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case in point. Strictly stationary while not weakly stationary processes can arise while trying to design empirically relevant processes as well. Such a well-known (in the econometrics literature) example is an integrated GARCH process (D.B. Nelson, 1990, Stationarity and
Persistence in the GARCH(1,1) Model, Econometric Theory 6: 318-34). Thus the caveat (strict stationarity does not imply weak stationarity) is certainly in order. The new sentence "Any strictly stationary process which has a mean and a covariance is also WSS." is fine. However, I think that the present descriptions of strict and weak stationarity (in Knowledge) still inadequately suggest dominance of strict stationarity over weak stationarity (e.g. "A weaker form of stationarity commonly employed - - "). It is possible that a process is weakly stationary while not strictly stationary. This could happen if the first two moments were time invariant while the third or fourth, say, moment were not. This should be pointed out in the text as well IMO. (PJP)
1473:, then the process is not stationary, but is cyclostationary (although I am not sure we have a definition for that). This is an important point, and is the reason that these sorts of examples are included here and in text books. Possibly it illustrates a weakness in the usefulness of the concept of "stationarity". But it illustrates that "stationarity" (according to this definition) may not mean what one thinks it means. However there are instances where a time series or stochastic process is most naturally defined with respect to a fixed time point, such that the process is formally non-stationary, where adding a random time shift to the whole series (as in the example here) makes the process stationary and hence allows a much simpler description of its properties (moments, correlations) than would otherwise be possible: an example here is the "broken line process".
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248:(EDIT - DELETED NONSENSE) Alternative examples of non-stationary processes are stock prices, economic aggregates (e.g. GDP) and the position of a gas particle in space. In all of these examples the long run distribution of the level/position depends on the current level/position violating the constancy of the (EDIT) unconditional (EDIT) distribution required for stationarity.
851:"However, the sound of a cymbal crashing is not stationary because the acoustic power of the crash (and hence its variance) diminishes with time." I don't think this is correct: The sound is not the process; the process is one that always reverts to zero decibels. Seems like a stationary AR(1) process to me, with the occasional crashing being the additive stochastic term.
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1530:"Stationarity, is defined as a quality of a process in which the statistical parameters (mean and standard deviation) of the process do not change with time. The most important property of a stationary process is that the auto-correlation function (ACF) depends on lag alone and does not change with the time at which the function was calculated."
952:= Y example, the article says "...realisations consist of a series of constant values, with a different constant value for each realisation." But to me "constant values" means "values that do not change over time", which presumably is not what's meant here. Would it be better to say "non-stochastic values" instead of "constant values"?
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Is it just me, or are the examples involving Y misleading? I understand the context in which they are stationary, but I think they might do more to confuse than inform. For evidence, just look at the debate on this page. I'd suggest either removing them entirely, or adding some sectioning above them
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A recent revision (no real objection to this) has diverted an attempt to make the intro material more accurate ... the present version seems to me to imply that only the marginal distribution need be fixed. Also, the rearrangement into sections means that weak stationarity is not mentioned till much
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No, the original statement was correct. Just take the n'th product of the of the Markov chain, where n is the length of the longest period. I agree, though, additional detail would be useful. In short, though, the
Bernoulli shift is isomorphic to pretty much everything, including many/most dynamical
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you're right about the definition, apologies - i've edited out the nonsensical parts of my previous post. I was mis-interpreting your statement of "distribution" as the "conditional distribution" (which is not necessarily constant). since i am finding it necessary at work to brush up on this stuff i
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Also, the definition in
Section "Weak or wide-sense stationarity" says that weak-sense stationarity, wide-sense stationarity (WSS), covariance stationarity, and second-order stationarity are just different names of a single notion. But Section "N-th Order Stationarity" does not require any moments.
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This was my first revision of the material in
Knowledge. Hence please excuse my ignorance and reverting "each other's edits". I hope editing this talk page is the proper method to discuss revisions. (Is it?) Now to the topic itself: The above mentioned iid Cauchy series is a well-known theoretical
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something akin to this is right consider something that continues over time, like your daily commute. The time varies from day to day in a random (stochastic) manner we say the process (commuting) is stationary if the variation doesn't change. By variation, we mean something like the average time.
