A note on invariance of the Cauchy and related distributions

Greg Markowsky; Paul Jung; Wooyoung Chin

REVIEW 3 major objections 6 minor 14 references

A note on invariance of the Cauchy and related distributions

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A log-sinh map leaves the hyperbolic secant law exactly invariant.

desk verdict Corollary 9 is true, but the proof as written doesn't establish it — the example doesn't satisfy the transfer principle's hypotheses, and the strip is scaled wrong. read the letter →

arxiv 1908.04006 v2 pith:L5HAVWF7 submitted 2019-08-12 math.PR math.CV

Wooyoung Chin , Paul Jung , Greg Markowsky This is my paper · ORCID

classification math.PRmath.CV MSC 60J6560E1037A0530C35

keywords CauchydistributionhyperbolicsecantBooletransformationNewton'smethodBrownianmotionconformalinvarianceoptionalstoppinginvariantdistributions

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This note establishes a transfer principle for invariant distributions: if a symmetric domain has a known Brownian-exit law and a conformal automorphism fixing the real axis, that automorphism produces an explicit transformation preserving the law. Applying the principle to a strip yields a new exact invariant law: the density $\frac{1}{2}\operatorname{sech}(\frac{\pi}{2}x)\,dx$ is invariant under $g(x)=\frac{2}{\pi}\log|\sinh(\frac{\pi}{2}x)|$. The same methods give Brownian proofs of the Cauchy exit law, a proof that Newton's method on $x^2+1$ asymptotically yields Cauchy-distributed iterates, and a Brownian proof of a known identity for the log-modulus of a Cauchy variable. A sympathetic reader would care because exactly invariant laws under nonlinear maps are rare, and the transfer principle suggests a general way to build them.

What carries the argument

The load-bearing object is Proposition 8, a transfer principle for exit laws: for a symmetric simply connected domain $W$, the distribution $\Delta_a$ of the real part of Brownian motion's exit point is pushed forward by the real part of any conformal automorphism $f$ fixing the real axis, with $f$ applied to the starting point. The proof uses conformal invariance of Brownian motion, Schwarz reflection ($f(\bar z)=\overline{f(z)}$), and Carathéodory's theorem to extend $f$ to the boundary. The other engine is optional stopping for martingales $e^{\theta B_t}$, which computes the Cauchy exit law from a half-plane, the hyperbolic secant law from a strip, and the identity $\mathbb{E}[e^{i\lambda \frac{2}{\pi}\ln|C|}]= \operatorname{sech}\lambda$ for standard Cauchy $C$.

What would settle it

Simulate planar Brownian motion started at $0$ and stopped on leaving the strip $\{-1<\Re z<1\}$, then estimate $\mathbb{E}[e^{i\theta \Im B_T}]$ and the independence of $\Re B_T$ and $\Im B_T$. If the characteristic function is not $\operatorname{sech}\theta$, or if the product of $\mathbb{E}[e^{\theta\Re B_T}]$ and $\mathbb{E}[e^{i\theta\Im B_T}]$ does not equal $1$, then Proposition 3 fails. Directly, sample $X$ from $\frac{1}{2}\operatorname{sech}(\frac{\pi}{2}x)\,dx$, apply $g(x)=\frac{2}{\pi}\log|\sinh(\frac{\pi}{2}x)|$, and compare the empirical law with the same density.

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Extended reading notes

Core claim

The central new assertion is Corollary 9: the distribution $\frac{1}{2}\operatorname{sech}(\frac{\pi}{2}x)\,dx$ is invariant under $g(x)=\frac{2}{\pi}\log|\sinh(\frac{\pi}{2}x)|$. This follows from Proposition 8, which states that for a simply connected domain $W$ symmetric about $\mathbb{R}$, if $X$ has the law of $\operatorname{Re}(B_\tau)$ for Brownian motion starting at $a$ and $f$ is a conformal automorphism of $W$ sending $\mathbb{R}$ into itself, then $\operatorname{Re}(f(\pi(X)))$ has the corresponding law for $f(a)$; here $\pi(x)$ is the upper boundary point above $x$. With $W$ a strip, the exit law is the hyperbolic secant law, and the automorphism built from $z\mapsto \frac{1}{2}(z-z^{-1})$ conjugated by $z\mapsto i e^z$ gives the stated invariant map.

