A note on invariance of the Cauchy and related distributions

Journal Pre-proof A note on invariance of the Cauchy and related distributions Wooyoung Chin, Paul Jung, Greg Markowsky

PII: DOI: Reference:

S0167-7152(19)30314-1 https://doi.org/10.1016/j.spl.2019.108668 STAPRO 108668

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Statistics and Probability Letters

Received date : 30 September 2019 Revised date : 21 October 2019 Accepted date : 21 October 2019 Please cite this article as: W. Chin, P. Jung and G. Markowsky, A note on invariance of the Cauchy and related distributions. Statistics and Probability Letters (2019), doi: https://doi.org/10.1016/j.spl.2019.108668. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2019 Elsevier B.V. All rights reserved.

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A note on invariance of the Cauchy and related distributionsI Wooyoung China,∗, Paul Jungb , Greg Markowskyc a KAIST,

291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea c Monash University, Wellington Rd, Clayton VIC 3800, Australia

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b KAIST,

Abstract

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It is known that if f is an analytic self map of the complex upper half-plane which also maps R ∪ {∞} to itself, and f (i) = i, then f preserves the Cauchy distribution. This note concerns three results related to the above fact. Keywords: Cauchy distribution, hyperbolic secant distribution, Boole transformation, Newton’s method

Contents

1 Optional stopping and exit distributions of planar Brownian motion 2 4

3 Invariance of the hyperbolic secant distribution

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2 Trying to find an imaginary root using Newton’s method

In [PW67], it was noticed that if C has a standard Cauchy distribution and the finite or infinite sum

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f (z) = az −

an bn b X bn + − 2 z an − z an + 1 n

(1)

is a meromorphic function with all real constants such that (i) an 6= 0,

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I W. Chin and P. Jung are supported in part by (South Korean) National Research Foundation grant N01170220. This project began while G. Markowsky was visiting KAIST, and he would like to thank the mathematics department there for their kind hospitality. We also thank Davar Khoshnevisan and Edson de Faria for helpful conversations. ∗ Corresponding author Email addresses: [email protected] (Wooyoung Chin), [email protected] (Paul Jung), [email protected] (Greg Markowsky)

Preprint submitted to Elsevier

October 14, 2019

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(ii) if any of the constants a, b, bn are nonzero, then they all have the same sign, and P bn (iii) a2 < ∞ with |an | → ∞,

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n

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P then (a + b + n a2bn+1 )−1 f (C) also has a standard Cauchy distribution. The n proof was a direct calculation of the characteristic function of f (C) using a contour integral. Later, [Let77] gave a full characterization of measurable functions mapping R to R which preserve the set of all Cauchy distributions.

Theorem 1 (Letac). Suppose f : R → R is measurable. Then f preserves the set of Cauchy distributions if and only if there is ε = ±1, α ∈ R, k ≥ 0 and a bounded singular positive measure µ on R such that for Lebesgue almost every x ∈ R, Z (x + iy)t + 1 εf = kx + α − lim µ(dt). y→0 R (x + iy) − t

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At the heart of these results is the following idea: if f is an analytic self map of the upper half of the complex plane, which also maps the extended real line to itself, then f , appropriately centered and scaled, should preserve the standard Cauchy distribution due to the fact that the Cauchy density is the Poisson kernel in the upper half-plane. This connection can be realized via the conformal invariance of Brownian motion and the fact that the exit distribution of Brownian motion from the upper half-plane is Cauchy (this was pointed out in [Let77]). This note is based on our investigation into these results, and contains three sections which are related, but relatively independent of each other. In the first, we use the optional stopping theorem to calculate exit distributions of planar Brownian motion. In the second, we show how a particular invariant map of the standard Cauchy distribution, known as the Boole transformation,1 and the ergodic theorem can be combined to give a quick proof of an interesting fact addressed in [Hor] and pointed out to us by D. Khoshnevisan: when one tries in vain to use Newton’s method to find real roots of x2 + 1, the empirical distribution of the iterates converges to a Cauchy distribution. In the third, we show how recent ideas from [Gro19] can be used together with conformal invariance in order to find invariant maps for more general distributions. We then use this technique to provide an example of a map which leaves the hyperbolic secant distribution invariant.

