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956:{\displaystyle \mathbb {P} _{i_{1}\dots i_{k}}^{X}(A_{1}\times \cdots \times A_{k}):=\mathbb {P} \left\{\omega \in \Omega \left|X_{i_{j}}(\omega )\in A_{j}\mathrm {\,for\,} 1\leq j\leq k\right.\right\}.}
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748:{\displaystyle \mathbb {P} _{i_{1}\dots i_{k}}^{X}(S):=\mathbb {P} \left\{\omega \in \Omega \left|\left(X_{i_{1}}(\omega ),\dots ,X_{i_{k}}(\omega )\right)\in S\right.\right\}.}
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1341:{\displaystyle f:\mathbb {X} ^{I}\to \mathbb {X} ^{k}:\sigma \mapsto \left(\sigma (t_{1}),\dots ,\sigma (t_{k})\right)}
394:
145:. A lot of information can be gained by studying the "projection" of a measure (or process) onto a finite-dimensional
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1131:. In general, this is an infinite-dimensional space. The finite dimensional distributions of
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to the corresponding finite-dimensional distributions of some probability measure
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The definition of the finite-dimensional distributions of a process
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Finite-dimensional distributions of a stochastic process
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380:{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}
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and all the finite-dimensional distributions of the
1193:{\displaystyle f_{*}\left({\mathcal {L}}_{X}\right)}
506:{\displaystyle \mathbb {P} _{i_{1}\dots i_{k}}^{X}}
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758:Very often, this condition is stated in terms of
424:{\displaystyle X:I\times \Omega \to \mathbb {X} }
153:Finite-dimensional distributions of a measure
8:
986:is related to the definition for a measure
53:. Unsourced material may be challenged and
1452:{\displaystyle (\mu _{n})_{n=1}^{\infty }}
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117:Learn how and when to remove this message
1200:on the finite-dimensional product space
300:{\displaystyle f:X\to \mathbb {R} ^{k}}
192:{\displaystyle (X,{\mathcal {F}},\mu )}
1405:It can be shown that if a sequence of
1006:in the following way: recall that the
7:
51:adding citations to reliable sources
1444:
1390:{\displaystyle t_{1},\dots ,t_{k}}
1351:is the natural "evaluate at times
1033:{\displaystyle {\mathcal {L}}_{X}}
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915:
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566:{\displaystyle k\in \mathbb {N} }
328:{\displaystyle k\in \mathbb {N} }
149:(or finite collection of times).
66:"Finite-dimensional distribution"
1222:{\displaystyle \mathbb {X} ^{k}}
1082:{\displaystyle \mathbb {X} ^{I}}
538:{\displaystyle \mathbb {X} ^{k}}
437:finite-dimensional distributions
205:finite-dimensional distributions
135:finite-dimensional distributions
23:
1060:is a measure on the collection
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1151:are the push forward measures
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459:are the push forward measures
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1:
1124:{\displaystyle \mathbb {X} }
259:{\displaystyle f_{*}(\mu )}
137:are a tool in the study of
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1570:Law (stochastic processes)
1533:{\displaystyle \mu _{n}}
1483:{\displaystyle \mu _{n}}
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1089:of all functions from
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1401:Relation to tightness
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1590:Stochastic processes
1553:{\displaystyle \mu }
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1540:converges weakly to
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1407:probability measures
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999:{\displaystyle \mu }
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229:pushforward measures
220:{\displaystyle \mu }
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143:stochastic processes
47:improve this article
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16:Mathematics concept
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433:stochastic process
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1144:{\displaystyle X}
1102:{\displaystyle I}
1053:{\displaystyle X}
979:{\displaystyle X}
452:{\displaystyle X}
389:probability space
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1585:Measure theory
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515:product space
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201:measure space
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107:December 2009
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68: –
67:
63:
62:Find sources:
56:
52:
48:
42:
41:
37:
32:This article
30:
26:
21:
20:
1404:
1397:" function.
1350:
965:
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436:
342:
204:
156:
147:vector space
134:
128:
113:
104:
94:
87:
80:
73:
61:
45:Please help
33:
573:defined by
131:mathematics
1579:Categories
763:rectangles
760:measurable
77:newspapers
1548:μ
1522:μ
1501:μ
1472:μ
1445:∞
1420:μ
1372:…
1315:σ
1309:…
1287:σ
1279:↦
1276:σ
1258:→
1164:∗
994:μ
936:≤
930:≤
901:∈
895:ω
867:Ω
864:∈
861:ω
832:×
829:⋯
826:×
793:…
728:∈
717:ω
691:…
682:ω
649:Ω
646:∈
643:ω
601:…
556:∈
484:…
414:→
411:Ω
408:×
354:Ω
318:∈
283:→
251:μ
243:∗
215:μ
184:μ
34:does not
1564:See also
1229:, where
391:and let
266:, where
227:are the
139:measures
1513:, then
513:on the
91:scholar
55:removed
40:sources
435:. The
203:. The
93:
86:
79:
72:
64:
1461:tight
1109:into
431:be a
387:be a
199:be a
98:JSTOR
84:books
545:for
343:Let
157:Let
141:and
70:news
38:any
36:cite
1459:is
1040:of
1008:law
439:of
207:of
129:In
49:by
1581::
1560:.
848::=
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307:,
133:,
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95:·
88:·
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57:.
43:.
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