

2022-12-11T21:28:13Z

← Older revision		Revision as of 23:28, 11 December 2022
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	$\text{Example 2:}$  Natural languages are often describable by Markov chains,  although the order  $k$  tends to infinity.		$\text{Example 2:}$  Natural languages are often describable by Markov chains,  although the order  $k$  tends to infinity.

	In this example,  however,  texts are only approximated by second-order Markov chains.		In this example,  however,  texts are only approximated by second order Markov chains.

	The graphic shows two synthetically generated texts:		The graphic shows two synthetically generated texts:

Hwang at 21:24, 11 December 2022

2022-12-11T21:24:19Z

← Older revision		Revision as of 23:24, 11 December 2022
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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Definition:}$		$\text{Definition:}$
	A  $k$-th order  »'''Markov Chain'''«  serves as a model for time- and value-discrete processes,  where the event probabilities at time  $ν$		A  $k$-th order  »'''Markov Chain'''«  serves as a model for time and value-discrete processes,  where the event probabilities at time  $ν$
	#   depend on the previous events   $E_{ν\hspace{0.05cm}–1}, \hspace{0.15cm}\text{...}\hspace{0.15cm}, E_{ν\hspace{0.05cm}–k}$,  and		#   depend on the previous events   $E_{ν\hspace{0.05cm}–1}, \hspace{0.15cm}\text{...}\hspace{0.15cm}, E_{ν\hspace{0.05cm}–k}$,  and
	#  can be expressed by  $M^{k+1}$  conditional probabilities.}}		#  can be expressed by  $M^{k+1}$  conditional probabilities.}}

Hwang at 21:19, 11 December 2022

2022-12-11T21:19:21Z

← Older revision		Revision as of 23:19, 11 December 2022
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	This scenario is a special case of a discrete-time, discrete-value random process, which will be discussed in more detail in the chapter  [[Theory_of_Stochastic_Signals/Auto-~~Correlation_Function_(ACF)~~#~~Zufallsprozesse~~\|Random Processes]].		This scenario is a special case of a discrete-time, discrete-value random process, which will be discussed in more detail in the chapter  [[Theory_of_Stochastic_Signals/Auto-Correlation_Function#Random_processes\|"Random Processes"]].

	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Definition:}$		$\text{Definition:}$
	A  $k$-th order  '''Markov Chain'''  serves as a model for time- and value-discrete processes,  where the event probabilities at time  $ν$		A  $k$-th order  »'''Markov Chain'''«  serves as a model for time- and value-discrete processes,  where the event probabilities at time  $ν$
	#   depend on the previous events   $E_{ν\hspace{0.05cm}–1}, \hspace{0.15cm}\text{...}\hspace{0.15cm}, E_{ν\hspace{0.05cm}–k}$,  and		#   depend on the previous events   $E_{ν\hspace{0.05cm}–1}, \hspace{0.15cm}\text{...}\hspace{0.15cm}, E_{ν\hspace{0.05cm}–k}$,  and
	#  can be expressed by  $M^{k+1}$  conditional probabilities.}}		#  can be expressed by  $M^{k+1}$  conditional probabilities.}}
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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Definition:}$		$\text{Definition:}$
	*In a  '''~~First Order~~ Markov ~~Chain~~'''  only the statistical binding to the last event is considered,  which in practice is usually the strongest.		*In a  »'''first order Markov chain'''«  only the statistical binding to the last event is considered,  which in practice is usually the strongest.

	*A binary first order Markov chain   ⇒   universal set  $G = \{ A, \ B \}$  has the following event probabilities at time  $\nu$:		*A binary first order Markov chain   ⇒   universal set  $G = \{ A, \ B \}$  has the following event probabilities at time  $\nu$:
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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Definition:}$		$\text{Definition:}$
	*If all transition probabilities are independent of the time  $ν$,  the Markov chain is  '''homogeneous'''.		*If all transition probabilities are independent of the time  $ν$,  the Markov chain is  »'''homogeneous'''«.

