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A p p e a r a n c e
F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
p
(
k
)
=
−
log
2
(
1
−
1
(
1
+
k
)
2
)
.
{\displaystyle p(k )=-\log _{2}\left(1-{\frac {1}{(1+k)^{2}}}\right)~.}
Gauss–Kuzmin theorem [ edit ]
Let
x
=
1
k
1
+
1
k
2
+
⋯
{\displaystyle x={\cfrac {1}{k_{1}+{\cfrac {1}{k_{2}+\cdots }}}}}
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
)
.
{\displaystyle \lim _{n\to \infty }\mathbb {P} \left\{k_{n}=k\right\}=-\log _{2}\left(1-{\frac {1}{(k+1)^{2}}}\right)~.}
Equivalently, let
x
n
=
1
k
n
+
1
+
1
k
n
+
2
+
⋯
;
{\displaystyle x_{n}={\cfrac {1}{k_{n+1}+{\cfrac {1}{k_{n+2}+\cdots }}}}~;}
then
Δ
n
(
s
)
=
P
{
x
n
≤
s
}
−
log
2
(
1
+
s
)
{\displaystyle \Delta _{n}(s )=\mathbb {P} \left\{x_{n}\leq s\right\}-\log _{2}(1+s)}
tends to zero as n tends to infinity.
Rate of convergence [ edit ]
In 1928, Kuzmin gave the bound
|
Δ
n
(
s
)
|
≤
C
exp
(
−
α
n
)
.
{\displaystyle |\Delta _{n}(s )|\leq C\exp(-\alpha {\sqrt {n}})~.}
In 1929, Paul Lévy [8] improved it to
|
Δ
n
(
s
)
|
≤
C
0.7
n
.
{\displaystyle |\Delta _{n}(s )|\leq C\,0.7^{n}~.}
Later, Eduard Wirsing showed[9] that, for λ = 0.30366... (the Gauss–Kuzmin–Wirsing constant ), the limit
Ψ
(
s
)
=
lim
n
→
∞
Δ
n
(
s
)
(
−
λ
)
n
{\displaystyle \Psi (s )=\lim _{n\to \infty }{\frac {\Delta _{n}(s )}{(-\lambda )^{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 .[10]
See also [ edit ]
References [ edit ]
^ Vepstas, L. (2008), Entropy of Continued Fractions (Gauss-Kuzmin Entropy) (PDF)
^ Weisstein, Eric W. "Gauss–Kuzmin Distribution" . MathWorld .
^ Gauss, Johann Carl Friedrich . Werke Sammlung . Vol. 10/1. pp. 552–556.
^ Kuzmin, R. O. (1928). "On a problem of Gauss". Dokl. Akad. Nauk 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 dépendant les quotients complets et incomplets d'une fraction continue" . Bulletin de la Société Mathématique de France . 57 : 178–194. doi :10.24033/bsmf.1150 . 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 (5 ): 507–528. doi :10.4064/aa-24-5-507-528 .
^ Babenko, K. I. (1978). "On a problem of Gauss". Soviet Math. Dokl . 19 : 136–140.
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Gauss–Kuzmin_distribution&oldid=1172033953 "
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