7. Probability and Statistics Soviet Essays - Sheynin, Oscar

distribution is a corollary of the limit theorem extended onto the appropriate sums of smallvariables.I shall not tire you with the precise formulation of the results concerning the necessary andsufficient conditions for the applicability of the limit theorem as obtained by variousmathematicians. Research in this field distinguished by extreme subtlety and deepness islinked with the main problem in analysis and makes use of two externally different methods.One of these, the method of expectations of the consecutive powers, or of moments, whoseidea is due to {Bienaymé and} Chebyshev, underlies Markov’s fundamental works. Itconsists in solving a system of an infinite number of equations in an infinite number ofunknowns by the algorithm of continued fractions; the solution is directly connected with theproblem of summing everywhere divergent Taylor series. The second method applied byLiapunov, that of characteristic functions, is based on the Dirichlet discontinuity factor thatconnects the calculation of the limiting probability with the theory of improper integrals andtrigonometric series 11 .The just mentioned scholars had investigated the case of sums of independent variableswith an exhaustive completeness, and the later work of Lindeberg, Pólya and others, withoutintroducing essentially new ideas, has only simplified some proofs and provided another,sometimes more general formulations for the results of Liapunov and Markov.Here, I shall only note one corollary of the Liapunov theorem especially important for thestatistical practice and, in particular, for justifying the method of sampling: For anydistribution of the values of some main {parent} population, the arithmetic mean of thesevalues, when the number of observations is sufficiently large, always obeys the Gauss law.The investigation of sums of dependent variables, an example of which I have consideredabove, presents special difficulties. However, in this field rather considerable findings hadalso been already obtained. In particular, they allow to explain why most of the curves ofdistribution of indications occurring in more or less uniform biological populations, asalready noticed by Quetelet, obey in the first approximation the Gauss law. By similarmethods {?} it became also possible to substantiate mathematically the theory of normalcorrelation whose main formulas were indicated by Bravais and applied by Galton forstudying the phenomena of heredity. I shall not expound here the Galtonian statistical theoryof heredity which Pearson developed later in detail. Its essence consists in his law ofhereditary regression according to which normal correlation exists between the sizes of somequantitatively measured indication in parents and offspring. At present, owing to theexperiments connected with the Mendelian theory, it should be considered as experimentallyestablished that the Galtonian theory is not as universal as Pearson, who based his opinion onhis numerous statistical observations, thought it was.However, the abovementioned mathematical investigations enable us to prove that, even ifthe Mendelian law is not the sole regulator of the inheritance of elementary indications, theGaltonian law of hereditary regression must be applicable to all the complicated indications(for example, to the stature of man) made up of a large number of elementary ones. The sametheorems explain why Pearson and his students could have also statistically revealed, inmany cases, the existence of normal correlation between the sizes of various organs inindividuals of one and the same race. The investigations also show that both the Gaussiannormal curve and the normal correlation are only the limiting cases of some generaltheoretical patterns so that the actually observed more or less considerable deviations fromthem are quite natural.[10] Thus we approach a new cycle of problems in the theory of probability whichcomprises the theories of distribution and of the general non-normal correlation. From thepractical viewpoint the Pearsonian British school is occupying the most considerable place inthis field. Pearson fulfilled an enormous work in managing statistics; he also has great