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A process with cycles has a mean or expected value that is dependent on time and is therefore not strictly stationary? I'm tempted to edit it. But maybe the individual who wrote this had something in mind which I don't understand and it can be clarified why those 2 sentences don't contradict each
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there are many ways to extend this article with brief discusison of: history, applications, differencing to induce stationarity (orders of non-stationarity), linear systems (ARMA processes, VARs, GARCH processes etc), difference between trending and non-stationary series, "unit roots", spurious
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I don't understand how the first paragraph of the article can state that a strictly stationary process "may have 'seasonal' cycles" and the third paragraph states "An important type of non-stationary process that does not include a trend-like behavior is a cyclostationary process, which is a
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Is it proper to equate wide-sense stationarity (WSS) with second-order stationarity. I think there is a difference. A process can be WSS without being second-order stationary. The definition in the article is that of WSS; second order stationary is different. (of course it depends on your
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Or maybe the article "Stationary distribution" should be changed? I never saw that notion applied to non-stationary processes, but someone could say that a nonlinear deterministic time change of a stationary process leads to a non-stationary process with a stationary (marginal, single-time)
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Thanks -- apologies for the revert, I wanted to get an example to be sure that what you were saying was correct. You are absolutely right that strict stationarity does not imply weak stationarity -- I had not realised this until now. Thanks for providing the example here.
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A process with mean 0 and standard deviation 1 may have very different ACFs. Thus, such a process may switch at some (nonrandom) time from one ACF to another ACF. Then it does satisfy the first phrase above, but violate the second one. Not a valid definition. I revert.
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the mean and corelation functions are finite. A process can be WSS without being Second Order
Stationary. The definition of Second Order Stationarity can be generalised to Nth order and strict stationary means stationary of all orders.
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The change you made converted the example to an entirtely different case, with independent values for each time-point ... in which case the end of the example (about the "law of large numbers" would be incorrect, as it would usuualy
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I am puzzled with
Section "Relation between types of stationarity". There I read: "If a stochastic process is wide-sense stationary and has finite second moments, it is wide-sense stationary." Does it make sense? Just
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No
Knowledge article should begin by saying "A stationary process is one in which...". Some context setting is needed first, at least by saying which discipline is being discussed; e.g., is this about
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The introduction strongly suggests that all stationary discrete-time random processes are
Bernoulli. I don't think that is correct. For example, isn't a random process where the distribution of X
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About the first issue. This is an error. I fixed it. Thank you for reporting. About the second issue, the problem here is that the terminology depends on the author as mentioned in the section
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The three examples all have the same character of being dependent on a single random variable. This is a very specialized case. It would be nice to have some more realistic cases.
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as of 25 july 2018, the intro is way way to technical for a general encylopedia remember, your mom, or dad, has to be able to understand it
Michael Hardy is 110% correct
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should be accurate, but not verbose. I tweaked it about just now -- tried to emphasize the fact that we're talking about the joint distribution. What do you think? --
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There is a difference between second order stationarity and WSS. A process is second order ( according to this def) if the second order density function satisfies
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A. Bochner's theorem requires continuity of the positive definite function. It is not obvious that this criterion is satisfied. A reference would be helpful.
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As for notation, it might be slightly clearer to those who already know the notation if we made use of the probability-space notation that might appear as {
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Early in the article we have: "An important type of non-stationary process that does not include a trend-like behavior is the cyclostationary process"
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You're probably right then. Sounds like this is related to the previous discussion (above). I removed second-order stationarity from the definition. --
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later ... it would be good to have both weak and strict refered to in the intro. Can someone find some appropriate wording to deal with these points.
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I think the intro is still pretty poor. "If you draw a line through the middle of a stationary process ..." What does "through the middle" mean?
341:). However, I think this is something of a pathological case; in most cases of interest, the moments exist and so stationary implies WSS. So perhaps
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However this may just be confusing to those not familiar with the notation, and the notation seems not to be used much in other
Knowledge articles.
236:"in signal processing, a stationary signal is a signal whose frequency content does not change over time." Is this true? Is it the same thing? -
1454:, then these distributions satisfy the conditions for the process to be strictly stationary. For example the distribution function of a single
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I realize now what you were trying to say. Sorry for reverting you, but I think the sentence you added was more confusing than helpful. The
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is technically correct: A process cannot be said to be WSS if its mean or variance do not exist, but it could still be stationary (e.g. an
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What version of opera (and what platform, I guess) are you running? I just downloaded it and it looks the same to me (latest, on a mac).
