# Gauss–Kuzmin distribution

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Ex. kurtosis | (not defined) |

Entropy |
3.432527514776...^{[1]}^{[2]}^{[3]} |

In mathematics, the **Gauss–Kuzmin distribution** is a discrete probability distribution that arises as the limit probability distribution of the coefficients in the continued fraction expansion of a random variable uniformly distributed in (0, 1).^{[4]} The distribution is named after Carl Friedrich Gauss, who derived it around 1800,^{[5]} and Rodion Kuzmin, who gave a bound on the rate of convergence in 1929.^{[6]}^{[7]} It is given by the probability mass function

## Gauss–Kuzmin theorem

Let

be the continued fraction expansion of a random number *x* uniformly distributed in (0, 1). Then

Equivalently, let

then

tends to zero as *n* tends to infinity.

## Rate of convergence

In 1928, Kuzmin gave the bound

In 1929, Paul Lévy^{[8]} improved it to

Later, Eduard Wirsing showed^{[9]} that, for *λ*=0.30366... (the Gauss-Kuzmin-Wirsing constant), the limit

exists for every *s* in [0, 1], and the function *Ψ*(*s*) is analytic and satisfies *Ψ*(0)=*Ψ*(1)=0. Further bounds were proved by K.I.Babenko.^{[10]}

## See also

## References

- ↑ Blachman, N. (1984). "The continued fraction as an information source (Corresp.)".
*IEEE Transactions onInformation Theory*.**30**(4): 671–674. doi:10.1109/TIT.1984.1056924. - ↑ Kornerup, P.; Matula, D. (July 1995). "LCF: A lexicographic binary representation of the rationals".
*Journal of Universal Computer Science*.**1**: 484–503. doi:10.1007/978-3-642-80350-5_41. - ↑ Vepstas, L. (2008),
*Entropy of Continued Fractions (Gauss-Kuzmin Entropy)*(PDF) - ↑ Weisstein, Eric W. "Gauss–Kuzmin Distribution".
*MathWorld*. - ↑ Gauss, C.F.
*Werke Sammlung*.**10/1**. pp. 552–556. - ↑ Kuzmin, R.O. (1928). "On a problem of Gauss".
*DAN SSSR*: 375–380. - ↑ Kuzmin, R.O. (1932). "On a problem of Gauss".
*Atti del Congresso Internazionale dei Matematici, Bologna*.**6**: 83–89. - ↑ Lévy, P. (1929). "Sur les lois de probabilité dont dépendent les quotients complets et incomplets d'une fraction continue".
*Bulletin de la Société Mathématique de France*.**57**: 178–194. JFM 55.0916.02. - ↑ Wirsing, E. (1974). "On the theorem of Gauss–Kusmin–Lévy and a Frobenius-type theorem for function spaces".
*Acta Arithmetica*.**24**: 507–528. - ↑ Babenko, K.I. (1978). "On a problem of Gauss".
*Soviet Math. Dokl*.**19**: 136–140.