38:
1742:
This discrepancy shows that the probability distribution for tomorrow's color depends not only on the present value, but is also affected by information about the past. This stochastic process of observed colors doesn't have the Markov property. Using the same experiment above, if sampling "without
154:
Note that there is a subtle, often overlooked and very important point that is often missed in the plain
English statement of the definition. Namely that the statespace of the process is constant through time. The conditional description involves a fixed "bandwidth". For example, without this
907:
1424:
155:
restriction we could augment any process to one which includes the complete history from a given initial condition and it would be made to be
Markovian. But the state space would be of increasing dimensionality over time and does not meet the definition.
1687:
Suppose you know that today's ball was red, but you have no information about yesterday's ball. The chance that tomorrow's ball will be red is 1/2. That's because the only two remaining outcomes for this random experiment are:
128:
of future states of the process (conditional on both past and present values) depends only upon the present state; that is, given the present, the future does not depend on the past. A process with this property is said to be
638:
701:
1683:
Assume that an urn contains two red balls and one green ball. One ball was drawn yesterday, one ball was drawn today, and the final ball will be drawn tomorrow. All of the draws are "without replacement".
1039:
1320:
437:
1269:
1816:. Imprint Moscow, Academy of Sciences of the USSR, 1954 Added t.p. in Russian Translation of Works of the Mathematical Institute, Academy of Sciences of the USSR, v. 42. Original title:
273:
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1216:
214:
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1503:
1667:, the Markov property is considered desirable since it may enable the reasoning and resolution of the problem that otherwise would not be possible to be resolved because of its
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extends this property to two or more dimensions or to random variables defined for an interconnected network of items. An example of a model for such a field is the
1471:
1292:
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902:{\displaystyle P(X_{n+1}=x_{n+1}\mid X_{n}=x_{n},\dots ,X_{1}=x_{1})=P(X_{n+1}=x_{n+1}\mid X_{n}=x_{n}){\text{ for all }}n\in \mathbb {N} .}
45:
for times 0 ā¤ t ā¤ 2. Brownian motion has the Markov property, as the displacement of the particle does not depend on its past displacements.
125:
1739:
On the other hand, if you know that both today and yesterday's balls were red, then you are guaranteed to get a green ball tomorrow.
1889:
1845:
1768:
221:
83:
is similar to the Markov property, except that the meaning of "present" is defined in terms of a random variable known as a
923:
30:
This article is about the property of a stochastic process. For the class of properties of a finitely presented group, see
1419:{\displaystyle {\mathcal {F}}_{\tau }=\{A\in {\mathcal {F}}:\forall t\geq 0,\{\tau \leq t\}\cap A\in {\mathcal {F}}_{t}\}}
378:
1743:
replacement" is changed to sampling "with replacement," the process of observed colors will have the Markov property.
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The strong Markov property implies the ordinary Markov property since by taking the stopping time
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69:, which means that its future evolution is independent of its history. It is named after the
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is used to describe a model where the Markov property is assumed to hold, such as a
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A discrete-time stochastic process satisfying the Markov property is known as a
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633:{\displaystyle P(X_{t}\in A\mid {\mathcal {F}}_{s})=P(X_{t}\in A\mid X_{s}).}
1922:"Example of a stochastic process which does not have the Markov property"
1921:
1911:. Wiley Series in Probability and Mathematical Statistics, 1986, p. 158.
70:
1882:
Stochastic
Differential Equations: An Introduction with Applications
1746:
An application of the Markov property in a generalized form is in
36:
917:
Alternatively, the Markov property can be formulated as follows.
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is said to have the strong Markov property if, for each
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1034:{\displaystyle \operatorname {E} =\operatorname {E} }
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A stochastic process has the Markov property if the
1909:Markov Processes: Characterization and Convergence
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432:{\displaystyle X=\{X_{t}:\Omega \to S\}_{t\in I}}
1264:{\displaystyle \{{\mathcal {F}}_{t}\}_{t\geq 0}}
1651:, the ordinary Markov property can be deduced.