284:. And you seem to contradict yourself concerning the cymbal: if it decays, then its probability distribution approaches a point mass as
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But then later on referencing a random variable Y: "...Let Y have a uniform distribution on (0,2π] and define the time series { Xt } by
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I believe the {{technical}}, which has been up for two years, is outdated. The comments by Michael Hardy have already been corrected. --
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but you get the drift this may offend your elitist academic haughty academic sensibility, but it is correct for a general encylopedia
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Continuity of the (positive definite) autocovariation function can be violated, but only if the Hilbert space is nonseparable.
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So it seems to me that the passage in this article contradicts its own link. I'll remove it unless there is an objection.
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885:= Y example. Y seems to stand for many different things: the name of a random variable; the Y value that determines
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This certainly looks like a mistake to me. I've gone ahead and fixed it, and also the related mistake in the page
971:"Values that do not change over time" is exactly what is meant here" and any single realisation of the process {
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155:, etc.? The first sentence seems confused to me. But I'm not sure what the person who wrote it had in mind.
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1. This new paragraph treats the discrete-time case, while the rest of the article is about continuous time.
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might offer some additional content for the page (as per the list) if you are interested in editing it --
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other. As it currently reads, my reaction is, either a process with cycles is stationary or it is not?
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woo-hoo my first constructive wikipedia edit, up until now i've just vandalised things ocasionally.
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I also find it contradictory that they claim seasonality may be stationary and not cyclostationary.
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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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Please try to work things out in the talk page rather than reverting each other's edits. IMO,
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Does it mean that "2-th order stationarity" is not the same as "second-order stationarity"?
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to give a clearer indication that they are (in my opinion) "pathological examples".
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by all fucked up I mean lots of visual noise making the image almost unreadable.
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not all stationary discrete-time processes on {0,1} are bernoulli processes?!
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for one statement of this. Err, well, when its infinite, you just get the
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2. If the discrete time runs from -∞ to +∞ then "the Hilbert subspace of
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in the whole article except in the section about other terminology.
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I stand corrected on seasonal data — you're right. Sorry about that.
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regressioon, link to cointegration, link to brownian motion etc. --
1549:"Stationary process" and "process with a stationary distribution"
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Could you quote that definition for us? I don't have the book. --
1905:{\displaystyle F_{X}(x_{t_{1}+\tau },\ldots ,x_{t_{k}+\tau })}
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how is it not cyclostationary and therefore non-stationary?
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definition, but that must be clarified.) See Section 6.2 of
162:... maybe a stochastic process with stationary increments??
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are more-or-less, in a hand-waving way, are special cases.
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embedded maths notation formatting (possible Opera bug)
