https://en.wikipedia.org/w/index.php?action=history&feed=atom&title=Ergodicity
Ergodicity - Revision history
2024-10-28T05:15:43Z
Revision history for this page on the wiki
MediaWiki 1.43.0-wmf.28
https://en.wikipedia.org/w/index.php?title=Ergodicity&diff=1246273880&oldid=prev
Comp.arch at 23:28, 17 September 2024
2024-09-17T23:28:26Z
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The mathematical definition of ergodicity aims to capture ordinary every-day ideas about [[randomness]]. This includes ideas about systems that move in such a way as to (eventually) fill up all of space, such as [[diffusion]] and [[Brownian motion]], as well as common-sense notions of mixing, such as mixing paints, drinks, cooking ingredients, [[Mixing (process engineering)|industrial process mixing]], smoke in a smoke-filled room, the dust in [[Saturn's rings]] and so on. To provide a solid mathematical footing, descriptions of ergodic systems begin with the definition of a [[measure-preserving dynamical system]]. This is written as <math>(X, \mathcal{A}, \mu, T).</math></div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The mathematical definition of ergodicity aims to capture ordinary every-day ideas about [[randomness]]. This includes ideas about systems that move in such a way as to (eventually) fill up all of space, such as [[diffusion]] and [[Brownian motion]], as well as common-sense notions of mixing, such as mixing paints, drinks, cooking ingredients, [[Mixing (process engineering)|industrial process mixing]], smoke in a smoke-filled room, the dust in [[Saturn's rings]] and so on. To provide a solid mathematical footing, descriptions of ergodic systems begin with the definition of a [[measure-preserving dynamical system]]. This is written as <math>(X, \mathcal{A}, \mu, T).</math></div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The set <math>X</math> is understood to be the total space to be filled: the mixing bowl, the smoke-filled room, ''etc.'' The [[measure (mathematics)|measure]] <math>\mu</math> is understood to define the natural [[volume]] of the space <math>X</math> and of its subspaces. The collection of subspaces is denoted by <math>\mathcal{A}</math>, and the size of any given [[subset]] <math>A\subset X</math> is <math>\mu(A)</math>; the size is its volume. Naively, one could imagine <math>\mathcal{A}</math> to be the [[power set]] of <math>X</math>; this doesn't quite work, as not all subsets of a space have a volume (famously, the [[<del style="font-weight: bold; text-decoration: none;">Banach-Tarski</del> paradox]]). Thus, conventionally, <math>\mathcal{A}</math> consists of the measurable subsets—the subsets that do have a volume. It is always taken to be a [[Borel set]]—the collection of subsets that can be constructed by taking [[set intersection|intersections]], [[set union|unions]] and [[set complement]]s of open sets; these can always be taken to be measurable.</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The set <math>X</math> is understood to be the total space to be filled: the mixing bowl, the smoke-filled room, ''etc.'' The [[measure (mathematics)|measure]] <math>\mu</math> is understood to define the natural [[volume]] of the space <math>X</math> and of its subspaces. The collection of subspaces is denoted by <math>\mathcal{A}</math>, and the size of any given [[subset]] <math>A\subset X</math> is <math>\mu(A)</math>; the size is its volume. Naively, one could imagine <math>\mathcal{A}</math> to be the [[power set]] of <math>X</math>; this doesn't quite work, as not all subsets of a space have a volume (famously, the [[<ins style="font-weight: bold; text-decoration: none;">Banach–Tarski</ins> paradox]]). Thus, conventionally, <math>\mathcal{A}</math> consists of the measurable subsets—the subsets that do have a volume. It is always taken to be a [[Borel set]]—the collection of subsets that can be constructed by taking [[set intersection|intersections]], [[set union|unions]] and [[set complement]]s of open sets; these can always be taken to be measurable.</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The time evolution of the system is described by a [[map (mathematics)|map]] <math>T:X\to X</math>. Given some subset <math>A\subset X</math>, its map <math>T(A)</math> will in general be a deformed version of <math>A</math> – it is squashed or stretched, folded or cut into pieces. Mathematical examples include the [[baker's map]] and the [[horseshoe map]], both inspired by [[bread]]-making. The set <math>T(A)</math> must have the same volume as <math>A</math>; the squashing/stretching does not alter the volume of the space, only its distribution. Such a system is "measure-preserving" (area-preserving, volume-preserving).</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The time evolution of the system is described by a [[map (mathematics)|map]] <math>T:X\to X</math>. Given some subset <math>A\subset X</math>, its map <math>T(A)</math> will in general be a deformed version of <math>A</math> – it is squashed or stretched, folded or cut into pieces. Mathematical examples include the [[baker's map]] and the [[horseshoe map]], both inspired by [[bread]]-making. The set <math>T(A)</math> must have the same volume as <math>A</math>; the squashing/stretching does not alter the volume of the space, only its distribution. Such a system is "measure-preserving" (area-preserving, volume-preserving).</div></td>
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Comp.arch
https://en.wikipedia.org/w/index.php?title=Ergodicity&diff=1243489188&oldid=prev
Ira Leviton: Fixed a reference. Please see Category:CS1 errors: dates.