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1. Optional stopping and exit distributions of planar Brownian motion In this section, we use the optional stopping theorem in order to calculate various exit distributions of planar Brownian motion via their characteristic or 1 The

name of this transformation is taken from [AW73].

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moment generating functions. The results in this section are all well-known and have been previously proved by other methods; however, our method produces short and intuitive proofs, and we hope that the method may be found to be useful in other settings as well. To the best of our knowledge, this technique for proving our results is not present in the literature, even though it is a classical technique in the context of one dimensional Brownian motion.

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Proposition 2 (Cauchy law for Brownian exit of the half-plane). Let (Bt )t≥0 be a complex-valued Brownian motion starting at i. If T is the time when (Bt )t≥0 exits the upper half plane, then the distribution of BT is the standard Cauchy distribution.

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Proof. We use the optional stopping theorem to compute the characteristic function of BT . Let θ ≥ 0 be given. Since z 7→ eiθz is holomorphic, Mt = (eiθBt )t≥0 is a (complex-valued) martingale, and it is bounded on the upper half-plane since there |Mt | = e−θ Im(Bt ) ≤ 1. Thus, by the optional stopping theorem, E[eiθBT ] = E[eiθB0 ] = E[eiθi ] = e−θ . This argument fails when θ < 0, because in that case Mt is unbounded, however symmetry allows us to conclude easily that E[eiθBT ] = e−|θ| for all real θ, which is the characteristic function of the standard Cauchy distribution.

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Remark 3. One can obtain the result regarding (1) in the introduction as follows. Suppose B0 = i and T is the first time (Bt )t hits R. L´evy’s theorem on the conformal invariance of Brownian motion says that if f is analytic, then (f (Bt ))t is a time-changed Brownian motion (see [Dur84] or [Bas95]). Note that the f described in (1) is indeed analytic on the upper half-plane. One can also check that f maps the complex upper half-plane to itself and maps R ∪ {∞} to itself, thus f (BT ) is precisely the place where the time-changed Brownian motion first hits R. Therefore, if f (B0 ) = f (i) = i, then f (BT ) has a standard Cauchy distribution. This final fact is gauranteed by the normalizing constant P (a + b + n a2bn+1 )−1 in the statement of (1).

Proposition 4 (Sech law for Brownian exit of the strip). Let (Bt )t≥0 be a complex-valued Brownian motion starting at 0. If T is the time when (Bt )t≥0 exits the infinite strip {−1 < Re(z) < 1}, then the distribution of Im(BT ) has a hyperbolic secant law, characterized by the density sech( π2 y) dy. 2

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Proof. We again use the optional stopping theorem to compute the characteristic function of BT . Let θ ≥ 0 be given. Since z 7→ eθz is holomorphic, Mt = (eθBt )t≥0 is a martingale, and it is bounded on {−1 < Re(z) < 1} since |Mt | = eθ Re(Bt ) ≤ e|θ| . Thus, by the optional stopping theorem, 1 = E[eθB0 ] = E[eθBT ] = E[eθ Re(BT ) eiθ Im(BT ) ]. 3

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θ

−θ

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= cosh θ, and furthermore the random By symmetry, E[eθ Re(BT ) ] = e +e 2 variables Re(BT ) and Im(BT ) are independent so that the final expectation factors. From this we obtain E[eiθ Im(BT ) ] = sech(θ), and inverting the Fourier transform gives the result.

