Extreme Value Theory on Generalized Continued Fractions

02 Sep 2022

This post is a wrap-up of the research internship, advised by Prof. S. Lim. We review previous studies that are related to the extreme value problem on generalized continued fractions.

Extreme Value Theory

Extreme value theory is a field of statistics regarding the distribution of the largest value. Given a sequence $\seq{X\subscr{n}}$ of random variables, consider the maximum

\[M\subscr{n} = \max\{X\subscr{1},\dots,X\subscr{n}\}.\]

The extreme value problem asks what the vague limit of $\seq{M\subscr{n}}$ under a suitable normalization is. The easiest case is when $\seq{X\subscr{n}}$ is i.i.d.; the limiting distribution is one of three types by the Fisher-Tippett-Gnedenko theorem. Then, the next case we should study would be the sequences that are stationary and weakly independent. The entries $\seq{a_n}$ of the continued fraction are one of those.

The Classic

Galambos [1] is the one who first showed the following extreme value theorem on continued fractions.

\[\lim_{N\to\infty} \mu\left\{ x\in[0,1] : \max_{1\le n\le N} a_n \le \frac{y}{\log 2}N \right\} = e^{-\frac{1}{y}}\]

As we have seen in an earlier post, the proof of the theorem mostly relies on the weak independency of continued fraction—the $\psi$-mixing property:

For any $A\in\mathcal{M}^{1,t}$ and $B\in\mathcal{M}^{t+n,\infty}$, we have

\[\left\lvert \mu(A\cap B) - \mu(A)\mu(B) \right\rvert \le \psi(n)\mu(A)\mu(B),\]

where $\psi:\N\to\R$ is a function for which $\psi(n)\to0$ as $n\to0$.

Hence, the extreme value problem of a stationary sequence is essentially the problem of proving its $\psi$-mixing property.

A commonly used way [6] to prove the $\psi$-mixing property is to use the Kuzmin theorem. The theorem dates back to the era of Gauss. In 1812, Gauss made a conjecture on the metric theory of continued fractions.

For $x = [a_1, a_2, \dots]$ in $I = [0,1]$, let $x_n = [a\subscr{n+1}, a\subscr{n+2}, \dots]$. Then

\[e_n(s) = m(x_n \in [0,s]) - \mu([0,s])\]

tends to zero as $n\to\infty$, where $m$ is the Lebesgue measure and $\mu$ is the invariant measure of the Gauss map given by

\[\mu(A) = \frac{1}{\log 2} \int_A \frac{1}{1+s} ds.\]

It was solved independently by Kuzmin (in 1928) and Lévy (in 1929). Kuzmin gave a subexponential decay

\[e_n(s) = O(\theta^{\sqrt{n}}),\]

and Lévy showed the exponential rate of decay. Since the Kuzmin theorem implies the $\psi$-mixing property of continued fraction, we need to state and prove a Kuzmin-type theorem to solve other extreme value problems.

Into the Complex Field

We have broken down the problem on continued fractions, so it is time to look for more generalized stuff. One way to achieve this is to consider a continued fraction on other fields, for example, a complex field. However, there is no way to canonically generalize the continued fraction into the complex field. The simplest way we can think to define a new Gauss map—defining the complex floor function as

\[\floor{x+iy} := \floor{x} + i \floor{y}\]

—does not result in having some good properties. Hence, there are several different generalizations of continued fractions, which differ in ways of defining the lattice (the floor function) or the inversion (the inverse). One commonly used way is the Hurwitz complex continued fraction. We have the usual inversion, but the floor function is given by

\[\floor{x+iy} := \floor{x}\subscr{near} + i \floor{y}\subscr{near},\]

where $\floor{\,\cdot\,}\subscr{near}$ is the nearest integer function. Then it is known that the entries $\seq{a\subscr{n}}$ generated by the new Gauss map $T$ provides a continued fraction representation of complex numbers. Also, there exists a unique $T$-invariant measure $\nu$ which is absolutely continuous with respect to the Lebesgue measure. Here, we consider the measures on the lattice cell

\[B = \left\{x+iy \in \C : x,y \in \left[-\frac{1}{2},\frac{1}{2}\right)\right\}\]

induced by the floor function.

Nakada [5] firstly stated the $\psi$-mixing property of the Hurwitz complex continued fraction. The proof relied on the Kuzmin-type theorem given by Schweiger, which unfortunately turned out to be a flawed proof. In 2000, Schweiger [7] was able to restore the theorem, yielding a subexponential decay of $e_n(s)$. His theorem deals with a wider class of stochastic processes: fibred system, but its definition is quite lengthy so we will omit it here. Schweiger [8] improved his theorem to exponential decay in 2011. Gathering these results, Kirsebom [4] stated and proved the following extreme value theorem on the Hurwitz complex continued fraction: there exists $C>0$ so that

\[\lim_{N\to\infty} \nu\left\{ x+iy\in B : \max_{1\le n\le N} \abs{a_n} \le Cr\sqrt{N} \right\} = e^{-\frac{1}{r^2}}.\]

References

[1] J. Galambos, The distribution of the largest coefficient in continued fraction expansions, The Quarterly Journal of Mathematics 23 (1972), 147-151.

[2] M. Iosifescu, Doeblin and the metric theory of continued fractions: A functional theoretic solution to Gauss’ 1812 problem, Contemporary Mathematics 149 (1993), 97-110.

[3] A. Khintchine, Kettenbrüche, Teubner Verlagsgesellschaft, 1956.

[4] M. Kirsebom, Extreme Value Theory for Hurwitz Complex Continued Fractions, Entropy 23.7 (2021), 840.

[5] H. Nakada, On the Kuzmin’s theorem for complex continued fractions, Keio Engineering Reports 29.9 (1976), 93-108.

[6] W. Philipp, Some metrical theorems in number theory, Pacific Journal of Mathematics 20.1 (1967), 109-127.

[7] F. Schweiger, Kuzmin’s theorem revisited, Ergodic Theory and Dynamical Systems 20.2 (2000), 557-565.

[8] F. Schweiger, A new proof of Kuzmin’s theorem, Rev. Roum. Math. Pures Appl 56.3 (2011), 229-234.