1090:. It follows from this that the marginal distribution
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Say you compute the averge time by week; the process
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Fixed to be consistent with rest of article. My bad.
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stochastic process that varies cyclically with time."
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Thanks for all your good work on the stat articles!
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I still think there's a notational problem with the X
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Wide sense stationarity and Hilbert space techniques
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1186:
1185:
1167:
1165:
1164:
1159:
1156:
1146:
1145:
1116:
1114:
1113:
1108:
1106:
1105:
1085:
1083:
1082:
1077:
1069:
1068:
1046:
1044:
1043:
1038:
1030:
1029:
1001:
999:
998:
993:
990:
989:
988:
944:
942:
941:
936:
934:
933:
911:
909:
908:
903:
901:
900:
761:Koopman operator
758:
725:Bernoulli scheme
702:
649:
647:
646:
641:
635:
624:
623:
622:
610:
609:
593:
591:
590:
585:
582:
572:
571:
553:
552:
540:
539:
527:
526:
514:
513:
498:
497:
485:
484:
472:
471:
459:
458:
446:
445:
435:
376:
357:
125:importance scale
107:
106:
103:
100:
97:
76:
69:
68:
63:
55:
48:
31:
25:
24:
16:
2236:
2235:
2231:
2230:
2229:
2227:
2226:
2225:
2201:
2200:
2185:
2155:
2116:Boris Tsirelson
2080:
2079:
2075:
2047:
2025:
2020:
2019:
1998:
1993:
1992:
1973:
1972:
1969:
1921:
1881:
1876:
1849:
1844:
1831:
1826:
1825:
1819:
1790:
1786:
1781:
1780:
1773:as an example:
1771:
1749:
1733:Boris Tsirelson
1705:
1675:Boris Tsirelson
1646:
1610:Boris Tsirelson
1551:
1536:Boris Tsirelson
1528:
1459:
1452:
1356:
1351:
1350:
1342:
1265:
1260:
1259:
1177:
1172:
1171:
1137:
1130:
1129:
1097:
1092:
1091:
1054:
1049:
1048:
1021:
1016:
1015:
980:
973:
972:
951:
919:
914:
913:
892:
887:
886:
884:
872:
869:
849:
795:
743:
700:
696:
692:
688:
683:
614:
601:
596:
595:
563:
544:
531:
518:
505:
489:
476:
463:
450:
437:
430:
429:
358:
327:
246:
234:
183:
144:
121:High-importance
104:
101:
98:
95:
94:
62:High‑importance
61:
32:on Knowledge's
29:
12:
11:
5:
2234:
2232:
2224:
2223:
2218:
2213:
2203:
2202:
2184:
2181:
2154:
2151:
2150:
2149:
2099:
2096:
2093:
2090:
2087:
2074:
2071:
2032:
2028:
2005:
2001:
1980:
1968:
1965:
1964:
1963:
1912:represent the
1901:
1896:
1893:
1888:
1884:
1879:
1875:
1872:
1869:
1864:
1861:
1856:
1852:
1847:
1843:
1838:
1834:
1823:
1802:
1797:
1793:
1789:
1779:Formally, let
1778:
1770:
1767:
1748:
1745:
1744:
1743:
1701:
1700:
1645:
1642:
1641:
1640:
1639:
1638:
1621:
1620:
1590:
1589:
1565:
1564:
1550:
1547:
1527:
1524:
1523:
1522:
1521:
1520:
1519:
1518:
1489:
1465:. However, if
1457:
1450:
1410:
1406:
1402:
1399:
1389:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1363:
1359:
1341:
1338:
1337:
1336:
1335:
1334:
1311:
1308:
1298:
1295:
1292:
1289:
1286:
1283:
1280:
1277:
1272:
1268:
1253:
1252:
1239:
1236:
1235:
1234:
1223:
1220:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1184:
1180:
1155:
1152:
1149:
1144:
1140:
1126:
1122:
1104:
1100:
1075:
1072:
1067:
1064:
1061:
1057:
1036:
1033:
1028:
1024:
987:
983:
954:
953:
949:
946:
932:
929:
926:
922:
899:
895:
882:
879:
874:Hi, Melcombe!
871:
867:
864:
848:
845:
844:
843:
829:
794:
791:
790:
789:
788:
787:
769:shift operator
698:
694:
690:
686:
682:
679:
678:
677:
676:
675:
663:
662:
639:
634:
631:
628:
621:
617:
613:
608:
604:
581:
578:
575:
570:
566:
562:
559:
556:
551:
547:
543:
538:
534:
530:
525:
521:
517:
512:
508:
504:
501:
496:
492:
488:
483:
479:
475:
470:
466:
462:
457:
453:
449:
444:
440:
425:
424:
423:
422:
410:
409:
395:
394:
393:
392:
379:
378:
326:
323:
322:
321:
320:
319:
306:
305:
304:
303:
245:
242:
233:
230:
182:
179:
178:
177:
143:
140:
137:
136:
133:
132:
129:
128:
117:
111:
110:
108:
91:the discussion
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
2233:
2222:
2219:
2217:
2214:
2212:
2209:
2208:
2206:
2199:
2198:
2194:
2190:
2182:
2180:
2178:
2174:
2170:
2166:
2162:
2152:
2148:
2144:
2140:
2136:
2132:
2128:
2127:
2126:
2125:
2121:
2117:
2111:
2097:
2091:
2088:
2085:
2072:
2070:
2067:
2063:
2059:
2055:
2051:
2030:
2026:
2003:
1999:
1978:
1966:
1962:
1958:
1954:
1950:
1949:
1948:
1946:
1942:
1938:
1934:
1927:
1924:
1919:
1915:
1894:
1891:
1886:
1882:
1877:
1873:
1870:
1867:
1862:
1859:
1854:
1850:
1845:
1836:
1832:
1822:
1817:
1800:
1795:
1791:
1787:
1777:
1774:
1768:
1766:
1765:
1761:
1757:
1753:
1746:
1742:
1738:
1734:
1730:
1729:
1728:
1725:
1721:
1717:
1713:
1709:
1699:
1695:
1691:
1687:
1686:
1685:
1684:
1680:
1676:
1672:
1671:), isn't it?