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We note that both of the preceding results can be obtained using the conformal invariance of Brownian motion and harmonic measure (see [Mar18]), and Theorem 2 can also be deduced by a direct calculation, using the Poisson kernel, or by properties of stable distributions; see for instance [Dur84, Sec. 1.9] or [Fel08, Ch. VI.2]. To conclude this section, we give a new proof of a result from [BFY07], which was proved there by a different argument which also involved planar Brownian motion. Proposition 5. If C is a standard Cauchy random variable, then 2

E[eiλ π log |C| ] = sech(λ). 2

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Proof. The function z 7→ eiλ π log(z) is holomorphic and bounded on {Re(z) > 0}, where log(z) = log |z| + i Arg(z) and Arg is the principal branch of the argument function, taking values in (−π, π) on {Re(z) > 0}. Thus, if (Bt )t≥0 is a complex-valued Brownian motion starting at 1 and T is the time at which 2 Bt exits {Re(z) > 0}, then eiλ π log(Bt ) is a bounded martingale for 0 ≤ t ≤ T and the optional stopping theorem gives 2

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2

1 = E[eiλ π log(B0 ) ] = E[eiλ π log(BT ) ] = E[eiλ π (log |BT |+i Arg(BT )) ].

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Now, by symmetry, the random variables log |BT | and Arg(BT ) are independent, and BT is standard Cauchy (Theorem 2), while Arg(BT ) is uniform on {− π2 , π2 }, so we obtain

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and the result follows.

 eλ + e−λ  2 1 = E[eiλ π log |C| ] , 2

2. Trying to find an imaginary root using Newton’s method

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In this subsection we show that when one tries to use Newton’s method to find real roots of x2 + 1, one will asymptotically end up with a Cauchy distribution. Recall that Newton’s method finds real roots of a differentiable f finds by using a sequence of approximations which are trying to successively get closer to a root. Starting with an arbitrary initial guess x0 , Newton’s method computes the next approximation using xn+1 = x −

f (xn ) , f 0 (xn )

while hoping that we would never have f 0 (xn ) = 0. 4

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For x2 + 1, Newton’s method computes the approximations using xn+1 = ϕ(xn ),

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where ϕ : R \ {0} → R is given by the Boole transformation   x2 + 1 1 1 ϕ(x) = x − 2 = x− . (x + 1)0 2 x

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If we let

N := {x ∈ R : ϕn (x) = ∞ for some n ∈ N},

then N is countable, and ϕ maps the extended real line into itself. Restrict the domain and codomain of ϕ to R \ N .

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Proposition 6 (Newton’s method applied to x2 + 1). For Lebesgue-almost every x and for any measurable function f : R → R such that f (x)/(1 + x2 ) is integrable, we have Z n−1 f (t) 1X f (ϕk (x)) → dt. n π(1 + t2 ) k=0

In particular, for Lebesgue-almost every x, we have

n−1 1 1X δϕk (x) ⇒ dt, n π(1 + t2 ) k=0

where ⇒ denotes weak convergence.

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Remark 7. Newton’s method has several variants such as a trapezoidal-Newton’s method or Simpson-Newton’s method, to which the above proposition also applies. One simply replaces ϕ by x3 − 3x , 3x2 − 1

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ψ(x) :=

√ √ x ∈ R \ {1/ 3, −1/ 3}.

All the arguments in the proof of the proposition can then be extended to the map ψ.

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In order to prove the proposition, we make use of Birkhoff’s ergodic theorem and the ergodicity of ϕ. The ergodicity of ϕ is well-known, but for completeness we include an argument here. We first consider a related map on S 1 , namely z 7→ −z 2 , and show that this map on S 1 is ergodic.

Lemma 8 (Ergodicity of the doubling map). Assume that S 1 is equipped with the uniform distribution. The map ρ : S 1 → S 1 given by ρ(z) := −z 2 is ergodic.

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n∈Z

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Proof. Clearly the map ρ is measure preserving with respect to the uniform distribution. Now assume that A ⊂ S 1 is a measurable set with 1ρ−1 (A) = 1A a.s. Let P 1A (z) = n∈Z cn z n where the infinite summation is in the L2 sense. It is not difficult to see that X X cn (−z 2 )n = (−1)n cn z 2n , 1ρ−1 (A) = 1A (−z 2 ) =

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again in the L2 sense. Since 1ρ−1 (A) = 1A a.s., the coefficients of the two series must match and so c2n = (−1)n cn for each n ∈ Z. Since |cn | = |c2n | = |c4n | = · · · and limn→∞ cn = limn→−∞ cn = 0, we have cn = 0 for all n 6= 0. Thus 1A must be constant a.s., and this shows that ρ is ergodic.

