1 Studies in the History of Statistics and Probability ... - Sheynin, Oscar

distribution of the maximal term |s k |, 1 ≤ k ≤ n, presents a specialproblem which we will not discuss.For any fixed n the deviation of ξ 1 + ... + ξ n from its expectation nais only possible by a magnitude of the order of √n. This means that thenon-random magnitude na, certainly if a ≠ 0, plays a predominant partas compared with random deviations of the order √n. Now, dividing(4.10) by n, we getξ + ... + ξ 3σ− ≤ (4.11)nn1n| a |which is valid with probability 0.997 (if we believe in the normaldistribution). When replacing 3σ by 4σ or 5σ, this inequality will bevalid even with a higher probability. Given a large n, the difference inthe left side of (4.11) is practically certainly small which is thecelebrated law of large numbers.It is interesting to dwell on the history of its proof andinterpretation. It was note long ago that the results of separateobservations, physical, meteorological, demographic or other, fluctuateessentially whereas the mean values of a large number of observationsreveal a remarkable stability. The first statisticians had seen heredivine intervention, but, as a scientific understanding of the world wasbeing established, that stability became a scientific fact.In the 18 th , the century of reason, mathematics became very trusted;it was believed that the main laws of natural sciences and even ofeconomics, moral philosophy and politics can be derived by thatscience. A desire to regard the stability of mean values as amathematical theorem had been established and that opinion persistedin the 19 th century. Exactly in that sense did Poisson interpret thediscovery of his form of the law of large numbers and he thought thathe had succeeded in proving that the mean of really made observationsshould be stable.Chebyshev essentially developed the mathematical form of the lawof large numbers by reducing its proof to the application of the[Bienaymé –] Chebyshev inequality. His proof can be found in anytextbook on the theory of probability, and after him that law began tobe considered as a very simple theorem independent of, andexpounded before the central limit theorem. Students are now eventaught to apply the inequalityP2ξ + ... + ξ σ− > ≤nnε1n[| a | ε]2for estimating the probability that the mean will deviate from a morethan by ε. However, such an application (of the Bienaymé –Chebyshev inequality) is absolutely absurd because the central limittheorem provides a much more precise result. True, the Chebyshevform of the law of large numbers demands less mathematicalrestrictions to be imposed on the random variables ξ i as compared withthe central limit theorem, but it is just the same practically impossible39