1670:
1666:
1662:
1658:
1654:
1649:
1643:
1637:
1633:
1629:
1628:178.38.78.134
1625:
1624:
1623:
1622:
1619:
1615:
1611:
1606:
1605:
1604:
1603:
1599:
1595:
1588:
1586:
1582:
1578:
1574:
1573:
1572:
1570:
1567:But the link
1563:
1561:
1556:
1555:
1554:
1548:
1546:
1545:
1541:
1537:
1531:
1525:
1517:
1513:
1509:
1505:
1504:
1503:
1499:
1495:
1490:
1486:
1485:
1484:
1480:
1476:
1472:
1468:
1464:
1460:
1453:
1446:
1442:
1441:
1440:
1439:
1435:
1431:
1427:
1424:
1421:
1408:
1400:
1397:
1384:
1381:
1378:
1372:
1369:
1366:
1361:
1357:
1348:
1345:
1339:
1333:
1329:
1325:
1309:
1306:
1293:
1287:
1284:
1278:
1270:
1266:
1257:
1256:
1255:
1254:
1251:
1247:
1243:
1240:
1237:
1221:
1218:
1205:
1199:
1196:
1190:
1182:
1178:
1170:
1169:
1150:
1142:
1138:
1127:
1123:
1120:
1102:
1098:
1089:
1073:
1070:
1065:
1062:
1059:
1055:
1034:
1031:
1026:
1022:
1013:
1009:
1005:
985:
981:
970:
969:
968:
967:
963:
959:
947:
930:
927:
924:
920:
897:
893:
880:
877:
876:
875:
865:
863:
862:
858:
854:
846:
842:
838:
834:
830:
828:
824:
820:
816:
812:
811:
810:
809:
805:
801:
792:
786:
782:
778:
774:
770:
766:
762:
757:
753:
752:
747:
740:
739:
738:
734:
730:
726:
722:
721:
720:
718:
714:
710:
706:
680:
674:
671:
667:
666:
665:
664:
661:
658:
653:
632:
629:
626:
619:
615:
611:
606:
602:
573:
568:
564:
560:
554:
549:
545:
541:
536:
532:
528:
523:
519:
510:
506:
502:
494:
490:
486:
481:
477:
473:
468:
464:
460:
455:
451:
442:
438:
427:
426:
421:
418:
414:
413:
412:
411:
408:
405:
402:
397:
396:
391:
388:
387:Richard Clegg
383:
382:
381:
380:
374:
370:
366:
365:128.214.205.4
362:
354:
353:
352:
351:
348:
344:
340:
336:
332:
324:
318:
315:
310:
309:
308:
307:
302:
299:
298:Michael Hardy
295:
291:
287:
283:
279:
275:
271:
267:
263:
262:
261:
260:
259:
258:
255:
249:
243:
241:
239:
231:
229:
227:
223:
219:
218:50.245.17.105
215:
208:
204:
202:
198:
194:
193:50.245.17.105
190:
180:
176:
173:
169:
168:
167:
165:
164:Michael Hardy
160:
158:
157:Michael Hardy
154:
150:
141:
126:
122:
116:
113:
112:
109:
92:
88:
84:
83:
78:
75:
71:
70:
66:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
2186:
2159:— Preceding
2156:
2134:
2112:
2076:
2048:— Preceding
1970:
1931:— Preceding
1928:
1925:
1923:looks fine.
1922:
1820:
1775:
1772:
1754:
1750:
1706:— Preceding
1702:
1673:
1668:
1664:
1660:
1656:
1652:
1650:
1647:
1591:
1576:
1575:
1566:
1557:
1552:
1532:
1529:
1494:70.19.91.190
1470:
1466:
1462:
1455:
1448:
1444:
1428:
1425:
1422:
1349:
1346:
1343:
1118:
1087:
1011:
1007:
1003:
955:
873:
850:
796:
749:
709:77.162.102.4
684:
651:
400:
328:
293:
289:
285:
281:
277:
273:
264:In books on
250:
247:
235:
212:— Preceding
209:
205:
187:— Preceding
184:
161:
145:
120:
80:
40:WikiProjects
2165:Nanoukaplus
703:—Preceding
359:—Preceding
270:time series
2205:Categories
657:Dakshayani
404:Dakshayani
96:Statistics
87:statistics
59:Statistics
1594:Duoduoduo
1324:Duoduoduo
958:Duoduoduo
853:Duoduoduo
756:EMS Press
296:gave us.