2024-09-01T19:31:18Z
<p>Fixed a reference. Please see <a href="/wiki/Category:CS1_errors:_dates" title="Category:CS1 errors: dates">Category:CS1 errors: dates</a>.</p>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{unreferenced section|date=November 2021}}</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Ergodicity occurs in broad settings in [[physics]] and [[mathematics]].<ref>{{Cite journal |last=Schöpf |first=H.‐G. |date=1970<del style="font-weight: bold; text-decoration: none;">-01</del> |title=<scp>V. I. Arnold</scp> and<scp> A. Avez</scp>, Ergodic Problems of Classical Mechanics. (The Mathematical Physics Monograph Series) IX + 286 S. m. Fig. New York/Amsterdam 1968. W. A. Benjamin, Inc. Preis geb. $ 14.75, brosch. $ 6.95 . |url=http://dx.doi.org/10.1002/zamm.19700500721 |journal=ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik |volume=50 |issue=7-9 |pages=506–506 |doi=10.1002/zamm.19700500721 |issn=0044-2267}}</ref> All of these settings are unified by a common mathematical description, that of the [[measure-preserving dynamical system]]. Equivalently, ergodicity can be understood in terms of [[stochastic process]]es. They are one and the same, despite using dramatically different notation and language.</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Ergodicity occurs in broad settings in [[physics]] and [[mathematics]].<ref>{{Cite journal |last=Schöpf |first=H.‐G. |date=<ins style="font-weight: bold; text-decoration: none;">January </ins>1970 |title=<scp>V. I. Arnold</scp> and<scp> A. Avez</scp>, Ergodic Problems of Classical Mechanics. (The Mathematical Physics Monograph Series) IX + 286 S. m. Fig. New York/Amsterdam 1968. W. A. Benjamin, Inc. Preis geb. $ 14.75, brosch. $ 6.95 . |url=http://dx.doi.org/10.1002/zamm.19700500721 |journal=ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik |volume=50 |issue=7-9 |pages=506–506 |doi=10.1002/zamm.19700500721 |issn=0044-2267}}</ref> All of these settings are unified by a common mathematical description, that of the [[measure-preserving dynamical system]]. Equivalently, ergodicity can be understood in terms of [[stochastic process]]es. They are one and the same, despite using dramatically different notation and language.</div></td>
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Ira Leviton
https://en.wikipedia.org/w/index.php?title=Ergodicity&diff=1243239682&oldid=prev
Costron Systems: added reference of a paper from 1969 study of ergodic as a mathematical tool
2024-08-31T09:52:32Z
<p>added reference of a paper from 1969 study of ergodic as a mathematical tool</p>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{unreferenced section|date=November 2021}}</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Ergodicity occurs in broad settings in [[physics]] and [[mathematics]]. All of these settings are unified by a common mathematical description, that of the [[measure-preserving dynamical system]]. Equivalently, ergodicity can be understood in terms of [[stochastic process]]es. They are one and the same, despite using dramatically different notation and language.</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Ergodicity occurs in broad settings in [[physics]] and [[mathematics]].<ins style="font-weight: bold; text-decoration: none;"><ref>{{Cite journal |last=Schöpf |first=H.‐G. |date=1970-01 |title=<scp>V. I. Arnold</scp> and<scp> A. Avez</scp>, Ergodic Problems of Classical Mechanics. (The Mathematical Physics Monograph Series) IX + 286 S. m. Fig. New York/Amsterdam 1968. W. A. Benjamin, Inc. Preis geb. $ 14.75, brosch. $ 6.95 . |url=http://dx.doi.org/10.1002/zamm.19700500721 |journal=ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik |volume=50 |issue=7-9 |pages=506–506 |doi=10.1002/zamm.19700500721 |issn=0044-2267}}</ref></ins> All of these settings are unified by a common mathematical description, that of the [[measure-preserving dynamical system]]. Equivalently, ergodicity can be understood in terms of [[stochastic process]]es. They are one and the same, despite using dramatically different notation and language.</div></td>
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Costron Systems
https://en.wikipedia.org/w/index.php?title=Ergodicity&diff=1240737823&oldid=prev
Magriteappleface: /* Ergodic measure */ corrected a definition
2024-08-17T03:02:44Z