Load-bearing premise

The derivation of the hyperbolic secant exit law assumes, without proof, that the real and imaginary parts of Brownian motion's exit point from a strip are independent and that $\mathbb{E}[e^{\theta \operatorname{Re}(B_T)}]=\cosh\theta$; if this factorization fails, Proposition 3 and hence the new invariant map in Corollary 9 are not established, and the example's stated width must also be rescaled to match the corollary.

Editorial extensions

If this is right

The hyperbolic secant law now has a new, explicitly written invariant map, so iterates of $g$ preserve the law exactly rather than approximately.
Any other symmetric domain with a computable Brownian-exit law and a conformal automorphism fixing the real axis yields an invariant map by the same Proposition 8, enlarging the known catalogue of invariant distributions.
Newton's method applied to $x^2+1$ has Cauchy-distributed empirical iterates for almost every starting point, with the Boole transformation ergodic under the Cauchy law.
The optional-stopping method reproduces the Cauchy and hyperbolic secant exit laws and the $\log|C|$ identity with short martingale proofs, suggesting the technique may work for other holomorphic exponentials.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The example's horizontal strip of width $\pi$ is connected to the corollary's density by the change of variable $x\mapsto \frac{\pi}{2}x$; the unscaled invariant statement is that $\frac{1}{\pi}\operatorname{sech} x$ is preserved by $\ln|\sinh x|$, and Corollary 9 is its rescaling, though the paper does not spell out this step.
If the unproved independence assertion in Proposition 3 is supplied, the same transfer principle likely extends to other one-parameter families of exit distributions, giving invariant laws for other special functions.
The map $g$ is an explicit nonlinear dynamical system preserving a probability measure; the paper does not analyze its mixing or periodic points, but the invariance alone invites a dynamical study.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: Sections 1 and 2 are clean and worth reading; Corollary 9 is true, but Section 3's proof as printed has a real gap. I checked the invariant map directly by change of variables, so the result is right, but it does not follow from the stated Proposition 8.

The transfer principle (Prop 8) is a nice observation and is likely correct for genuine automorphisms. The optional stopping proofs in Section 1 are neat, and the ergodic proof for the Boole transformation in Section 2 is elegant. The paper is honest about re-proving known results, and the new sech-law map is a legitimate extension.

Now the soft spots, concentrated in Section 3. First, W is defined as {−π/2 < Im z < π/2}, but the exit distribution of Re(B_τ) from that strip is (1/π)sech(x), not the (1/2)sech(πx/2) used in Corollary 9. That scaling mismatch is concrete and easy to fix. Second, the map f(z)=Log cosh z is not a conformal automorphism of W — f'(0)=0 — so Proposition 8 does not apply. The composite f = ψ^{-1}∘ϕ∘ψ maps W into itself, but it is not one-to-one. The corollary therefore does not follow from the stated proposition; it is saved only by a direct calculation that the text does not provide.

There are smaller gaps too: the independence of Re(B_T) and Im(B_T) in Prop 3, and of ln|B_T| and Arg(B_T) in Prop 4, is asserted without proof. These are standard facts and can be supplied, but they are load-bearing.

All of this is fixable. The authors should either replace the example with one where f is a genuine automorphism, or prove a more general version of Prop 8 for non-injective conformal maps with the needed boundary behavior, or simply include the direct change-of-variables proof. As is, I would not accept without revision, but I would send it to a referee rather than desk reject: the central theorem is true and the earlier sections are solid.

Referee Report

3 major / 6 minor

Summary. The paper studies invariance of probability distributions under analytic maps via planar Brownian motion. Section 1 uses optional stopping to compute exit distributions of Brownian motion, giving short proofs of the Cauchy law for half-plane exit, a hyperbolic secant law for strip exit, and a characteristic-function identity for log|Cauchy|. Section 2 proves that Newton's method applied to x^2+1 yields, for almost every starting point, empirical measures converging weakly to the standard Cauchy distribution, using the ergodicity of the Boole transformation. Section 3 proposes a general transfer principle (Prop. 8) for symmetric simply connected domains and applies it to find a nonlinear map g(x)=(2/\pi)\log|\sinh(\pi x/2)| that supposedly preserves the distribution (1/2)\sech(\pi x/2)dx. The central new assertion is Corollary 9.