143:. Two famous classes of Markov process are the
268:{\displaystyle ({\mathcal {F}}_{s},\ s\in I)}
8:
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1103:{\displaystyle f:S\rightarrow \mathbb {R} }
27:Memoryless property of a stochastic process
1834:The Oxford Dictionary of Statistical Terms
1211:{\displaystyle (\Omega ,{\mathcal {F}},P)}
209:{\displaystyle (\Omega ,{\mathcal {F}},P)}
41:A single realisation of three-dimensional
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695:, this can be reformulated as follows:
1588:{\displaystyle {\mathcal {F}}_{\tau }}
7:
1498:{\displaystyle \{\tau <\infty \}}
126:conditional probability distribution
469:{\displaystyle A\in {\mathcal {S}}}
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368:{\displaystyle (S,{\mathcal {S}})}
328:{\displaystyle (S,{\mathcal {S}})}
184:
25:
1162:{\displaystyle X=(X_{t}:t\geq 0)}
1861:Probability: Theory and Examples
1750:computations in the context of
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688:{\displaystyle I=\mathbb {N} }
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1:
1671:. Such a model is known as a
1069:{\displaystyle t\geq s\geq 0}
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1769:ChapmanāKolmogorov equation
1557:{\displaystyle X_{\tau +t}}
1473:, conditional on the event
663:is a discrete set with the
375:-valued stochastic process
165:Markov chain Ā§ History
1973:
1865:Cambridge University Press
162:
29:
1665:probabilistic forecasting
1615:{\displaystyle X_{\tau }}
440:adapted to the filtration
1748:Markov chain Monte Carlo
1505:, we have that for each
1110:bounded and measurable.
913:Alternative formulations
501:{\displaystyle s,t\in I}
1904:Ethier, Stewart N. and
1838:Oxford University Press
1789:Markov decision process
1764:Causal Markov condition
1644:{\displaystyle \tau =t}
1524:{\displaystyle t\geq 0}
1307:{\displaystyle \Omega }
442:is said to possess the
1812:Markov, A. A. (1954).
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1114:Strong Markov property
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665:discrete sigma algebra
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527:{\displaystyle s<t}
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81:strong Markov property
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18:Strong Markov property
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1466:{\displaystyle \tau }
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1884:. Springer, Berlin.
1814:Theory of Algorithms
1661:predictive modelling
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1752:Bayesian statistics
882: for all
103:Markov random field
96:hidden Markov model
32:AdianāRabin theorem
1878:Ćksendal, Bernt K.
1863:. Fourth Edition.
1818:Teoriya algorifmov
1641:
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1564:is independent of
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1220:natural filtration
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1171:stochastic process
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643:In the case where
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67:stochastic process
51:probability theory
47:
1737:
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1659:In the fields of
1443:{\displaystyle X}
1175:probability space
883:
656:{\displaystyle S}
292:{\displaystyle I}
252:
218:probability space
92:Markov assumption
16:(Redirected from
1964:
1957:Markov processes
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1906:Kurtz, Thomas G.
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1271:. Then for any
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444:Markov property
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277:totally ordered
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149:Brownian motion
137:and known as a
122:
59:Markov property
43:Brownian motion
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28:
23:
22:
15:
12:
11:
5:
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1926:Stack Exchange
1913:
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1849:
1830:Dodge, Yadolah
1822:
1804:
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1801:
1798:
1797:
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1779:Markov blanket
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1669:intractability
1656:
1655:In forecasting
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163:Main article:
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140:Markov process
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65:property of a
61:refers to the
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14:
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10:
9:
6:
4:
3:
2:
1969:
1958:
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1952:Markov models
1950:
1949:
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1927:
1923:
1917:
1914:
1910:
1907:
1901:
1898:
1893:
1891:3-540-04758-1
1887:
1883:
1879:
1873:
1870:
1866:
1862:
1858:
1857:Durrett, Rick
1853:
1850:
1847:
1846:0-19-850994-4
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1453:
1452:stopping time
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1281:
1274:
1273:stopping time
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1199:
1189:
1176:
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1128:
1125:
1118:Suppose that
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1111:
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919:
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885:
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859:
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836:
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