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Using Lemma 8, we now prove that ϕ is ergodic. Let C+ be the upper half plane, and D the unit open disk. Let F : C+ ∪R → D\{−1} and G : D\{−1} → C+ ∪ R be defined by i−x F (x) := i+x and 1−z G(x) := i . 1+z Then F and G are inverses to each other.

is ergodic.

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Lemma 9 (Ergodicity of the Boole transformation). With respect to the standard Cauchy distribution, the Boole transformation,   1 1 ϕ(x) = x− , x ∈ R \ N, 2 x

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Proof. Note that (1) implies that ϕ is measure preserving with respect to the Cauchy distribution. Now, let D := F (R \ N ). Restrict the domains and codomains of F and G by F : R \ N → D and G : D → R \ N . For each z ∈ D, we have      1+z 1 1−z i (1 − z)2 + (1 + z)2 i +i =F · (F ◦ ϕ ◦ G)(z) = F 2 1+z 1−z 2 1 − z2 2   1+z 1 − 1−z2 1 + z2 −2z 2 = = −z 2 . =F i 2 = 2 1+z 1−z 2 1 + 1−z2

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Assume that A ⊂ R \ N is a measurable set with 1ϕ−1 (A) = 1A a.s. Since F and G map measure-zero sets to measure-zero sets, we have 1F (A) = 1F (ϕ−1 (A)) = 1(F ◦ϕ◦G)−1 (F (A))

a.s.

By Lemma 8, the measure of F (A) should be either 0 or 1, and therefore the measure of A is either 0 or 1. This shows that ϕ is ergodic. 6

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Proof of Proposition 6. Note that ϕ(i) = i. Also, if z ∈ C+ , then 1/z ∈ −C+ , and thus −1/z ∈ C+ . Therefore,   1 1 ϕ(z) = z− ∈ C+ 2 z

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and ϕ maps C+ into itself. Thus, by (1) (see also Remark 3) we have that ϕ preserves the standard Cauchy distribution. With respect to the standard Cauchy distribution, the map ϕ is measurepreserving and ergodic, thus the desired conclusion follows from the ergodic theorem (see for instance, [Dur10, Theorem 7.2.1]). 3. Invariance of the hyperbolic secant distribution

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Invariant maps of the Cauchy distribution can also be used to find functions that preserve distributions other than the Cauchy, for instance [PY04] applies this theorem to find a family of rational maps under which the arc-sine law is invariant. We can adapt this technique to more general distributions, as follows. Suppose that W is a simply connected domain in C which is symmetric about R; that is, z ∈ W if, and only if, z¯ ∈ W . Suppose further that the boundary components of W in the upper and lower half-planes can each be viewed as the graph of a continuous function with dependent variable y and independent variable x; equivalently, any vertical line z(t) = x + it intersects ∂W at exactly two points, which are necessarily conjugates of each other. Let τ = inf{t ≥ 0 : Bt ∈ ∂W },

(2)

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and let ∆a , for a ∈ R, denote the distribution of Re(Bτ ) under the condition B0 = a a.s. It was shown in a recent elegant paper [Gro19], that any distribution satisfying certain moment conditions can be realized by a simply connected domain in this manner. For x ∈ R, let π(x) be the unique point z in the upper half-plane with z ∈ ∂W and Re(z) = x. We have the following simple proposition. Proposition 10. If X ∼ ∆a for some a ∈ R and f is a conformal automorphism of W which maps R into itself, then Re(f (π(X))) ∼ ∆f (a) .

