# A. Markov's papers?

A. Markov published several papers on his chains, starting in 1906, so it is written, in the journal: (1) Извѣстія Физико-математического общества при Казанском университете

I am surprised by the difficulty of finding the papers to read, surprised because of just how well-know this material is today. The only work by Markov I can find is his 1900 book "Calculating Variation" and an earlier book in German.

The journal (1) still exists today, having published without interruption. They have back issues available, but only for the last decade. They have an archive 1834-2014! It consists of important papers and is a WIP. Ah, I see, Markov's work is apparently not sufficiently important (because it's not there), although a paper by his son without mathematical content is.

1a) Did Markov invent the concept of Markov chains? I ask whomever had read his papers, not ABOUT his papers. 1b) Was the general idea---if not the various theorems proved using the approach---know earlier, perhaps by Karl Pearson (or William Clifford) or John Strutt. 1c) If you did read Markov's papers, where are they available, at least the 15th volume, if not the later or earlier volumes? (My university, for instance, has a great Imperial and Soviet collection, but not this; it may be obscure.) If the papers might be in some paywall database, not necessarily scanned and archived, please suggest where to look; I have access probably.

2) What influential papers began the use of Markov chains outside the USSR? I presume this occurred in the 1920's. Because the use of such concepts in the ergodic context came soon after, I know having read those papers.

3) What are the OLDEST papers of important results proved using Markov chains in $$\mathbb{C}^n$$?

EDIT A: Following the suggestions in the answers below I found links to both volumes of collected works: http://publ.lib.ru/ARCHIVES/M/MARKOV_Andrey_Andreevich/_Markov_A.A..html

EDIT B: The attribution of Markov's paper, Распространение закона больших чисел на величины, зависящие друг от друга, the usually cited origin of Markov chains, to (1) in the year 1906, volume 15(4),135-156, is strange. In the collected works, it's dated by Markov: 25 March 1907, apparently this being the final version printed in 1907 as a booklet.

• I should add the motivation for the above: I prefer to read all the papers I cite, even if they are standard literature cited simply to fill an introductory paragraph or name something. – Gottfried William Sep 29 '14 at 15:09
• I'd like to mention that an Area 51 StackExchange site on History of Science and Mathematics is in the middle of a Commitment stage: area51.stackexchange.com/proposals/65204/… – Todd Trimble Sep 29 '14 at 15:42
• For historical account, and possibly some answers to your questions, look at First Links in the Markov Chain" by Brian Hayes at americanscientist.org/libraries/documents/… – Margaret Friedland Sep 29 '14 at 16:10
• Todd, I committed :) Margaret, I read the article, unfortunately it has no citations to Markov, only anecdotes; А. А. Марков. "Распространение закона больших чисел на величины, зависящие друг от друга". "Известия (really Извѣстія) Физико-математического общества при Казанском университете", 2-я серия, том 15, ст. 135–156, 1906. is the reference in Wikipedia. I believe, in addition to this paper, he has two (or three) more papers prior the Onegin speech in Petersburg, in the same journal. Only the archaeology section, not the physics-mathematics volumes, are present in Google books or Archive. – Gottfried William Sep 29 '14 at 16:47
• From where you gathered that the journal (1) still exists today? My impression was that it was discontinued (not to be confused with "Ученые Записки Казанского Университета"). – Pasha Zusmanovich Oct 17 '14 at 1:18

Obtaining old issues of this journal is difficult, though perhaps some American libraries have it, and you can try Interlibrary Loan (ILL). Alternatively, there are two volumes of collected papers Markov's Collected papers, MR0050525 the papers you are looking for must be in this volume. Our library has this, so you can surely obtain this via ILL.

BTW, historians say that Markov chains were introduced for the first time in his paper of 1913, here is an account available online:

http://www.americanscientist.org/libraries/documents/201321152149545-2013-03Hayes.pdf

It gives a reference to an English translation which must be easily available. Also: http://www.alpha60.de/research/markov/ which has links to some translations.

• English translation of the paper in question is available here (unfortunately, not for free): An Example of Statistical Investigation of the Text Eugene Onegin Concerning the Connection of Samples in Chains, trans. David Link. Science in Context 19.4 (2006): 591–600. Online: journals.cambridge.org/production/action/… Contrary to OP's first impression, Hayes (whose article I linked to in my earlier comment) does cite this paper by Markov. – Margaret Friedland Sep 30 '14 at 0:38

There is a volume of collected works of Markov published in 1951 (in Russian), which is available online:

http://pyrkov.professorjournal.ru/c/document_library/get_file?uuid=dee29674-473c-4c73-a9f2-cdf58abb182b&groupId=996446

EDIT The claim that "Markov chains were introduced for the first time in his paper of 1913" is completely misleading (as well as several other claims made by Hayes). Moreover, in my opinion, his popular "essay" does not qualify as a scholarly publication at all (to begin with, as he himself admits, he was not able to access Markov's originals - as he does not read Russian - and is based entirely on secondary sources significantly embellished by him).