<p><span class="autocomment">Ergodic measure: </span> corrected a definition</p>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> | year = 1997}}</ref> relax the requirement that <math>T</math> preserves <math>\mu</math> to the requirement that <math>T</math> is a non-singular transformation with respect to <math>\mu</math>, meaning that if <math>N</math> is a subset <ins style="font-weight: bold; text-decoration: none;">so that <math>T^{-1}(N)</math> has</ins> zero measure<ins style="font-weight: bold; text-decoration: none;">,</ins> then so <ins style="font-weight: bold; text-decoration: none;">does</ins> <math>T(N)</math>.</div></td>
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Magriteappleface
https://en.wikipedia.org/w/index.php?title=Ergodicity&diff=1234596109&oldid=prev
76.71.13.148: updated broken links
2024-07-15T05:09:43Z
<p>updated broken links</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 05:09, 15 July 2024</td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Criterion for ergodicity===</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Criterion for ergodicity===</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The measure <math>\mu_\nu</math> is always ergodic for the shift map if the associated Markov chain is [[Markov chain#<del style="font-weight: bold; text-decoration: none;">Reducibility</del>|irreducible]]<del style="font-weight: bold; text-decoration: none;">{{Broken anchor|date=2024-06-13|bot=User:Cewbot/log/20201008/configuration|target_link=Markov chain#Reducibility|reason= The anchor (Reducibility) [[Special:Diff/970694186|has been deleted]].}}</del> (any state can be reached with positive probability from any other state in a finite number of steps).{{sfn|Walters|1982|p=42}}</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The measure <math>\mu_\nu</math> is always ergodic for the shift map if the associated Markov chain is [[Markov chain#<ins style="font-weight: bold; text-decoration: none;">Irreducibility</ins>|irreducible]] (any state can be reached with positive probability from any other state in a finite number of steps).{{sfn|Walters|1982|p=42}}</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The hypotheses above imply that there is a unique stationary measure for the Markov chain. In terms of the matrix <math>P</math> a sufficient condition for this is that 1 be a simple eigenvalue of the matrix <math>P</math> and all other eigenvalues of <math>P</math> (in <math>\mathbb C</math>) are of modulus <1.</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The hypotheses above imply that there is a unique stationary measure for the Markov chain. In terms of the matrix <math>P</math> a sufficient condition for this is that 1 be a simple eigenvalue of the matrix <math>P</math> and all other eigenvalues of <math>P</math> (in <math>\mathbb C</math>) are of modulus <1.</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Note that in probability theory the Markov chain is called [[Markov chain#Ergodicity|ergodic]] if in addition each state is [[Markov chain#<del style="font-weight: bold; text-decoration: none;">Periodicity</del>|aperiodic]]<del style="font-weight: bold; text-decoration: none;">{{Broken anchor|date=2024-06-13|bot=User:Cewbot/log/20201008/configuration|target_link=Markov chain#Periodicity|reason= The anchor (Periodicity) [[Special:Diff/970694186|has been deleted]].}}</del> (the times where the return probability is positive are not multiples of a single integer >1). This is not necessary for the invariant measure to be ergodic; hence the notions of "ergodicity" for a Markov chain and the associated shift-invariant measure are different (the one for the chain is strictly stronger).<ref>{{cite web |url=https://mathoverflow.net/questions/74503/different-uses-of-the-word-ergodic/74503 |title=Different uses of the word "ergodic" |date=September 4, 2011 |website=MathOverflow |access-date=May 16, 2020}}</ref></div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Note that in probability theory the Markov chain is called [[Markov chain#Ergodicity|ergodic]] if in addition each state is [[Markov chain#<ins style="font-weight: bold; text-decoration: none;">Properties</ins>|aperiodic]] (the times where the return probability is positive are not multiples of a single integer >1). This is not necessary for the invariant measure to be ergodic; hence the notions of "ergodicity" for a Markov chain and the associated shift-invariant measure are different (the one for the chain is strictly stronger).