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There is a theorem due to Markov, called Markov's Law of Large Numbers which goes by:

The weak law of large numbers holds if for some $\delta > 0,$ all the mathematical expectations $\mathbb E\left(|X_i|^{1+\delta}\right);~ i = 1,2,\ldots$ exist and are bounded.

But I didn't find any proof. So, I started sieving down the sources. Seneta (cf.$\rm [I]$) mentions that but didn't provide any proof. It was pinpointed to Markov's Ischislenie Veroiatnostei $(1913)$ chapter $\rm III.$ Unfortunately, it was written in Russian (is there any English translation?) and I couldn't make anything of it. Also, I was wondering how he derived it without using measure theoretic language. Nevertheless, I checked some of my probability books like Rosenthal, Klenke but couldn't find any explicit result and its derivation. Feller though mentioned Markov in a footnote but again didn't provide any explicit elaboration.

So, that leaves me wondering: who was the first English author that incorporated the result with a proof in their treatise?


$\rm [I]$ A Tricentenary history of the Law of Large Numbers, Eugene Seneta, $2013, $ p. $27.$

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  • $\begingroup$ Markov's Calculus of Probabilities (1913) is the third edition of his book that originally came out in 1900. There was a German translation in 1912 and posthumous edition in 1924. He does prove the law of large numbers there under various assumptions including the OP one. Measure theoretic language is not really necessary as one can express everything equivalently by using probabilities of events and their expected values, which is what he does. $\endgroup$
    – Conifold
    Oct 6 at 3:45
  • $\begingroup$ Yes @Conifold. But did he publish any English translation? I saw the name Calculus of Probabilities but only could get Russian edition. $\endgroup$ Oct 6 at 3:46
  • $\begingroup$ There was no English translation, as far as I know. A well sourced survey of pre-Kolmogov history that discusses Markov's contributions at length is Sheynin, Theory of Probability. A Historical Essay. $\endgroup$
    – Conifold
    Oct 6 at 3:55
  • $\begingroup$ Thanks @Conifold for the info. I guess it was his student Uspensky who first wrote the result in detail in his treatise. $\endgroup$ Oct 6 at 3:57

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In 1937, Markov's student J. Uspensky published his English-language Introduction to Mathematical Probability, which has the result in question on p.191. It seems hard to believe this is the first English publication, however.

It is not present in the 3rd (1979) edition of Cramer's Random Variables, and so it might not be in the 1st (1936) edition, but I don't have one at hand to check.

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    $\begingroup$ Thanks, @J. W. Tanner, for the edit. $\endgroup$ Oct 6 at 13:05

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