In mathematics Lévy's constant (sometimes known as the Khinchin–Lévy constant) occurs in an expression for the asymptotic behaviour of the denominators of the convergents of continued fractions. In 1935, the Soviet mathematician Aleksandr Khinchin showed that the denominators qn of the convergents of the continued fraction expansions of almost all real numbers satisfy
The term "Lévy's constant" is sometimes used to refer to (the logarithm of the above expression), which is approximately equal to 1.1865691104…. The value derives from the asymptotic expectation of the logarithm of the ratio of successive denominators, using the Gauss-Kuzmin distribution. In particular, the ratio has the asymptotic density function
for and zero otherwise. This gives Lévy's constant as
- A. Ya. Khinchin; Herbert Eagle (transl.) (1997), Continued fractions, Courier Dover Publications, p. 66, ISBN 978-0-486-69630-0
- [Reference given in Dover book] "Zur metrischen Kettenbruchtheorie," Compositio Matlzematica, 3, No.2, 275–285 (1936).
- [Reference given in Dover book] P. Levy, Théorie de l'addition des variables aléatoires, Paris, 1937, p. 320.