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1124:. The switching of quantifiers makes convergence in the Wasserstein metric stronger than weak convergence. As for strong convergence, I have never really seen the definition before, but if it is what it is, then as User Hairer points out below, it is strictly weaker than TV (on all measurable spaces), for roughly the same quantifier switching reason as above.
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After many many years on WP, I have realized that one of the greatest things we can do here is to provide a 'rosetta stone' translating across different notations. I've had this problem, where theorems I might recognize, when written from a measure-theoretic point of view, become utterly foreign and
1501:
Speaking of the weak-* topology, why specialize to the Borel sigma-algebra in the definition? For any measurable space, the dual of L is the family of measures of finite total variation---the total variation norm should coincide with the operator norm on functionals, I believe. (L is definitely not
1409:
Hmm, I agree with Piyush, I think this article could use a section of equivalences and counter-examples: so e.g. "on Polish spaces, blah is equivalent to blah, but a counter-example is provided by blah which shows that x is not equivalent to y when z." I admit that I'm too lazy to read two volumes
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Also I need to apologize for an error in my original response: in the definition of TV, you probably also need to restrict f to being integrable with respect to both measures (for otherwise the definition does not make sense). This restriction is not needed when Āµ and Ī½ are probability measures,
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This isn't completely true. In "nice" spaces (Polish will do), convergence in
Wasserstein is equivalent to weak convergence + convergence of first moment. If you choose a bounded metric (or if you modify the definition of Wasserstein so that you force the test functions to also be bounded by 1, in
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defines it in terms of the Radon metric, which would be strictly stronger than the definition on this page, as you point out, in case of Polish spaces ( I confess to not being very familiar with measure theory in general, but I believe the proof for continuous functions being dense in the set of
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Agreed. Even as a physicist I don't understand the contents of this entry. As a scientist, I found this explanation utterly useless as an every-day "pedestrian" type definition, and it doesn't belong in this form in an encyclopedia (Knowledge is not an encyclopedia written by mathematicians to
1803:
is the compact and non-compact. In the compact case, this is just the "obvious" weak-* topology and standard functional analytic results apply (like weak-* compactness by Banach-Alaoglu and weak-* density of convex-hull by Krein-Milman). In the noncompact case, the probability measures on the
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Thanks! So I see further down in the article that the
Lipschitz version in fact gives weak convergence. (This is correct, as far as you know?) Perhaps it is the case that the radon distance is identical to strong convergence? (which I think is what the article used to say, but when it was
1846:
Not sure, my abstract measure theory isn't that great, but if the topology isn't metrisable, it might break some of the other characterisations. Come to think of it, I am not even 100% sure that the equivalent characterisations stated on this page are really all equivalent without assuming
1178:
I am not sure what you mean by "convergence of first moment" in your first sentence above. Is this the first moment of all integrable functions? or the first moment of the measures themselves (that is, the measures are finite?). Further, isn't it the case that the equivalence of the
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is a variant definition of weak convergence. But if I remove the word 'continuity' in the above, then I get the definition of strong convergence. I'm trying to think of a good example where the two are inequivalent, and where its obvious that the continuity sets were the cause of the
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I am not sure this would be very useful to the casual reader, I wouldn't expect anyone to know what the Stone-Cech compactification isĀ ;-) I suppose that it would however make sense to add a remark/warning stating that while weak convergence is defined by using the dual pairing with
1437:? I'm reading about this now; its used in the theory of dynamical systems, and it seems to be maybe related to the Wasserstein metric, if not maybe identical to it in some or another case. I'll see if I can figure this all out in the next few...days? months? Its needed to define a
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integrable functions generalizes to Polish spaces easily). Also, do you think we should add the caveat that in the definition of TV, f needs to be restricted further to be integrable wrt both Āµ and Ī½? (This is not a problem in the probabilistic setting (where bounded =: -->
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I agree with you that this section isn't great. A number of your questions/remarks might however be answered by the following observation. While it is true that the space of probability measures consists of the positive elements of norm 1 in the dual of
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themselves ...). You should at least add an intuitive explanation of the concept. (BTW, so far I have not found any good mathematician who would be short of giving an intuitive explanation for a definition - even mathematicians need those at times!)
