Gauss–Kuzmin distribution - Alchetron, the free social encyclopedia

Parameters (none)
Support k ∈ { 1 , 2 , … } {\displaystyle k\in \{1,2,\ldots \}} pmf − log 2 ⁡ [ 1 − 1 ( k + 1 ) 2 ] {\displaystyle -\log _{2}\left[1-{\frac {1}{(k+1)^{2}}}\right]} CDF 1 − log 2 ⁡ ( k + 2 k + 1 ) {\displaystyle 1-\log _{2}\left({\frac {k+2}{k+1}}\right)} Mean + ∞ {\displaystyle +\infty } Median 2 {\displaystyle 2\,}

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). The distribution is named after Carl Friedrich Gauss, who derived it around 1800, and Rodion Kuzmin, who gave a bound on the rate of convergence in 1929. It is given by the probability mass function

Gauss–Kuzmin theorem

Let

x = 1 k 1 + 1 k 2 + ⋯

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

lim n → ∞ P { k n = k } = − log 2 ⁡ ( 1 − 1 ( k + 1 ) 2 ) .

Equivalently, let

x n = 1 k n + 1 + 1 k n + 2 + ⋯ ;

then

Δ n ( s ) = P { x n ≤ s } − log 2 ⁡ ( 1 + s )

tends to zero as n tends to infinity.

In 1928, Kuzmin gave the bound

| Δ n ( s ) | ≤ C exp ⁡ ( − α n ) .

In 1929, Paul Lévy improved it to

| Δ n ( s ) | ≤ C 0.7 n .

Later, Eduard Wirsing showed that, for λ=0.30366... (the Gauss-Kuzmin-Wirsing constant), the limit

Ψ ( s ) = lim n → ∞ Δ n ( s ) ( − λ ) n

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.

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