Difference between revisions of "Markov moment"
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A notion used in probability theory for random variables having the property of independence of the "future" . More precisely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m0624801.png" /> be a [[Measurable space|measurable space]] with a non-decreasing family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m0624802.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m0624803.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m0624804.png" />-algebras of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m0624805.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m0624806.png" /> in the case of continuous time and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m0624807.png" /> in the case of discrete time). A [[Random variable|random variable]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m0624808.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m0624809.png" /> is called a Markov moment or Markov time (relative to the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248011.png" />) if for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248012.png" /> the event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248013.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248014.png" />. In the case of discrete time this is equivalent to saying that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248015.png" /> the event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248016.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248017.png" />. | A notion used in probability theory for random variables having the property of independence of the "future" . More precisely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m0624801.png" /> be a [[Measurable space|measurable space]] with a non-decreasing family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m0624802.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m0624803.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m0624804.png" />-algebras of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m0624805.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m0624806.png" /> in the case of continuous time and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m0624807.png" /> in the case of discrete time). A [[Random variable|random variable]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m0624808.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m0624809.png" /> is called a Markov moment or Markov time (relative to the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248011.png" />) if for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248012.png" /> the event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248013.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248014.png" />. In the case of discrete time this is equivalent to saying that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248015.png" /> the event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248016.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248017.png" />. |
Revision as of 16:10, 10 March 2012
Markov time; stopping time
2020 Mathematics Subject Classification: Primary: 60G40 [MSN][ZBL]
A notion used in probability theory for random variables having the property of independence of the "future" . More precisely, let be a measurable space with a non-decreasing family
,
, of
-algebras of
(
in the case of continuous time and
in the case of discrete time). A random variable
with values in
is called a Markov moment or Markov time (relative to the family
,
) if for each
the event
belongs to
. In the case of discrete time this is equivalent to saying that for any
the event
belongs to
.
Examples.
1) Let ,
, be a real-valued right-continuous random process given on
, and let
. Then the random variables
![]() |
and
![]() |
that is, the (first and first after ) times of hitting the (Borel) set
, form Markov moments (in the case
it is assumed that
).
2) If ,
, is a standard Wiener process, then the Markov moment
![]() |
has probability density
![]() |
Here , but
.
3) The random variable
![]() |
being the first time after which remains in
, is an example of a non-Markov moment (a random variable depending on the "future" ).
Using the idea of a Markov moment one can formulate the strong Markov property of Markov processes (cf. Markov process). Markov moments and stopping times (that is, finite Markov moments) play a major role in the general theory of random processes and statistical sequential analysis.
References
[1] | I.I. [I.I. Gikhman] Gihman, A.V. [A.V. Skorokhod] Skorohod, "The theory of stochastic processes" , 2 , Springer (1975) (Translated from Russian) |
Comments
References
[a1] | R.M. Blumenthal, R.K. Getoor, "Markov processes and potential theory" , Acad. Press (1968) |
[a2] | J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1984) pp. 390 |
[a3] | E.B. Dynkin, "Markov processes" , 1 , Springer (1965) (Translated from Russian) |
[a4] | A.D. [A.D. Ventsel'] Wentzell, "A course in the theory of stochastic processes" , McGraw-Hill (1981) (Translated from Russian) |
[a5] | L.P. Breiman, "Probability" , Addison-Wesley (1968) |
Markov moment. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Markov_moment&oldid=17905