Review of the life and times of the central limit theorem by William J. Adams

The Life and Times of the Central Limit Theorem by William J. Adams American Math Society, Second Edition Review by Miklós Bóna Intuitively speaking, the Central Limit Theorem describes the behavior of the sum Sn = X1 + X2 + · · · + Xn of n mutually independent random variables, each of which has only a slight effect on the sum. The result is that if n is large, then the distribution of Sn is approximately normal. The name “Central Limit Theorem” was coined in 1920 by George Pólya. In the world of precise mathematical statements, there is a large collection of theorems that are called Central Limit Theorems; they describe the conditions under which the conclusion discussed in the previous paragraph is valid. In this book, the author discusses the historical process from the first appearances of the Central Limit Theorem in its imprecise form to modern survey papers. The story starts with Jacob Bernoulli, who, at the end of the seventeenth century, in his book Ars Conjectandi, discussed questions like the following one. Let us assume that a box contains 5000 pebbles, 3000 of which are white and 2000 of which are black. If we remove 500 pebbles, how sure can we be that the number of white pebbles removed is close to 300? He then proved an upper bound for the probability of the event that the number of white pebbles will be outside the interval [290, 310]. What is important about Bernoulli’s work is that he was not interested in the a priori probability of an event, but instead he was interested in the level of precision at which results of experiments will approach that a priori probability. The next significant figure in the development of the field was Abraham de Moivre, who, like Bernoulli, was interested in error term estimates. This quest led him to look for approximations for n! since factorials regularly appear in arguments using involving choices of subsets. He essentially rediscovered Stirling’s formula, that is, the fact that n! ∼ nne−n √ 2πn. His most frequently used technique involved computations of ∫ e−x 2 , often with power series. As a consequence of De Moivre’s work, by the end of the eighteenth century, the use of integrals of e−x 2 was well established. Besides a tool in the estimation of errors in probabilistic calculations, it was also used in mathematical astronomy. In fact, in 1732, the Academy of Sciences in Paris solicited papers explaining why planetary orbits were not completely circular and why they were closer to elliptical curves. No prize was awarded. However, two years later, the prize was given to John Bernoulli and his son Daniel. These two authors, in different papers, argued that the small differences between the orbits and circular curves could not be attributed to chance. Their work was continued by Pierre-Simon Laplace, who extended his focus to cometary orbits. There were 97 known comets at that time, and Laplace assumed that the orbit of each of them is the result of a variable of the same distribution. In modern form, we would say that he used a Central Limit Theorem for identically distributed mutually independent random variables. Then comes a short chapter on the history of the concept that an error of observation is composed of elementary errors, and the effect of these elementary errors is diminished by their independence and multitude. This gets us to the abstract theory of the central limit theorem, and the three Russian mathematicians who brought that about, namely Pafnuty Lvovics Chebyshev, and his students Michail Vasilevich Lyapunov and Andrei Andreevich Markov. Their papers read reasonably close to contemporary research articles, as is illustrated by four papers of Lyapunov that can be found in the Appendix. The inclusion of two of them is new to the second edition. Chebyshev started his career in the field by noticing that earlier work by Poisson lacked in rigor in its error estimates. Later Markov criticized one of Chebyshev’s papers for the same reason. When Chebyshev could not provide a rigorous argument with the methods of classical analysis,