Significance. The intended result, if properly proved, is a nice addition to the short list of exactly invariant distributions for explicit nonlinear maps, and the method connecting conformal invariance, Brownian exit laws, and distribution preservation is attractive. Section 2's ergodic-theoretic proof of the Cauchy empirical law for Newton's method is clean and self-contained. The paper's claims appear to be true: Corollary 9 can be verified directly by a change of variables, and the optional stopping arguments are essentially correct. However, the proof of the main new result as printed does not follow from the stated Proposition 8, because the map used is not a conformal automorphism, and there is a scaling mismatch between the domain and the corollary's constants.

major comments (3)

[Section 3 (construction of f)] The map f(z)=Log cosh z is not a conformal automorphism of W={-\pi/2<Im z<\pi/2}. Its derivative is f'(z)=tanh z, so f'(0)=0 and f is not locally injective at 0; moreover f(-z)=f(z), so f is not injective globally. The composition f=\psi^{-1}\circ \phi\circ\psi inherits the failure of univalence from \phi, whose derivative vanishes at i. Therefore Proposition 8 cannot be applied to this f, and Corollary 9 does not follow from the printed argument. Since a direct change-of-variables calculation does establish the invariant law, the gap is fixable, but the manuscript must either supply that calculation or prove a suitable extension of Proposition 8 to non-injective analytic self-maps with the required boundary behavior.
[Section 3 (scaling)] There is a scaling mismatch between the domain W and the constants in Corollary 9. For W={-\pi/2<Im z<\pi/2}, the upper boundary point is \pi(x)=x+i\pi/2, and the calculation in the text gives g(x)=Re f(\pi(x))=Log|\sinh x|, not g(x)=(2/\pi)Log|\sinh(\pi x/2)|. Moreover, after rotating and scaling the exit law from Proposition 3, the distribution of Re(B_\tau) for this W is (1/\pi)\sech x dx, not (1/2)\sech(\pi x/2)dx. The corollary corresponds instead to the strip {|Im z|<1} (or to a scaled version of the calculation), and this rescaling must be stated explicitly and carried through consistently.
[Section 1, Proof of Proposition 3] The proof asserts without justification that E[e^{\theta Re(B_T)}]=cosh\theta and that Re(B_T) and Im(B_T) are independent. Both facts are true here: Re(B_T) takes the values \pm 1 with equal probability, and the sign is independent of |Im(B_T)| by reflection symmetry across the imaginary axis. But no argument is supplied, and this factorization is the only derivation of the sech law used in Section 3. The proof should include the symmetry argument explicitly.

minor comments (6)

[Section 1, Proposition 4 proof] The phrase 'Arg(B_T) is uniform on {-\pi/2,\pi/2}' should say that Arg(B_T) takes the two values \pm\pi/2 with probability 1/2 each; as written it could suggest a continuous uniform distribution on the interval.
[Section 1, Proposition 4 proof] The independence of ln|B_T| and Arg(B_T) is asserted rather than proved; a one-sentence reflection-symmetry argument across the imaginary axis would suffice.
[Section 1, Proposition 4 proof] The reference to 'Theorem 1' in the proof of Proposition 4 should be 'Proposition 1'.
[Section 3, Corollary 9] The notation \sinh(\pi/2 x) is ambiguous; \sinh(\pi x/2) would be clearer.
[References] The reference [Hor] lacks a year and venue information; if it is an online source, a URL and access date would be helpful.
[Section 2, Lemma 7] The proof implicitly uses that F sends the standard Cauchy distribution to normalized Lebesgue measure on the circle; stating this fact explicitly would make the ergodicity transfer fully transparent.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the paper's claims are derived from external standard theorems and contain no fitted inputs, so score 0.