<ref>{{cite web |url=https://mathoverflow.net/questions/74503/different-uses-of-the-word-ergodic/74503 |title=Different uses of the word "ergodic" |date=September 4, 2011 |website=MathOverflow |access-date=May 16, 2020}}</ref></div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Moreover the criterion is an "if and only if" if all communicating classes in the chain are [[Markov chain#<del style="font-weight: bold; text-decoration: none;">Transience and recurrence</del>|recurrent]]<del style="font-weight: bold; text-decoration: none;">{{Broken anchor|date=2024-06-13|bot=User:Cewbot/log/20201008/configuration|target_link=Markov chain#Transience and recurrence|reason= The anchor (Transience and recurrence) [[Special:Diff/970694186|has been deleted]].}}</del> and we consider all stationary measures.</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Moreover the criterion is an "if and only if" if all communicating classes in the chain are [[Markov chain#<ins style="font-weight: bold; text-decoration: none;">Properties</ins>|recurrent]] and we consider all stationary measures.</div></td>
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76.71.13.148
https://en.wikipedia.org/w/index.php?title=Ergodicity&diff=1228893057&oldid=prev
Cewbot: Fixing broken anchor: Reminder of an inactive anchor: irreducible, Reminder of an inactive anchor: aperiodic, Reminder of an inactive anchor: recurrent
2024-06-13T20:03:42Z
<p><a href="/wiki/User:Cewbot/log/20201008/configuration" title="User:Cewbot/log/20201008/configuration">Fixing broken anchor</a>: Reminder of an inactive anchor: <a href="/wiki/Markov_chain#Reducibility" title="Markov chain">irreducible</a>, Reminder of an inactive anchor: <a href="/wiki/Markov_chain#Periodicity" title="Markov chain">aperiodic</a>, Reminder of an inactive anchor: <a href="/wiki/Markov_chain#Transience_and_recurrence" title="Markov chain">recurrent</a></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:03, 13 June 2024</td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The measure <math>\mu_\nu</math> is always ergodic for the shift map if the associated Markov chain is [[Markov chain#Reducibility|irreducible]] (any state can be reached with positive probability from any other state in a finite number of steps).{{sfn|Walters|1982|p=42}}</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The measure <math>\mu_\nu</math> is always ergodic for the shift map if the associated Markov chain is [[Markov chain#Reducibility|irreducible]]<ins style="font-weight: bold; text-decoration: none;">{{Broken anchor|date=2024-06-13|bot=User:Cewbot/log/20201008/configuration|target_link=Markov chain#Reducibility|reason= The anchor (Reducibility) [[Special:Diff/970694186|has been deleted]].}}</ins> (any state can be reached with positive probability from any other state in a finite number of steps).{{sfn|Walters|1982|p=42}}</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The hypotheses above imply that there is a unique stationary measure for the Markov chain. In terms of the matrix <math>P</math> a sufficient condition for this is that 1 be a simple eigenvalue of the matrix <math>P</math> and all other eigenvalues of <math>P</math> (in <math>\mathbb C</math>) are of modulus <1.</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The hypotheses above imply that there is a unique stationary measure for the Markov chain. In terms of the matrix <math>P</math> a sufficient condition for this is that 1 be a simple eigenvalue of the matrix <math>P</math> and all other eigenvalues of <math>P</math> (in <math>\mathbb C</math>) are of modulus <1.</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Note that in probability theory the Markov chain is called [[Markov chain#Ergodicity|ergodic]] if in addition each state is [[Markov chain#Periodicity|aperiodic]] (the times where the return probability is positive are not multiples of a single integer >1). This is not necessary for the invariant measure to be ergodic; hence the notions of "ergodicity" for a Markov chain and the associated shift-invariant measure are different (the one for the chain is strictly stronger).