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Gauss–Kuzmin distribution

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Gauss–Kuzmin distribution
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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

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Gauss–Kuzmin theorem

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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.

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Rate of convergence

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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]

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References

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