[Paper Review] Some remarks on invariant maps of the Cauchy distribution

Wooyoung Chin, Paul Jung+1 more|arXiv (Cornell University)|Aug 12, 2019

Stochastic processes and financial applications5 references4 citations

TL;DR

This paper leverages the conformal invariance of planar Brownian motion to identify a class of functions under which the Cauchy distribution remains invariant. It then applies this invariance to explain the emergence of the Cauchy distribution when Newton's method is applied to $ f(x) = x^2 + 1 $, revealing a deep connection between stochastic processes and nonlinear iteration dynamics.

ABSTRACT

We use the conformal invariance of planar Brownian motion to present a class of functions under which the Cauchy distribution is invariant. We then use this invariance to explain why one obtains the Cauchy distribution when trying to use Newton's method on the function $f(x)=x^2+1$.

Motivation & Objective

• To identify a class of functions under which the Cauchy distribution is invariant, using geometric and stochastic properties.
• To explain the persistent appearance of the Cauchy distribution in numerical analysis, particularly in Newton's method applied to $ f(x) = x^2 + 1 $.
• To establish a link between conformal invariance in stochastic processes and the invariant behavior of certain probability distributions.
• To provide a probabilistic explanation for the heavy-tailed convergence behavior observed in iterative methods involving rational functions.

Proposed method

• Utilizes the conformal invariance property of planar Brownian motion to derive transformations that preserve the Cauchy distribution.
• Identifies a specific class of Möbius transformations (real linear fractional transformations) that act as invariance maps for the standard Cauchy distribution.
• Analyzes the dynamics of Newton's method on $ f(x) = x^2 + 1 $, showing that the iteration map corresponds to a composition of such invariant transformations.
• Demonstrates that the invariant measure of the Newton iteration map coincides with the Cauchy distribution due to the underlying conformal symmetry.
• Applies results from stochastic processes and complex analysis to connect the geometric invariance of Brownian motion with the statistical invariance of the Cauchy law.
• Uses the fact that the Cauchy distribution is stable under rational transformations of a specific type, linking it to the iteration function of Newton's method.

Experimental results

Research questions

• RQ1 Which classes of functions preserve the Cauchy distribution under transformation, and what geometric or stochastic properties ensure this invariance?
• RQ2 Why does Newton's method applied to $ f(x) = x^2 + 1 $ lead to iterates that are distributed as Cauchy, despite the function having no real roots?
• RQ3 How does the conformal invariance of planar Brownian motion relate to the invariance of the Cauchy distribution under certain rational maps?
• RQ4 Can the heavy-tailed behavior observed in Newton's method for $ f(x) = x^2 + 1 $ be explained through the invariance of the Cauchy distribution under specific dynamical systems?
• RQ5 What is the role of Möbius transformations in preserving the Cauchy distribution, and how do they emerge naturally in iterative numerical schemes?

Key findings

• The Cauchy distribution is invariant under a specific class of real linear fractional transformations, which are conformal maps arising from planar Brownian motion invariance.
• Newton's method applied to $ f(x) = x^2 + 1 $ generates a dynamical system whose invariant measure is the standard Cauchy distribution.
• The iteration map $ x_{n+1} = x_n - rac{x_n^2 + 1}{2x_n} $ simplifies to $ x_{n+1} = rac{x_n^2 - 1}{2x_n} $, a Möbius transformation that preserves the Cauchy distribution.
• The conformal invariance of planar Brownian motion provides a stochastic foundation for the observed invariance of the Cauchy law under such transformations.
• The emergence of the Cauchy distribution in Newton's method is not coincidental but a consequence of the underlying symmetry in the iteration map.
• The paper establishes a direct correspondence between the invariant measures of stochastic processes and the invariant distributions in deterministic nonlinear iterations.