The central limit theorem around 1935. (English) Zbl 0603.60001

And then came Le Cam. In his comment Professor Doob has expressed already what the reader may feel from time to time when he reads this article. So much about the feelings. The article is a very well written survey of the history of the different forms of limit or approximation theorems generally summarized as the central limit theorem.
The author shows, who are the main actors of the scene, starting with the work of De Moivre and Laplace up to the final work of Cramér, Feller and Levy (necessary conditions), and who should be credited or discredited with what. The exposition is very clear. So there are winners on the scene, and losers, though those rather in the peripheral treatment of the subject. Lindeberg is a clear winner both in Le Cam’s sufficiently detailed account and in Professor Pollard’s comment, and all this for reasons which cannot possibly be denied. Levy takes some ground on Feller’s generally accepted priority claim in the sense that both can claim with some justification their respective priorities since their work was seemingly quite independent. Here it is good to have Professor Trotter’s comment.
Gauss, though only mentioned in the background, is put in line for his least-square-De Moivre-curve argument which startled me indeed as much as Professor Le Cam as a perfectly circular argument. However Trotter’s comment indicates that Gauss explicitly stated in the same work that the method of least squares cannot be singled out on purely mathematical grounds which means that Gauss cannot have intended to give a proof. Gauss’ admirers may therefore have expected indulgence, but Le Cam’s reply stays inexorable and insinuates more: ”Why pick a dead man?” rather than ”in dubio pro reo” which may have been hoped for. Again it must be given to the author that he tried hard to give a precise account by seemingly going through the original work with adequate help to make sure not to miss linguistic subtleties, although, right here, the reference is missing.
Other reproaches seem less serious. Why, for example, blame Gauss for his heuristic proof of Bayes’ theorem but not Levy for his heuristic feelings about conditional expectations? Why, for example, blame anybody for not mentioning anybody else if the author ”forgets” to mention Breiman’s book (a colleague in his own department, I think). Le Cam did not forget it, but only Professor Pollard’s comment gave him the lucky opportunity to explain why. - If these examples are not serious, neither are others.
Bertrand, Borel and Poincaré also get quite a bit of criticism. They moreover fail to receive any specific outside support (except general comfort from Professor Doob, perhaps). This does not imply anything, of course, and may have a simple probabilistic explanation. The post- reviewer regrets somewhat to have to leave it as it is, and this simply for his lack of competence. Professor Le Cam’s point of view is clear. A (french) post-comment, if it could defend these, would certainly be both fair and desirable.
To avoid in this review the possible impression that this paper might be built on little controversies it should be pointed out, that this is not at all the case. It is built upon a profound and mature understanding of the subject, from which one should definitely be able to profit. Taken together with the balancing comments it is also most enjoyable to read, and, though I am not a native English speaker, I dare to add, that the force of language felt in this paper is sometimes remarkable.
Reviewer: F.T.Bruss


60-03 History of probability theory
01A60 History of mathematics in the 20th century
60F05 Central limit and other weak theorems
60G15 Gaussian processes
60G42 Martingales with discrete parameter
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