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 | + | 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" />. |
===Examples.=== | ===Examples.=== | ||
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248034.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248034.png" /></td> </tr></table> | ||
| − | being the first time after which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248035.png" /> remains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248036.png" />, is an example of a non-Markov moment (a random variable depending on the | + | being the first time after which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248035.png" /> remains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248036.png" />, 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|Markov property]] of Markov processes (cf. [[Markov process|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|sequential analysis]]. | Using the idea of a Markov moment one can formulate the strong [[Markov property|Markov property]] of Markov processes (cf. [[Markov process|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|sequential analysis]]. | ||
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.I. [I.I. Gikhman] Gihman, A.V. [A.V. Skorokhod] Skorohod, "The theory of stochastic processes" , '''2''' , Springer (1975) (Translated from Russian) {{MR|0375463}} {{ZBL|0305.60027}} </TD></TR></table> |
| Line 47: | Line 47: | ||
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R.M. Blumenthal, R.K. Getoor, "Markov processes and potential theory" , Acad. Press (1968) {{MR|0264757}} {{ZBL|0169.49204}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1984) pp. 390 {{MR|0731258}} {{ZBL|0549.31001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> E.B. Dynkin, "Markov processes" , '''1''' , Springer (1965) (Translated from Russian) {{MR|0193671}} {{ZBL|0132.37901}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> A.D. [A.D. Ventsel'] Wentzell, "A course in the theory of stochastic processes" , McGraw-Hill (1981) (Translated from Russian) {{MR|0781738}} {{MR|0614594}} {{ZBL|0502.60001}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> L.P. Breiman, "Probability" , Addison-Wesley (1968) {{MR|0229267}} {{ZBL|0174.48801}} </TD></TR></table> |
Revision as of 10:31, 27 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) MR0375463 Zbl 0305.60027 |
Comments
References
| [a1] | R.M. Blumenthal, R.K. Getoor, "Markov processes and potential theory" , Acad. Press (1968) MR0264757 Zbl 0169.49204 |
| [a2] | J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1984) pp. 390 MR0731258 Zbl 0549.31001 |
| [a3] | E.B. Dynkin, "Markov processes" , 1 , Springer (1965) (Translated from Russian) MR0193671 Zbl 0132.37901 |
| [a4] | A.D. [A.D. Ventsel'] Wentzell, "A course in the theory of stochastic processes" , McGraw-Hill (1981) (Translated from Russian) MR0781738 MR0614594 Zbl 0502.60001 |
| [a5] | L.P. Breiman, "Probability" , Addison-Wesley (1968) MR0229267 Zbl 0174.48801 |
How to Cite This Entry:
Markov moment. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Markov_moment&oldid=21656
Markov moment. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Markov_moment&oldid=21656
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article