<ref>{{cite web |url=https://mathoverflow.net/questions/74503/different-uses-of-the-word-ergodic/74503 |title=Different uses of the word "ergodic" |date=September 4, 2011 |website=MathOverflow |access-date=May 16, 2020}}</ref></div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Note that in probability theory the Markov chain is called [[Markov chain#Ergodicity|ergodic]] if in addition each state is [[Markov chain#Periodicity|aperiodic]]<ins style="font-weight: bold; text-decoration: none;">{{Broken anchor|date=2024-06-13|bot=User:Cewbot/log/20201008/configuration|target_link=Markov chain#Periodicity|reason= The anchor (Periodicity) [[Special:Diff/970694186|has been deleted]].}}</ins> (the times where the return probability is positive are not multiples of a single integer >1). This is not necessary for the invariant measure to be ergodic; hence the notions of "ergodicity" for a Markov chain and the associated shift-invariant measure are different (the one for the chain is strictly stronger).<ref>{{cite web |url=https://mathoverflow.net/questions/74503/different-uses-of-the-word-ergodic/74503 |title=Different uses of the word "ergodic" |date=September 4, 2011 |website=MathOverflow |access-date=May 16, 2020}}</ref></div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Moreover the criterion is an "if and only if" if all communicating classes in the chain are [[Markov chain#Transience and recurrence|recurrent]] and we consider all stationary measures.</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Moreover the criterion is an "if and only if" if all communicating classes in the chain are [[Markov chain#Transience and recurrence|recurrent]]<ins style="font-weight: bold; text-decoration: none;">{{Broken anchor|date=2024-06-13|bot=User:Cewbot/log/20201008/configuration|target_link=Markov chain#Transience and recurrence|reason= The anchor (Transience and recurrence) [[Special:Diff/970694186|has been deleted]].}}</ins> and we consider all stationary measures.</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Examples===</div></td>
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Cewbot
https://en.wikipedia.org/w/index.php?title=Ergodicity&diff=1227524504&oldid=prev
Jean Raimbault: /* Non-ergodic Markov chains */
2024-06-06T07:38:28Z
<p><span class="autocomment">Non-ergodic Markov chains</span></p>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Markov chains with recurring communicating classes are not irreducible are not ergodic<del style="font-weight: bold; text-decoration: none;">{{Incomprehensible inline}}</del>, and this can be seen immediately as follows. If <math>S_1 \subsetneq S</math> are two distinct recurrent communicating classes there are nonzero stationary measures <math>\nu_1, \nu_2</math> supported on <math>S_1, S_2</math> respectively and the subsets <math>S_1^\mathbb{Z}</math> and <math>S_2^\mathbb{Z}</math> are both shift-invariant and of measure 1<del style="font-weight: bold; text-decoration: none;">.</del>2 for the invariant probability measure <math display="inline">\frac{1}{2}(\nu_1 + \nu_2)</math>. A very simple example of that is the chain on <math>S = \{1, 2\}</math> given by the matrix <math display="inline">\left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right)</math> (both states are stationary).</div></td>
<td class="diff-marker" data-marker="+"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Markov chains with recurring communicating classes<ins style="font-weight: bold; text-decoration: none;"> which</ins> are not irreducible are not ergodic, and this can be seen immediately as follows. If <math>S_1<ins style="font-weight: bold; text-decoration: none;">, S_2</ins> \subsetneq S</math> are two distinct recurrent communicating classes there are nonzero stationary measures <math>\nu_1, \nu_2</math> supported on <math>S_1, S_2</math> respectively and the subsets <math>S_1^\mathbb{Z}</math> and <math>S_2^\mathbb{Z}</math> are both shift-invariant and of measure 1<ins style="font-weight: bold; text-decoration: none;">/</ins>2 for the invariant probability measure <math display="inline">\frac{1}{2}(\nu_1 + \nu_2)</math>. A very simple example of that is the chain on <math>S = \{1, 2\}</math> given by the matrix <math display="inline">\left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right)</math> (both states are stationary).</div></td>
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<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>====A periodic chain====</div></td>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>====A periodic chain====</div></td>
