1 Oscar Sheynin History of Statistics Berlin, 2012 ISBN 978-3 ...

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1 Oscar Sheynin History of Statistics Berlin, 2012 ISBN 978-3 ...

in Russia) and the expressions normal law and coefficient of correlationwere likewise absent in his works. And, not wishing to leave his field(§13.3-1, letter to Chuprov), he never mentioned applications of his chainsto natural sciences. The structure of his Treatise became ever morecomplicated with each new edition.13.3. Markov: Main Investigations1) Mathematical treatment of observations. In spite of severalcommentators, I deny Markov’s accomplishments here. Neyman (1934, p.595) invented a non-existing Gauss – Markov theorem and F. N. David &Neyman (1938) repeated this mistake but finally Neyman (1938/1952, p.228) admitted it.In his Treatise (1900) Markov combined the treatment of observationswith the study of correlation, statistical series and interpolation, but hisinnovation was methodically doubtful. While discussing statistical series,Markov did not mention Chuprov’s relevant papers (1916; 1918 – 1919).When considering Weldon’s experiment with 26,306 throws of 12 dice (K.Pearson 1900), Markov (Treatise 1924, pp. 349 – 353) decided, afterapplying the CLT and the Bayes theorem with transition to the normal law,that the probability of a 5 or a 6 was higher than 1/3. Unlike Pearson, he hadnot used the chi-squared test and apparently left an impression that(although suitable for a small number of trials as well) it was not needed atall. Markov possibly followed here his own rigid principle (Ondar1977/1981, Letter 44 to Chuprov of 1910): I shall not go a step out of thatregion where my competence is beyond any doubt.The explication of the MLSq proper was involved; in a letter of 1910 toChuprov Markov (Ondar 1977/1981, p. 21) wrote: I have often heard thatmy presentation is not sufficiently clear. In 1893, his former student,Koialovitch (Sheynin 2006a, pp. 81 and 85), writing to Markov, formulatedsome puzzling questions about his university lectures.2) The LLN. Markov (1906/1951, p. 341) noted that the conditionlimE{[E∑ξ i – ∑Eξ i )] 2 /n 2 } = 0, n → ∞(1)was sufficient for the sequence ξ 1 , ξ 2 , …, ξ n , … of random variables to obeythe LLN; or to comply with the conditionlimP{(1/n)|(∑ξ i – ∑Eξ i )| < ε} = 1, n → ∞.Then Markov (Ibidem, pp. 342 – 344; Treatise, 1913, pp. 116 – 129)derived a few sufficient conditions for sequences of independent, and,especially, dependent random variables (1906/1951, p. 351; Treatise 1913,p. 119; 1924, p. 174), provided examples of sequences not obeying the law,and (Treatise, 1913, p. 129), proved that independent variables obeyed theLLN if, for every i, there existed the momentsEξ i = a i , E|ξ i – a i | 1+δ < C, 0 < δ < 1.Again, Markov (Treatise, 1900; p. 86 in the edition of 1924) had provedthat, for a positive random variable ξ,131

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