238:Omegatron
2183:Examples
2173:contribs
2161:unsigned
2139:Fvultier
2062:contribs
2050:unsigned
1933:unsigned
1824:and let
1756:briardew
1747:Examples
1720:contribs
1712:Dunham08
1708:unsigned
1475:Melcombe
1303:for all
1242:Melcombe
1215:for all
948:In the X
800:Melcombe
705:unsigned
697:, ..., X
594:for all
373:contribs
361:unsigned
214:unsigned
189:unsigned
153:software
151:, about
2073:Puzzled
1953:Protonk
1916:of the
1690:Mct mht
1508:Ionsme2
1125:apply).
268:and on
123:on the
30:C-class
2054:MMmpds
1920:of...
1571:says
1430:watson
314:Cripes
254:Cripes
36:scale.
1814:be a
1583:of a
1047:and
1014:...
819:Zvika
777:linas
729:Zvika
695:n-k+1
670:Zvika
417:Zvika
347:Zvika
181:style
172:Zvika
142:Style
2193:talk
2169:talk
2143:talk
2120:talk
2058:talk
1957:talk
1941:talk
1818:...
1760:talk
1737:talk
1716:talk
1694:talk
1679:talk
1632:talk
1614:talk
1598:talk
1587:....
1540:talk
1512:talk
1498:talk
1479:talk
1434:talk
1394:for
1328:talk
1246:talk
962:talk
857:talk
837:talk
823:talk
815:lead
804:talk
781:talk
733:talk
713:talk
369:talk
222:talk
197:talk
115:High
2045:.
1443:If
1370:cos
870:= Y
699:n-1
693:, X
691:n-k
335:iid
294:you
149:law
2207::
2195:)
2175:)
2171:•
2145:)
2122:)
2110:?
2095:⟹
2089:∧
2064:)
2060:•
1979:τ
1959:)
1943:)
1895:τ
1871:…
1863:τ
1762:)
1739:)
1722:)
1718:•
1696:)
1681:)
1634:)
1616:)
1600:)
1562:".
1542:)
1514:)
1500:)
1481:)
1436:)
1401:∈
1373:
1330:)
1294:ω
1279:ω
1248:)
1206:ω
1191:ω
1151:ω
1103:τ
1063:−
1060:τ
1027:τ
964:)
928:−
925:τ
898:τ
859:)
839:)
825:)
806:)
783:)
754:,
748:,
735:)
715:)
652:if
638:Δ
577:Δ
558:Δ
385:--
371:•
290:is
224:)
199:)
2191:(
2167:(
2141:(
2118:(
2098:A
2092:B
2086:A
2056:(
2031:X
2027:F
2004:X
2000:F
1955:(
1939:(
1900:)
1892:+
1887:k
1883:t
1878:x
1874:,
1868:,
1860:+
1855:1
1851:t
1846:x
1842:(
1837:X
1833:F
1801:}
1796:t
1792:X
1788:{
1758:(
1735:(
1714:(
1692:(
1677:(
1669:μ
1667:(
1665:L
1661:e
1657:μ
1655:(
1653:L
1630:(
1612:(
1596:(
1538:(
1510:(
1496:(
1477:(
1471:y
1467:Y
1463:t
1458:t
1456:X
1451:t
1449:X
1445:Y
1432:(
1409:.
1405:R
1398:t
1388:)
1385:Y
1382:+
1379:t
1376:(
1367:=
1362:t
1358:X
1326:(
1310:.
1307:t
1297:)
1291:(
1288:Y
1285:=
1282:)
1276:(
1271:t
1267:X
1244:(
1222:.
1219:t
1209:)
1203:(
1200:Y
1197:=
1194:)
1188:(
1183:t
1179:X
1154:)
1148:(
1143:t
1139:X
1119:Y
1099:X
1088:Y
1074:Y
1071:=
1066:1
1056:X
1035:Y
1032:=
1023:X
1012:X
1008:Y
1004:Y
986:t
982:X
960:(
950:t
931:1
921:X
894:X
883:t
868:t
855:(
835:(
821:(
802:(
779:(
731:(
711:(
687:n
633:d
630:n
627:a
620:2
616:t
612:,
607:1
603:t
580:)
574:+
569:2
565:t
561:,
555:+
550:1
546:t
542::
537:2
533:x
529:,
524:1
520:x
516:(
511:X
507:f
503:=
500:)
495:2
491:t
487:,
482:1
478:t
474::
469:2
465:x
461:,
456:1
452:x
448:(
443:X
439:f
367:(
286:t
282:t
278:t
276:(
274:X
220:(
195:(
127:.
42::
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