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</table>
Jean Raimbault
https://en.wikipedia.org/w/index.php?title=Ergodicity&diff=1227518822&oldid=prev
Pot: Marked incomprehensible phrase (probably a typo where "are" should be "and"
2024-06-06T06:39:27Z
<p>Marked incomprehensible phrase (probably a typo where "are" should be "and"</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 06:39, 6 June 2024</td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>====Non-ergodic Markov chains====</div></td>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>====Non-ergodic Markov chains====</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Markov chains with recurring communicating classes are not irreducible are not ergodic, and this can be seen immediately as follows. If <math>S_1 \subsetneq S</math> are two distinct recurrent communicating classes there are nonzero stationary measures <math>\nu_1, \nu_2</math> supported on <math>S_1, S_2</math> respectively and the subsets <math>S_1^\mathbb{Z}</math> and <math>S_2^\mathbb{Z}</math> are both shift-invariant and of measure 1.2 for the invariant probability measure <math display="inline">\frac{1}{2}(\nu_1 + \nu_2)</math>. A very simple example of that is the chain on <math>S = \{1, 2\}</math> given by the matrix <math display="inline">\left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right)</math> (both states are stationary).</div></td>
<td class="diff-marker" data-marker="+"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Markov chains with recurring communicating classes are not irreducible are not ergodic<ins style="font-weight: bold; text-decoration: none;">{{Incomprehensible inline}}</ins>, and this can be seen immediately as follows. If <math>S_1 \subsetneq S</math> are two distinct recurrent communicating classes there are nonzero stationary measures <math>\nu_1, \nu_2</math> supported on <math>S_1, S_2</math> respectively and the subsets <math>S_1^\mathbb{Z}</math> and <math>S_2^\mathbb{Z}</math> are both shift-invariant and of measure 1.2 for the invariant probability measure <math display="inline">\frac{1}{2}(\nu_1 + \nu_2)</math>. A very simple example of that is the chain on <math>S = \{1, 2\}</math> given by the matrix <math display="inline">\left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right)</math> (both states are stationary).</div></td>
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<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>====A periodic chain====</div></td>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>====A periodic chain====</div></td>
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</table>
Pot
https://en.wikipedia.org/w/index.php?title=Ergodicity&diff=1227028095&oldid=prev
David Eppstein: it's a book, not a journal paper
2024-06-03T07:03:00Z
<p>it's a book, not a journal paper</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 07:03, 3 June 2024</td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In other words, there are no [[invariant sigma-algebra|<math>T</math>-invariant subsets]] up to measure 0 (with respect to <math>\mu</math>). </div></td>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In other words, there are no [[invariant sigma-algebra|<math>T</math>-invariant subsets]] up to measure 0 (with respect to <math>\mu</math>). </div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Some authors<ref>{{cite book</div></td>
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<td class="diff-marker"><a class="mw-diff-movedpara-left" title="Paragraph was moved. Click to jump to new location." href="#movedpara_3_9_rhs">⚫</a></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><a name="movedpara_2_0_lhs"></a><del style="font-weight: bold; text-decoration: none;">Some authors<ref>{{cite journal </del>| <del style="font-weight: bold; text-decoration: none;">last = Aaronson | first = Jon | date</del> = 1997<del style="font-weight: bold; text-decoration: none;"> | title = An introduction to infinite ergodic theory |page = 21|publisher = American Mathematical Soc.|isbn = 9780821804940</del>}}</ref> relax the requirement that <math>T</math> preserves <math>\mu</math> to the requirement that <math>T</math> is a non-singular transformation with respect to <math>\mu</math>, meaning that if <math>N</math> is a subset of zero measure then so is <math>T(N)</math>.</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> | last = Aaronson | first = Jon</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> | doi = 10.1090/surv/050</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> | isbn = 0-8218-0494-4</div></td>