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strange looking, when written with using random variables. It gets even worse if there is some overlap to quantum stuff, or with projective varieties or whatever. So repeating a claim, in alternative notations, is a good thing.
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Yes, I was implicitly assuming all measures are finite. Otherwise you're opening a whole can of worms... By convergence of first moment, I mean that the integral of the distance function (measured from an arbitrary fixed point)
1736:, which is then indeed compact. In this space, some subsequence of the above sequence converges to some probability measure concentrated on the various "points at infinity" added by the compactification procedure.
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From the point of view of a general encyclopedia reader, this page is complete gibberish. If you have to be a statistician to understand an encyclopedia entry on a statistical item of interest, what's the point?
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The definition of strong convergence was complete rubbish. I have now replaced it by something that is at least correct, but references are still missing. I agree that intuitive explanations are also missing.
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Also, aren't "nice" and "Polish" essentially the same things: complete separable spaces which either already come with a metric (in the "nice" case), or can be endowed with a metric (in the "Polish" case).
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That makes sense, a little bit. Why don't we put what you said in the article, that this is the relative weak-* topology in the space of probability measures on the Stone-Cech compactification?
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is Polish, I've added that. TV convergence is strictly stronger than the strong convergence of measures, this is why I removed the corresponding incorrect statement from an earlier version.
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My impression is that the too many special assumptions are being made, some are unnecessary, and the article is not clear on exactly why and how they strengthen the intended statements.
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approach? I'm guessing these are all inverse limits on some kind of appropriately defined category, but my ability to guess stops there, and I have been too lazy to google, so far...
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addition to having
Lipschitz constant 1), then convergence in Wasserstein is exactly the same as weak convergence, even though at first glance it would appear to be stronger.
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Why not a simple explanation using two six-sided dice - showing how while the result on each die is random, the sum of the dice converges on a mean of 7?
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is a locally compact Polish space), what probabilists call "Weak convergence" is the weak-* convergence with respect to the pairing with
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or even continuous at all? Can it be a discontinuous-everywhere but still measurable function? (e.g. derivative of minkowski function)
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I rewrote this section a bit to, hopefully, make it more clear, as well as moved it further down because it uses vague convergence.
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Is it customary to consider only sequences in probability? That's a bit funny. The weak-* topology is not metrizable in general.
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Thanks for adding the clarification. I was wondering if you knew of a good source we could use for this article and the one on
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Also, what is the metric structure doing besides adding one (Lipschitz) characterization? Is it necessary for the definition?
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is an arbitrary measurable space, so doesn't come with a topology and even less with a metric. TV and Radon are the same when
1777:), the set of positive elements of norm one of its dual is in general strictly larger than the set of probability measures.
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Bounded-Lipschitz version of the
Wasserstein metric with weak convergence requires the measures themselves to be finite?
593:{\displaystyle \|\mu -\nu \|_{TV}=\sup {\Bigl \{}\int _{X}fd\mu -\int _{X}fd\nu \;\;{\Big |}\;\;f\colon X\to {\Bigr \}}.}
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As of
Bourbaki the vague convergence is a convergence resp. of space C_c, and not C_0. Bourbaki, Integration, Ch. III.
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re-written, that was dropped.) Two more questions, if you are up to it: Can anything be said with regards to the
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Stone-Cech compactification is tricky to identify concretely and the corresponding results are less trivial(?).
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In the list of equivalences, does E denote expectation with respect to P? This should be specified, I think.
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I'm not very familiar with this, because I mostly work with probability measures (and in that case taking
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This page is about the measure-theoretic concept. If you want a more lay-person explanation, see the ā
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Yes, fixed now (I am not sure how to find a reference for it though: this is basically the so called
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If one brings in a topology and insists on the Borel sigma-algebra, there is now a pairing between
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399:(sometimes called theorem) which would be presented using different notation in different books.