In addition to the aforementioned volume of Markov's collected works published in 1951 the originals of almost all Markov papers are available online including the very first paper on the subject, Распространеніе закона большихъ чиселъ на величины, зависящія другъ отъ друга (although published in the 1906 volume, the date at the end is indeed March 25, 1907, which just means that the 1906 volume was physically printed only in 1907 - this is pretty common). Most other originals are accessible on Markov's page: Изслѣдованiе замѣчательнаго случая зависимыхъ испытанiй (1907), О связанныхъ величинахъ, не образующихъ настоящей цѣпи (1911), Объ одномъ случаѣ испытанiй, связанныхъ въ сложную цѣпь (1911), Объ испытанiяхъ связанныхъ въ цѣпь не наблюдаемыми событiями (1912), and, finally , the "Onegin paper" Примѣръ статистическаго изслѣдованiя надъ текстомъ “Евгенiя Онѣгина”, иллюстрирующiй связь испытанiй въ цѣпь (1913). The original of the 1908 paper Распространеніе предѣльныхъ теоремъ исчисленія вѣроятностей на сумму величинъ связанныхъ въ цепь is available at the link just provided by yannis.

It is in this latter paper that he introduces what we now know as a Markov chain in a very clear and explicit form (not really different from what one can read in modern textbooks), by saying that it is determined by a transition matrix and an initial distribution (p. 2).

Concerning the history of the earlier stages of the theory of Markov chains, the most comprehensive source is, in my opinion, Souvenirs de Bologne by Bernard Bru (2003). To make it short.

The work of Markov was not as unaccessible as one might think. Translations of his papers were published in French in Acta Mathematica (1910) and in German in the 1912 edition of his probability course. However, it did not help much. Poincaré, who in Chapitre XVI ("questions diverses") of the second edition of his Calcul des Probabilités (published also in 1912) introduced random walks on finite groups for studying card shuffling, was not aware of it. As Fréchet and Hadamard put it in their 1933 Sur les probabilités discontinues des événements "en chaîne",

On peut considérer comme une preuve de l’importance de l’étude des probabilités des événements "en chaîne", le fait qu’un certain nombre de mathématiciens s’y sont trouvés conduits indépendamment sans connaître les travaux de leurs prédécesseurs. En particulier, comme Poincaré, comme M.M. Urban, Paul Lévy, Hostinsky, Chapman, comme nous, M. von Mises s’est engagé dans cette étude sans avoir eu connaissance des travaux de Markoff, travaux fondamentaux, mais publiés en russe.

For the first time the work of Markov was acknowledged in the West by Hadamard. In a footnote to his 1928 ICM talk Sur le battage des cartes et ses relations avec la mécanique statistique inspired by Poincaré's 1912 work and published in 1931 (pp.133-139) he very clearly describes how he learned about Markov's work:

Comme me l'a fait remarquer M. Polyà, la question des "grandeurs enchaînées" traitée par Markoff (Ac. Russe des Sciences 1908 et Wahrscheinlichkeitsrechnung, 1912, Anhang II) puis, plus récemment par M. Serge Bernstein (Math. Ann., t. 97), est voisine de celle du texte, qu'elle combine en quelque sorte avec celle de la loi des grands nombres. On y est conduit en faisant correspondre à chaque substitution $S$ une valeur d'une grandeur $x$ et considérant la distribution en probabilité de la somme des valeurs prises par $x$ dans les $n$ premiers coups.

It is notable that Hadamard does not even mention the 1913 Onegin paper :)

Concerning 2). In the book by T. Hawkins, "The Mathematics of Frobenius in Context. A Journey Through 18th to 20th Century Mathematics", Springer, 2013 (DOI: 10.1007/978-1-4614-6333-7; MR: 3099749), Markov chains are discussed in chapter 17. An excerpt from there:

"Even though Markov’s paper was translated into German and appended to the 1912 German translation of his book on the theory of probability [432], it is uncertain how widely read it was. Apparently, those who did discuss Markov chains in the period 1912–1930 limited their attention to the case $P > 0$ ..."

"In the late 1920s, there was a renewed interest in Markov chains on the part of a large number of mathematicians, who became more or less simultaneously interested in the subject. Some of them, including J. Hadamard and M. Fréchet, apparently reinvented aspects of the theory without knowing of Markov’s pioneering work [258, p. 2083, 2083n.3]. In the early 1930s, in the midst of the revival of interest in Markov chains, two applied mathematicians, R. von Mises and V.I. Romanovsky, independently applied Frobenius’ theory of nonnegative matrices in order to deal with chains corresponding to stochastic $P \not\gt 0$."

I couldn't find the original Распространение закона больших чисел на величины зависящие друг от друга in the Известия физико-математического общества при Казанском университете either, but I managed to find two other important papers:

Распространеніе предѣльныхъ теоремъ. Исчисленія вѣроятностей на сумму величинъ связанныхъ въ цѣпь in Записки Императорской Академіи Наукъ по физико-математическому отдѣленію, 1908 https://ia800608.us.archive.org/35/items/zapiskiimperator822impe/zapiskiimperator822impe.pdf

Объ одной задачѣ Лапласа, in Извѣстія императорской академіи наукъ, 1915 https://ia600709.us.archive.org/14/items/mobot31753003650469/mobot31753003650469.pdf

These facsimili are nice if you wish to read the papers in the original pre-Lenin spelling (with ѣ, ѳ, і, ъ and the such).