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Finite random sums (a historical essay)

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Summary

The study of the stochastic behavior of sums of independent random quantities, the number of which increases without limit, has been a most important problem of the classical theory of probability. However, finite random sums occurring in various fields of application of this theory were also studied again and again by a number of scholars and led to the development of mathematical methods and ideas in the theory of probability proper.

The subject of my article intersects that of Seal [32]. Although written skillfully, that article is too concise. Possibly for this reason Seal often did no more than describe the end results of various authors; moreover, he described those results only in modern mathematical language.

Section 1 is devoted to the 18th century developments in the context of games of chance and the theory of errors. Section 2 deals with La Grange, whose memoir [17] on the subject, though pertaining also to the theory of errors, actually paved the way for characteristic functions and thus was extremely important. Section 3 considers various early memoirs of Laplace and his main work on probability [21]. Section 4 is a synopsis and, finally, Section 5, an appendix, is devoted to the work of J. Arbuthnot [1].

What I claim to be new is the description of one manuscript of R. Boscovich in § 1.2 as well as the general subject-matter of §§ 2, 3.2, 3.5, and 5.

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    O. B. Sheynin

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Communicated by A. P. Youshkevitch

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Sheynin, O.B. Finite random sums (a historical essay). Arch. Hist. Exact Sci. 9, 275–305 (1973). https://doi.org/10.1007/BF00348365

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