kniga 7 - Probability and Statistics 1 - Sheynin, Oscar

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Among Romanovsky’s most fundamental and important studies in probability is his workon Markov chains (which he began in 1928) and their generalizations (correlation chains andpolycyclic chains) and on generalizing the central limit theorem onto the multidimensionalcase. He was the first to study exhaustively by algebraic methods the limiting (as n )behavior of the transition probabilities describing the change of state during n steps forhomogeneous Markov chains with a finite number of states [96].In the same paper and in his later work [112; 121; 126; 132; 142] Romanovsky wasengaged in proving a number of other limit theorems for the same kind of Markov chains.This research also became the starting point for many other studies of Markov chains andtheir various generalizations by algebraic methods. In [63], applying the method ofcharacteristic functions, he extended the central limit theorem onto sums of independentrandom vectors.In statistics, Romanovsky’s work cover an extremely wide range of problems. It is hardlypossible to point out any large section of this discipline, whether modern or classical, inwhose development he did not actively and authoritatively participate. Especially great areRomanovsky’s merits in widely popularizing mathematical-statistical methods in our countryas well as in heightening the mathematical level of statistical thought. Here, his course [144]published in 1924 and 1939 and his books [50; 105; 136; 137] played a very large part.I shall now briefly describe some of his important studies in mathematical statistics.Depending on the form of the theoretical law of distribution and on the organization ofobservations, there appear various methods of approximately estimating the differentcharacteristics of the parent population. The most prominent research in our country in thissphere was done by Romanovsky.A large cycle of his writings [38; 39; 45; 46; 48; 54; 60; 62] concerned with the theory ofsampling was generally recognized. With regard to their substance, they adjoin the studies ofthe British school of statistics, but they advantageously differ from the latter by rigor of theirmethodological principles. In addition, when choosing methods for solving his problems,Romanovsky exclusively used those developed by the Chebyshev school; to be sure, heperfected and adopted these methods for achieving new goals. That Romanovsky followedhere Chebyshev can partly be explained by his belonging to the latter’s school and havingattended the course in probability theory read by the celebrated Markov. Keeping in hisstudies to that mathematical rigor that distinguished his teacher, Markov, he used the theoryas the main tool for logically irreproachably justifying mathematical statistics. Such asubstantiation was indeed lacking in the constructions of British statisticians whose worksserved Romanovsky as a starting point for choosing his problems. The rigorous theoreticalunderpinning of mathematical statistics is one of his merits that promoted its development inour country.Romanovsky was the first to offer an analytical derivation of the laws of distribution of thewell-known criteria, of the Student – Fisher t and z, of empirical coefficients of regressionand other characteristics [105]. He also provided a more general theory of the Pearson chisquaredtest [65] and studied problems connected with checking whether two independentsamples belonged to one and the same normal population.From among Romanovsky’s work on probability and statistics deserving serious attentionI also mention [113; 120; 122; and 115]. In the first of these, he shows that the criterion,that he himself introduced in 1928 [60], is much easier to apply to all the problems where theFisherian z test based on the tables of that scholar is made use of; that it leads to the samequalitative solutions; and that it often solves these problems more precisely than the latter. Inaddition, the construction of the former is simpler.The second writing [120] is very interesting methodologically. There, Romanovskyattempts to review systematically the main statistical concepts and problems. Given the
variety and detachment of those latter, and the availability of a diverse set of methods appliedby statistics, his endeavor was absolutely necessary.The third paper provided an elementary and simple solution to a topical statistical problemconnected with objectively estimating unknown characteristics of parent populations bymeans of observation. In the last-mentioned work he calculated transition and other kinds ofprobabilities for Markov chains and offered their statistical estimates given the appropriateobservations.The classical theory of periodograms enables to analyze a number of random variablesunder the assumption that several periodic oscillations and additional perturbationsindependent from one trial to another one are superposed. Romanovsky devoted a series ofimportant studies [78; 79; 81] to the circumstances occurring when admitting dependencebetween random perturbations.Systematically and intensively carrying out his scientific work for half a century,Romanovsky, especially from the 1930s onwards, had been paying more attention than hedid before to problems directly connected with practical needs, e.g., in [86; 87; 98; 146]. Andduring the last period of his life he was much engaged in the important problem ofcontemporary statistics, – in the statistical estimation of the quality of production [134; 145;153; 151; and other ones].From among contributions not connected with probability or statistics I mention [68; 72;104; 92]. The first three of these, although originating from problems connected with Markovchains, provided findings of independent mathematical interest.In concluding this brief review, it is also necessary to indicate that, not restricting hisefforts to publishing scientific contributions, Romanovsky unceasingly counselled, verballyand in writing, most diverse productive establishments and scientific institutions andanswered questions arriving from all quarters of our country. Until his last days he combinedscientific studies with active social work. He was permanent chairman of TashkentMathematical Society and an active member of the Society for Dissemination of Political andScientific Knowledge. The people and the government of Uzbekistan estimated his merits attheir true worth. He was three times elected Deputy of the Republic’s Supreme Soviet anddecorated with three orders of Lenin and an Order of the Red Banner of Labor; he becameHonored Science Worker of Uzbekistan, and he wined a Stalin prize (1948).The sum totals of all his scientific work and teaching activities was the mathematicalschool that he created in Tashkent.Bibliography{A comprehensive bibliography of Romanovsky’s contributions is in Bogoliubov, A.N.,Matvievskaia, G.P. (1997), Romanovsky. M. (R), against which I checked and specified theentries below. With the exception of contributions mentioned by Sarymsakov in his maintext, I omitted those not belonging to probability or statistics. Nevertheless, I preserved hisnumeration; for example, the first entry below has number 6. Contributions 52, 57, 80, 92and 104 are reprinted/translated in Romanovsky’s (Sel. Works), vol. 2.Tashkent, 1964.}AbbreviationAN = Akademia Nauk (of Soviet Union if not stated otherwise)C.r. = C.r. Acad. Sci. ParisGIIA = G. Ist. Ital. AttuariIMM = Inst. Math. & Mech.SAGU = Srendeaziatsk. Gosudarstven. Univ.SSR = Soviet Socialist RepublicVS = Vestnik Statistiki
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of All Countries and to the Entire
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(Coll. Works), vol. 4. N.p., 1964,
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individuals of the third class, the
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From the theoretical point of view
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Second case: Each crossing can repr
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On the other hand, for four classes
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f i = i S + i , i = 1, 2, 3, 4, (
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f 1 = C 1 P(f 1 ; …; f n+1 ), C 1
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ut in this case f = 2 , f 1 = 2 ,
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I also note the essential differenc
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A 1 23n1 + 1 A 1 A 1 … A 11A 2 A
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coefficient of 2 in the right side
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h(A r h - c h A r 0 ) = - A r0we tr
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Notes1. Our formulas obviously pres
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Bernstein’s standpoint regarding
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Corollary 1.8. A true proposition c
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It is important to indicate that al
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ut for the simultaneous realization
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devoid of quadratic divisors and re
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propositions (B i and C j ) can be
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A ~ A 1 and B = B 1 , we will have
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included in a given totality as equ
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For unconnected totalities we would
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proposition given that a second one
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On the other hand, let x be a parti
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totality is perfect, but that the j
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In this case, all the finite or inf
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probabilities p 1 , p 2 , … respe
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where x is determined by the inequa
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totality of the second type (§3.1.
