7. Probability and Statistics Soviet Essays - Sheynin, Oscar

Smirnov elucidates the application of probability to mathematical statistics in a companionarticle {also translated below}. Applications to statistical physics would have been worthy ofa special paper since the appropriate problems are specific; we only treat some of them.1.Sums of Independent TermsChebyshev’s and Liapunov’s main research was almost entirely concentrated on studyingthe behavior of sums of a large number of such independent random variables that theinfluence of each of them on their sum was negligible. More special investigations connectedwith sequences of independent trials, which constituted the chief object of attention for JakobBernoulli, Laplace and Poisson, are reducible to studying such sums. This problem is of mainimportance for the probability-theoretic substantiation of the statistical methods of research(the sampling theory) and of the {Laplacean} theory of errors; the interest in it is thereforequite well founded.In addition, as it usually happens in the history of mathematics, this problem, that occupiedmost considerable efforts of the scholars of the highest caliber belonging to the precedinggeneration, became important as a touch-stone for verifying the power of new methods ofresearch. When only the laws of distribution of separate sums are involved, the method ofcharacteristic functions proved to be the most powerful. It gradually swallowed up theclassical Russian method of moments and superseded the direct methods which the newMoscow school had borrowed from the theory of functions of a real variable. Elementarydirect methods of the Moscow school are still providing the most for problems in which anestimation of the probabilities of events depending on many sums is needed. In future, thesemethods will possibly be replaced by the method of stochastic differential or integrodifferentialequations.1.1. The Law of Large Numbers. A sequence of random variables 1 , 2 , …, n , …is called stable if there exists such a sequence of constants C 1 , C 2 , …, C n , … that, for any >0,lim P(| n – C n | > ) = 0 as n . (1.1.1)Practically this means that, as n increases, the dependence of the variables n on randomnessbecomes negligible. If n have finite expectations E n = A n , it will be the most natural tochoose these as the constants C n . We shall therefore say that the sequence of n with finiteexpectations A n is normally stable if, for any > 0,lim P(| n – A n | > ) = 0 as n . (1.1.2)When stating that the sequence n obeys the law of large numbers, we mean that it is stable.Classical contributions always had to do with normal stability but in many cases it is morelogical and easier to consider stability in its general sense. If the variables n have finitevariances B n = var n = E( n –E n ) 2 the relationlim B n = 0 as n (1.1.3)