Borel and the St. Petersburg martingale. (Borel et la martingale de Saint-Pétersbourg.) (French) Zbl 0979.01018

Bru, Bernard; Bru, Marie-France; Chung, Kai Lai

Rev. Hist. Math. 5, No. 2, 181-247 (1999).

In addition to its main subject, this essay describes the related work and the biographies of Le Dantec (1869-1917) and Ville (1910-1989) and provides general information about Borel. It is based in part on archival sources.
Borel believed that the dissemination of mathematical knowledge was socially important even though his technique lagged behind his advanced ideas. In 1909, he non-rigorously studied the problem of the return to a draw in a long game of heads and tails which later gave rise to the arc sine law and which led him to the strong law of large numbers. In 1911 Borel noted the connection of this problem with the Petersburg paradox to which he turned his attention in 1939 by applying the notion of martingale and proved that, by regulating the stakes at each round and choosing the moment for stopping, a gambler can make a fair game advantageous for himself.
The authors also touch on Le Dantec’s non-recognition of the probability of a single event and his views on evolution theory, on Mises’s frequentist theory, and on Borel’s anticipation of the theory of games. When referring to books, they fail to mention the appropriate page numbers.

Reviewer: O.B.Cheinine (Berlin)

MSC:

01A60

History of mathematics in the 20th century

Keywords:

frequentist theory; martingales; Petersburg game; strong law of large numbers; Le Dantec; Ville