Gauss-Kuzmin distribution

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In mathematics, the Gauss-Kuzmin distribution gives the probability distribution of the occurrence of a given integer in the continued fraction expansion of an arbitrary real number. The distribution is named after Carl Friedrich Gauss, who first conjectured and studied the distribution around 1800, and R. O. Kuz'min, who, in 1928, along with Paul Lévy, in 1929, was able to prove Gauss's conjecture. Later, K. Ivan Babenko and Eduard Wirsing completely solved the problem, and were able to show that the speed of convergence of the continued fraction digits to the limiting distribution was exponential.

The probability that the nth term [a_n] in a continued fraction expansion is equal to k is given by

[\Pr(a_n=k)=-\log_2\left[ 1-fracright].]

References

Carl Friedrich Gauss, Recherches Arithmétiques, (1807), Blanchard, Paris.

Eric W. Weisstein. "[Gauss-Kuzmin Distribution]." From MathWorld--A Wolfram Web Resource.

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Gauss-Kuzmin distribution

See also

References