	*In the case  $M = 2$  we use the following abbreviations for this:		*In the case  $M = 2$  we use the following abbreviations for this:
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	We can also read from the Markov diagram:		We can also read from the Markov diagram:
	*The sum of the  '''outgoing arrows'''  of an event is always one		*The sum of the  »'''outgoing arrows'''«  of an event is always one
	:$${\rm Pr}(A \hspace{0.05cm} \| \hspace{0.05cm}A) + {\rm Pr}(B \hspace{0.05cm} \| \hspace{0.05cm}A) =1,$$		:$${\rm Pr}(A \hspace{0.05cm} \| \hspace{0.05cm}A) + {\rm Pr}(B \hspace{0.05cm} \| \hspace{0.05cm}A) =1,$$
	:$${\rm Pr}(A \hspace{0.05cm} \| \hspace{0.05cm}B) + {\rm Pr}(B \hspace{0.05cm} \| \hspace{0.05cm}B) =1.$$		:$${\rm Pr}(A \hspace{0.05cm} \| \hspace{0.05cm}B) + {\rm Pr}(B \hspace{0.05cm} \| \hspace{0.05cm}B) =1.$$

	*For the sums of the  '''incoming arrows'''  this restriction does not apply:		*For the sums of the  »'''incoming arrows'''«  this restriction does not apply:
	:$$ 0 < {\rm Pr}(A \hspace{0.05cm} \| \hspace{0.05cm}A) + {\rm Pr}(A \hspace{0.05cm} \| \hspace{0.05cm}B)< 2,$$		:$$ 0 < {\rm Pr}(A \hspace{0.05cm} \| \hspace{0.05cm}A) + {\rm Pr}(A \hspace{0.05cm} \| \hspace{0.05cm}B)< 2,$$
	:$$0 < {\rm Pr}(B \hspace{0.05cm} \| \hspace{0.05cm}A) + {\rm Pr}(B \hspace{0.05cm} \| \hspace{0.05cm}B)< 2.$$		:$$0 < {\rm Pr}(B \hspace{0.05cm} \| \hspace{0.05cm}A) + {\rm Pr}(B \hspace{0.05cm} \| \hspace{0.05cm}B)< 2.$$

	You can calculate and display the transient behavior of the event probabilities of such a binary Markov chain with the  (German language)  interactive SWF applet		You can calculate and display the transient behavior of the event probabilities of such a binary Markov chain with the  (German language)  interactive SWF applet
	: [[Applets:Markovketten\|Ereigniswahrscheinlichkeiten einer Markov-Kette erster Ordnung]]   ⇒   "Event Probabilities of a First Order Markov Chain".		: [[Applets:Markovketten\|"Ereigniswahrscheinlichkeiten einer Markov-Kette erster Ordnung"]]   ⇒   "Event Probabilities of a First Order Markov Chain".

	==Stationary probabilities==		==Stationary probabilities==
	<br>		<br>
	Important properties of random processes are  [[Theory_of_Stochastic_Signals/Auto-Correlation_Function_(ACF)#Stationary_random_processes\|stationarity]]  and  [[Theory_of_Stochastic_Signals/Auto-Correlation_Function_(ACF)#Ergodic_random_processes\|ergodicity]].  Although these terms are not defined until the fourth main chapter  "Random Variables with Statistical Dependence",  we apply them here already in advance to Markov chains.		Important properties of random processes are  [[Theory_of_Stochastic_Signals/Auto-Correlation_Function_(ACF)#Stationary_random_processes\|$\text{stationarity}$]]  and  [[Theory_of_Stochastic_Signals/Auto-Correlation_Function_(ACF)#Ergodic_random_processes\|$\text{ergodicity}$]].  Although these terms are not defined until the fourth main chapter  "Random Variables with Statistical Dependence",  we apply them here already in advance to Markov chains.