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x = /2 + /(23) + … + /(23… p n
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that the fall of a given die on any
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infinitely many digits only dependi
Page 71 and 72: 10. (§2.1.5). Such two proposition
Page 73 and 74: F(x + h) - F(x) = Mh, therefore F(x
Page 75 and 76: “confidence” probability is bas
Page 77 and 78: x1+ Lp n (x) x1− Lx1+ Lf(t)dt < x
Page 79 and 80: |(x 1 ; t 0 ; t 1 ) - 1 t0tf(t)dt|
Page 81 and 82: 5. The distribution ofξ , the arit
Page 83 and 84: P(x 1i < x) = F(x; a i ) = C(a i )
Page 85 and 86: egards his promises. Markov shows t
Page 87 and 88: other solely and equally possible i
Page 89 and 90: notion of probability and of its re
Page 91 and 92: However, already in the beginning o
Page 93 and 94: the revolution. My main findings we
Page 95 and 96: Nevertheless, Slutsky is not suffic
Page 97 and 98: path that would completely answer h
Page 99 and 100: on political economy as well as wit
Page 101 and 102: scientific merit. Borel was indeed
Page 103 and 104: [3] Already in Kiev Slutsky had bee
Page 105 and 106: different foundation. The difficult
Page 107 and 108: 5. On the criterion of goodness of
Page 109 and 110: --- (1999, in Russian), Slutsky: co
Page 111 and 112: Here also, the author considers the
Page 113 and 114: second, it is not based on assumpti
Page 115 and 116: experimentation and connected with
Page 117 and 118: Russian, and especially of the Sovi
Page 119 and 120: station in England. This book, as h
Page 121: Uspekhi Matematich. Nauk, vol. 10,
Page 125 and 126: 46. On the distribution of the regr
Page 127 and 128: 119. On the Markov method of establ
Page 129 and 130: No lesser difficulties than those e
Page 131 and 132: Separate spheres of work considerab
Page 133 and 134: 10. Anderson, O. Letters to Karl Pe
Page 135 and 136: Hier sind, im Allgemeinen, ganz ana
Page 137 and 138: Jedenfalls, glaube ich erwiesen zu
Page 139 and 140: werde ich das ganze Material in kur
Page 141 and 142: considered as the limiting case of
Page 143 and 144: and, inversely,] = m ...1 2 N[ ch h
Page 145 and 146: µ 2 2 = m 2 2 - 2m 2 m 1 2 + m 1 4
Page 147 and 148: (x k - x k+1 ) … (x k - x +) = E(
Page 149 and 150: the thus obtained relations as pert
Page 151 and 152: [1/S(S - 1)(S - 2)][(Si = 1Sx i ) 3
Page 153 and 154: ( N −1)((S − N )(2NS− 3S− 3
Page 155 and 156: µ 5 + 2µ 2 µ 3 = U [S/S] 5 + 2U
Page 157 and 158: case, the same property is true wit
Page 159 and 160: It follows that the question about
Page 161 and 162: then expressed my doubts). And Gned
Page 163 and 164: For Problem 1, formula (7) shows th
Page 165 and 166: Let us calculate now, by means of f
Page 167 and 168: ϕ′1(x)1E(a|x 1 ; x 2 ; …; x n
Page 169 and 170: Theorem 3. If the prior density 3
Page 171 and 172: P( ≤ ≤ |, 1 , 2 , …, s )
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6. A Sensible Choice of Confidence
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0 = A 0 n, = B2, = B2, 0 = C 0 n
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Note also that (95),(96), (83),(85)
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Γ(n / 2)Γ [( n −1) / 2]k = (1/2
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f (x 1 , x 2 , …, x n ) = 1 if x
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and the probability of achieving no
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E = kEµ. (14)In many particular ca
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a = np, b = np 2 = a 2 /n, = a/nand
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with number (2k - 2), we commit an
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(67)which is suitable even without
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" = 1/[1 - e - ], = - ln [1 - (1/
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Such structures are entirely approp
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11. As a result of its historical d
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exaggeration towards a total denial
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kniga 7 - Probability and Statistics 1 - Sheynin, Oscar

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