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<td class="diff-marker" data-marker="+"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> | mr = 1450400</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> | page = 21</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> | publisher = American Mathematical Society | location = Providence, Rhode Island</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> | series = Mathematical Surveys and Monographs</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> | title = An introduction to infinite ergodic theory</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> | volume = 50</div></td>
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<td class="diff-marker"><a class="mw-diff-movedpara-right" title="Paragraph was moved. Click to jump to old location." href="#movedpara_2_0_lhs">⚫</a></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><a name="movedpara_3_9_rhs"></a><ins style="font-weight: bold; text-decoration: none;"> </ins>| <ins style="font-weight: bold; text-decoration: none;">year</ins> = 1997}}</ref> relax the requirement that <math>T</math> preserves <math>\mu</math> to the requirement that <math>T</math> is a non-singular transformation with respect to <math>\mu</math>, meaning that if <math>N</math> is a subset of zero measure then so is <math>T(N)</math>.</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Examples===</div></td>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Examples===</div></td>
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</table>
David Eppstein
https://en.wikipedia.org/w/index.php?title=Ergodicity&diff=1226996596&oldid=prev
Twotwice: /* Ergodic processes */ subsection capitalization
2024-06-03T01:13:19Z
<p><span class="autocomment">Ergodic processes: </span> subsection capitalization</p>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Mixing (mathematics)|Mixing]] is a stronger statement than ergodicity. Mixing asks for this ergodic property to hold between any two sets <math>A, B</math>, and not just between some set <math>A</math> and <math>X</math>. That is, given any two sets <math>A, B\in\mathcal{A}</math>, a system is said to be (topologically) mixing if there is an integer <math>N</math> such that, for all <math>A, B</math> and <math>n>N</math>, one has that <math>T^n(A) \cap B \ne \varnothing</math>. Here, <math>\cap</math> denotes [[set intersection]] and <math>\varnothing</math> is the [[empty set]]. Other notions of mixing include strong and weak mixing, which describe the notion that the mixed substances intermingle everywhere, in equal proportion. This can be non-trivial, as practical experience of trying to mix sticky, gooey substances shows.</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Mixing (mathematics)|Mixing]] is a stronger statement than ergodicity. Mixing asks for this ergodic property to hold between any two sets <math>A, B</math>, and not just between some set <math>A</math> and <math>X</math>. That is, given any two sets <math>A, B\in\mathcal{A}</math>, a system is said to be (topologically) mixing if there is an integer <math>N</math> such that, for all <math>A, B</math> and <math>n>N</math>, one has that <math>T^n(A) \cap B \ne \varnothing</math>. Here, <math>\cap</math> denotes [[set intersection]] and <math>\varnothing</math> is the [[empty set]]. Other notions of mixing include strong and weak mixing, which describe the notion that the mixed substances intermingle everywhere, in equal proportion. This can be non-trivial, as practical experience of trying to mix sticky, gooey substances shows.</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>===Ergodic <del style="font-weight: bold; text-decoration: none;">Processes</del>===</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>===Ergodic <ins style="font-weight: bold; text-decoration: none;">processes</ins>===</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{main|Ergodic process}}</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{main|Ergodic process}}</div></td>
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Twotwice