	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Definition:}$		$\text{Definition:}$
	*If,  in addition to the transition probabilities,  all event probabilities of a Markov chain are independent of the time  $ν$,  then the Markov chain is called  '''stationary'''.  One then omits the index  $ν$  and writes in the binary case:		*If,  in addition to the transition probabilities,  all event probabilities of a Markov chain are independent of the time  $ν$,  then the Markov chain is called  »'''stationary'''«.  One then omits the index  $ν$  and writes in the binary case:
	:$${\rm Pr}(A_\nu ) = {\rm Pr}(A ), \hspace{0.5 cm} {\rm Pr}(B_\nu ) = {\rm Pr}(B).$$		:$${\rm Pr}(A_\nu ) = {\rm Pr}(A ), \hspace{0.5 cm} {\rm Pr}(B_\nu ) = {\rm Pr}(B).$$
	*These quantities are also called the  '''ergodic probabilities'''  of the Markov chain.}}		*These quantities are also called the  »'''ergodic probabilities'''«  of the Markov chain.}}


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	$\text{Definition:}$  Multiplying the transition matrix  ${\mathbf{P} }$  infinitely many times by itself and denoting the result by  ${\mathbf{P} }_{\rm erg}$,  the resulting matrix consists of  $M$  equal rows:		$\text{Definition:}$  Multiplying the transition matrix  ${\mathbf{P} }$  infinitely many times by itself and denoting the result by  ${\mathbf{P} }_{\rm erg}$,  the resulting matrix consists of  $M$  equal rows:
	:$${\mathbf{P} }_{\rm erg} = \lim_{n \to\infty} {\mathbf{P} }^n = \left[ \begin{array}{cccc} p_{1} & p_{2} & \cdots & p_{M} \\ p_{1} & p_{2}& \cdots & p_{M} \\ \dots & \dots & \dots & \dots \\ p_{1} & p_{2} & \cdots & p_{M} \end{array} \right] .$$		:$${\mathbf{P} }_{\rm erg} = \lim_{n \to\infty} {\mathbf{P} }^n = \left[ \begin{array}{cccc} p_{1} & p_{2} & \cdots & p_{M} \\ p_{1} & p_{2}& \cdots & p_{M} \\ \dots & \dots & \dots & \dots \\ p_{1} & p_{2} & \cdots & p_{M} \end{array} \right] .$$
	The probabilities  $p_1, \hspace{0.15cm}\text{...}\hspace{0.15cm}, p_M$  in each of these lines are called the  '''ergodic probabilities'''. }}		The probabilities  $p_1, \hspace{0.15cm}\text{...}\hspace{0.15cm}, p_M$  in each of these lines are called the  »'''ergodic probabilities'''«. }}

Noah at 14:35, 13 April 2022

2022-04-13T14:35:22Z

← Older revision		Revision as of 16:35, 13 April 2022
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	==Considered scenario==		==Considered scenario==
	<br>		<br>
	Finally,  we consider the case where one ~~runs~~ a random experiment continuously and that at each discrete time  $(ν = 1, 2, 3, \text{...})$  a new event  $E_ν$.  Here,  let hold:		Finally,  we consider the case where one conducts a random experiment continuously and that at each discrete time  $(ν = 1, 2, 3, \text{...})$  a new event  $E_ν$.  Here,  let hold:
	[[File:En_Sto_T_1_4_S1.png \|right\|frame\|Possible events and event sequence]]		[[File:En_Sto_T_1_4_S1.png \|right\|frame\|Possible events and event sequence]]
	:$$E_\nu \in G = \{ E_{\rm 1}, E_{\rm 2}, \hspace{0.1cm}\text{...}\hspace{0.1cm}, E_\mu , \hspace{0.1cm}\text{...}\hspace{0.1cm}, E_M \}.$$		:$$E_\nu \in G = \{ E_{\rm 1}, E_{\rm 2}, \hspace{0.1cm}\text{...}\hspace{0.1cm}, E_\mu , \hspace{0.1cm}\text{...}\hspace{0.1cm}, E_M \}.$$