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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One could on the other hand consider the space of all positive elements of norm 1 in the dual of
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I have made a (probably lousy) attempt to provide intuitive explanations. Please improve them.
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since then boundedness of f also implies integrability. 19:27, 26 July 2012 (UTC)
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Actually the weak convergence version is slightly weaker than convergence in the
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does indeed seem to be the most common one so I am going to fix the article.
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However, in the
Wasserstein case, you have the stronger version that for any
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1712:) on the other hand, this does not converge to any probability measure on
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Although that page is also pretty lousy, from a lay-person perspective.--
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Section Weak convergence of measures as an example of weak-* convergence
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norm in the finite setting), there are no constraints on the function
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More clarification is needed. The article currently states that
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inequivalence... having this example in the text would be nice.
864:{\displaystyle |\inf \int fd\mu _{n}-\int fd\mu |<\epsilon }
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says the unit-ball of a dual is always weak-* compact.
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except that it should be measurable. If you want the
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968:{\displaystyle \epsilon : -->
792:{\displaystyle \epsilon : -->
388:00:30, 15 February 2012 (UTC)
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303:06:29, 21 November 2009 (UTC)
198:and see a list of open tasks.
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1439:finitely-determined process
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1161:
1160:
1159:
1158:
1157:
1156:
1132:
1131:
1130:
1129:
1128:
1127:
1126:
1125:
1113:
1093:
1090:
1086:
1082:
1079:
1076:
1073:
1070:
1065:
1061:
1057:
1054:
1051:
1047:
1026:
1006:
986:
964:
961:
958:
939:
938:
937:
936:
935:
934:
933:
932:
920:
900:
880:
860:
857:
853:
849:
846:
843:
840:
837:
832:
828:
824:
821:
818:
815:
811:
788:
785:
782:
761:
696:
672:
650:
646:
601:
600:
589:
584:
579:
576:
573:
570:
567:
564:
561:
558:
555:
552:
543:
534:
531:
528:
523:
519:
515:
512:
509:
506:
501:
497:
491:
486:
483:
478:
475:
471:
467:
464:
461:
458:
443:
440:
439:
438:
437:
436:
435:
434:
414:
413:
412:
411:
375:
372:
328:
327:
322:
321:
306:
305:
248:
245:
242:
241:
238:
237:
234:
233:
226:Mid-importance
222:
216:
215:
213:
196:the discussion
182:
170:
169:
167:Midāimportance
161:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
2050:
2039:
2036:
2034:
2031:
2029:
2026:
2024:
2021:
2019:
2016:
2015:
2013:
2004:
2000:
1996:
1978:
1974:
1951:
1947:
1924:
1920:
1911:
1910:
1909:
1907:
1903:
1899:
1895:
1885:
1883:
1882:
1878:
1874:
1866:
1858:
1854:
1850:
1845:
1844:
1843:
1842:
1841:
1840:
1835:
1831:
1827:
1823:
1822:
1821:
1820:
1815:
1811:
1807:
1802:
1798:
1797:
1796:
1795:
1788:
1784:
1780:
1776:
1769:
1764:
1763:
1762:
1761:
1760:
1759:
1753:
1752:
1751:
1750:
1747:
1743:
1739:
1735:
1731:
1724:
1720:
1719:
1715:
1711:
1704:
1700:
1693:
1689:
1685:
1681:
1677:
1673:
1666:
1662:
1658:
1651:
1646:
1645:
1644:
1643:
1639:
1635:
1627:
1623:
1619:
1615:
1611:
1607:
1606:
1602:
1598:
1594:
1590:
1586:
1582:
1581:
1577:
1570:
1566:
1562:
1558:
1554:
1550:
1546:
1545:
1541:
1537:
1533:
1525:
1522:LCH, and the
1521:
1517:
1510:
1506:
1505:
1500:
1499:
1495:
1494:
1493:
1487:
1485:
1484:
1480:
1476:
1467:
1463:
1462:
1461:
1455:
1453:
1452:
1448:
1444:
1440:
1436:
1432:
1425:dbar distance
1424:
1422:
1421:
1417:
1413:
1401:
1397:
1393:
1389:
1388:
1387:
1386:
1385:
1384:
1379:
1375:
1371:
1365:
1364:Radon measure
1361:
1357:
1356:
1355:
1354:
1353:
1352:
1348:
1344:
1328:
1308:
1288:
1287:
1286:
1285:
1284:
1283:
1282:
1281:
1268:
1264:
1260:
1255:
1254:
1253:
1252:
1251:
1250:
1249:
1248:
1247:
1246:
1245:
1244:
1229:
1225:
1221:
1216:
1215:
1214:
1213:
1212:
1211:
1210:
1209:
1208:
1207:
1206:
1205:
1204:
1203:
1190:
1186:
1182:
1177:
1176:
1175:
1174:
1173:
1172:
1171:
1170:
1169:
1168:
1167:
1166:
1155:
1151:
1147:
1142:
1141:
1140:
1139:
1138:
1137:
1136:
1135:
1134:
1133:
1111:
1091:
1088:
1080:
1077:
1074:
1071:
1068:
1063:
1059:
1055:
1052:
1024:
1004:
984:
962:
959:
956:
947:
946:
945:
944:
943:
942:
941:
940:
918:
898:
878:
858:
855:
847:
844:
841:
838:
835:
830:
826:
822:
819:
816:
786:
783:
780:
759:
751:
747:
746:
745:
741:
737:
733:
729:
728:weak topology
724:
723:
722:
718:
714:
710:
694:
686:
670:
648:
644:
635:
634:
633:
632:
631:
630:
626:
622:
618:
614:
610:
606:
587:
574:
571:
568:
565:
556:
553:
550:
532:
529:
526:
521:
517:
513:
510:
507:
504:
499:
495:
481:
476:
473:
465:
462:
459:
449:
448:
447:
441:
433:
429:
425:
420:
419:
418:
417:
416:
415:
410:
406:
402:
398:
394:
393:
392:
391:
390:
389:
385:
381:
380:98.99.129.198
373:
371:
370:
366:
362:
361:98.99.129.198
357:
356:
352:
348:
342:
341:
337:
333:
332:134.76.219.87
324:
323:
320:
316:
312:
311:128.59.110.50
308:
307:
304:
299:
298:
286:
282:
281:
280:
277:
273:
269:
265:
261:
253:
246:
231:
227:
221:
218:
217:
214:
197:
193:
189:
188:
183:
180:
176:
175:
171:
165:
162:
159:
155:
142:
138:
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
1892:āĀ Preceding
1889:
1870:
1800:
1774:
1767:
1733:
1729:
1722:
1713:
1709:
1702:
1698:
1691:
1687:
1683:
1679:
1675:
1671:
1664:
1660:
1656:
1649:
1631:
1626:Krein-Milman
1621:
1617:
1613:
1609:
1596:
1592:
1588:
1584:
1575:
1568:
1564:
1560:
1556:
1552:
1548:
1539:
1535:
1527:
1519:
1515:
1508:
1491:
1471:
1459:
1435:d-bar metric
1428:
1408:
1360:Radon metric
1299:
612:
605:Radon metric
603:Is this the
602:
445:
377:
358:
343:
329:
290:
254:
250:
225:
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137:Mid-priority
136:
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287:ā article.
258:āPreceding
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192:statistics
164:Statistics
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707:to have
272:contribs
260:unsigned
1826:Mct mht
1806:Mct mht
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611:? Does
228:on the
139:on the
1849:Hairer
1779:Hairer
1738:Hairer
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1370:Piyush
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1259:Piyush
1220:Hairer
1181:Piyush
1146:Hairer
977:, and
713:Piyush
401:Piyush
347:Hairer
36:scale.
1475:linas
1443:linas
1412:linas
960:: -->
784:: -->
736:linas
621:linas
424:linas
296:pasha
1999:talk
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1877:talk
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1939:or
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300:Ā»
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289:ā¦
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270:ā¢
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1979:c
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