full rationale

The derivation chain is not circular in any of the senses enumerated. Proposition 1 is a direct optional-stopping calculation of the Cauchy exit law from the half-plane, using only the bounded martingale e^{iθB_t} and the external standard theorem of optional stopping. Proposition 3 uses the same technique for the strip, with the symmetry and independence assertions appearing as independent (though not fully proved) claims, not as consequences of the target invariant law. Proposition 4 is a separate optional-stopping calculation. Proposition 8 is a transfer principle proved from Lévy's conformal invariance of Brownian motion and Carathéodory's extension theorem; it does not assume the conclusion and contains no parameter to fit. The ergodicity results in Lemmas 6 and 7 are proved from Fourier series and the external Birkhoff ergodic theorem; the Cauchy-preserving property of the Boole map is cited from Pitman-Williams [PW67], an external source. Corollary 9 applies these ingredients to f(z)=Log cosh z, with no fitted parameter and no reliance on the desired invariant law in constructing the map. The only self-citation is [Mar18], and it appears as an aside offering an alternative route to previously known exit distributions; it is not load-bearing for any central claim. The scaling mismatch between the width-π domain and the width-2 constants in Corollary 9, and the separate concern that φ (hence f) is not a conformal automorphism because φ'(i)=0, are correctness or presentation gaps rather than circularity, since neither makes the conclusion definitionally equal to an input or to a fitted value. The manuscript therefore receives circularity score 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters or invented entities. The paper uses standard theorems plus one unproved symmetry assertion; the example is an explicit computation. The 2/π and π/2 constants come from the geometry of the strip, not from fitting.

assumptions (6)

standard math Optional stopping theorem applies to the bounded martingales used in Props 1,3,4 at the exit time T.
The stopped martingales are bounded on the domains (e.g., |e^{iθB_t}|≤1 in the half-plane), so E[M_T]=E[M_0] is valid.
standard math Lévy's conformal invariance of planar Brownian motion: f(B_t) is a time-changed Brownian motion for analytic f.
Used in Remark 2 and Prop 8 to transfer exit laws under conformal maps.
standard math Carathéodory's theorem that a conformal map between Jordan domains extends to a homeomorphism of closures.
Used in Prop 8 to define f on the boundary and use π(x).
standard math The analytic uniqueness principle: f(z)=overline(f(bar z)) follows from equality on the real axis.
Used in Prop 8 to replace Im(B_T) by its absolute value in the real part.
standard math Birkhoff's ergodic theorem applies to the Boole transformation with the Cauchy measure.
Used in Prop 5 to prove the empirical average of iterates converges to the Cauchy expectation.
domain assumption By symmetry, for the strip exit problem, Re(B_T) and Im(B_T) are independent and E[e^{θ Re(B_T)}]=cosh θ; for the half-plane exit, ln|B_T| and Arg(B_T) are independent.
Stated without proof in Props 3 and 4; load-bearing for the sech law and the new corollary.

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Cite this review

Pith. "Pith review of A note on invariance of the Cauchy and related distributions." pith.science (2026). https://pith.science/paper/L5HAVWF7

@misc{pith2026190804006,
  author       = {Pith},
  title        = {Pith review of: A note on invariance of the Cauchy and related distributions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L5HAVWF7}},
  note         = {Machine review of arXiv:1908.04006}
}

abstract

It is known that if $f$ is an analytic self map of the complex upper half-plane which also maps $\mathbb{R}\cup\{\infty\}$ to itself, and $f(i)=i$, then $f$ preserves the Cauchy distribution. This note concerns three results related to the above fact.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Adler and B

R. Adler and B. Weiss. The ergodic infinite measure preserving transformation of boole. Israel Journal of Mathematics , 16(3):263--278, 1973

work page 1973

[2] [2]

R. Bass. Probabilistic techniques in analysis . Springer Science & Business Media, 1995

work page 1995

[3] [3]

Bourgade, T

P. Bourgade, T. Fujita, and M. Yor. Euler's formulae for (2n) and products of C auchy variables. Electronic Communications in Probability , 12:73--80, 2007

work page 2007

[4] [4]

R. Durrett. Brownian motion and martingales in analysis . Belmont (Calif.): Wadsworth advanced books and software, 1984

work page 1984

[5] [5]

R. Durrett. Probability: theory and examples , volume 31 of Cambridge Series in Statistical and Probabilistic Mathematics . Cambridge University Press, Cambridge, fourth edition, 2010

work page 2010

[6] [6]

An introduction to probability theory and its applications , volume 2

Willliam Feller. An introduction to probability theory and its applications , volume 2. John Wiley & Sons, 2008

work page 2008

[7] [7]

G. Goluzin. Geometric theory of functions of a complex variable , volume 26. American Mathematical Society, 1969

work page 1969

[8] [8]

R. Gross. A conformal S korokhod embedding. arXiv:1905.00852 , 2019

work page Pith review arXiv 1905