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Proof. The conditions on the domain imply that W is conformally equivalent to a Jordan domain by a M¨obius transformation. By Carath´eodory’s theorem (see [Gol69]), f extends to a continuous bijection from the closure of W (in the Riemann sphere) to itself, thus f is defined on ∂W . The fact that f (W ∩R) ⊆ R z ) agree on R, and therefore by implies that the analytic functions f (z) and f (¯ the uniqueness principle for analytic functions (see [Rud06]) we have f (z) = f (¯ z) and also f (¯ z ) = f (z), for all z ∈ (W ∪ ∂W ). Now for τ as in (2), X ∼ ∆a ∼ Re(Bτ ), 7

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However, f (¯ z ) = f (z) implies

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and by L´evy’s theorem on the conformal invariance of Brownian motion, we have ∆f (a) ∼ Re(f (Bτ )) = Re(f (Re(Bτ ) + i Im(Bτ ))). Re(f (Re(Bτ ) + i Im(Bτ ))) = Re(f (Re(Bτ ) + i| Im(Bτ )|)) = Re(f (π(Re(Bτ )))).

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The result follows upon replacing Re(Bτ ) with X.

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Naturally, the difficulties in applying this result are (a) identifying the distribution of Re(Bτ ), and (b) finding a conformal automorphism of the required type. We can, however, give an example as follows. Let W = {− π2 < Im(z) < π2 }. The function ψ(z) = iez maps W
conformally onto the upper half-plane, so using the function ϕ(z) = 12 z − z1 from Section 2 we see that f = ψ −1 ◦ ϕ ◦ ψ is a conformal self-map of W of the type required for Proposition 10. Calculating, we have   z   e + e−z i z 1 , f (z) = log − (ie − z ) = log 2 ie 2 where log denotes the principal branch of the logarithm. If g(x) = Re(f (π(x))) then x x+iπ/2 e − e−x e + e−x−iπ/2 = log | sinh x|. g(x) = log = log 2 2

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Using the exit distribution of (Bt )t from W derived in Proposition 4, we obtain the following result. 1 2

sech( π2 x) dx

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Corollary 11 (An invariant map for the sech law). The distribution is invariant under the transformation  π  2 g(x) := log sinh x . π 2

[AW73] R. Adler and B. Weiss. The ergodic infinite measure preserving transformation of boole. Israel Journal of Mathematics, 16(3):263–278, 1973. [Bas95] R. Bass. Probabilistic techniques in analysis. Springer Science & Business Media, 1995.

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[BFY07] P. Bourgade, T. Fujita, and M. Yor. Euler’s formulae for ζ(2n) and products of Cauchy variables. Electronic Communications in Probability, 12:73–80, 2007. [Dur84] R. Durrett. Brownian motion and martingales in analysis. Belmont (Calif.): Wadsworth advanced books and software, 1984. 8

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[Dur10] R. Durrett. Probability: theory and examples, volume 31 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, fourth edition, 2010.

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[Fel08] Willliam Feller. An introduction to probability theory and its applications, volume 2. John Wiley & Sons, 2008.

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[Gol69] G. Goluzin. Geometric theory of functions of a complex variable, volume 26. American Mathematical Society, 1969. [Gro19] R. Gross. A conformal Skorokhod embedding. arXiv:1905.00852, 2019. [Hor] Zsuzsanna Horv´ath. Random number generators using dynamical systems. https://studylib.net/doc/12049153. NSF REU Project. [Let77] G. Letac. Which functions preserve Cauchy laws? Proceedings of the American Mathematical Society, 67(2):277–286, 1977.

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[Mar18] G. Markowsky. On the distribution of planar Brownian motion at stopping times. Ann. Acad. Sci. Fenn. Math., 43:597–616, 2018. [PW67] E. Pitman and E. Williams. Cauchy-distributed functions of cauchy variates. The Annals of Mathematical Statistics, pages 916–918, 1967. [PY04] J. Pitman and M. Yor. Some properties of the arc-sine law related to its invariance under a family of rational maps. In A festschrift for Herman Rubin, volume 45 of IMS Lecture Notes Monogr. Ser., pages 126–137. Inst. Math. Statist., Beachwood, OH, 2004.

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[Rud06] W. Rudin. Real and complex analysis. Tata McGraw-Hill, 2006.

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