NOLTA, IEICE Invited Paper
Ergodic transformations on R preserving Cauchy laws Ken Umeno 1 a) 1
a)
Graduate School of Informatics, Kyoto University Yoshidahonmachi, Sakyo-ku, Kyoto City, Japan
[email protected]
Received October 5, 2015; Revised November 4, 2015; Published January 1, 2016 Abstract: We consider a family of ergodic transformations on the real line R preserving Cauchy laws. A dualistic nature between the ergodic transformation and the associated transformation of the scale parameter of a Cauchy law is proven to be hold, which provides a systematic view of explicit mixing property with the ergodic transformation having the Cauchy law as the limiting distribution. Key Words: ergodic transformations, Cauchy law, mixing property, stable law, addition theorem
1. Introduction Ergodic transformations on the real line R having the power law are the topics of great interest. They explain the origin of the power law behavior which is widely observed in nature and social systems. Recalling that fading probability characteristic in communications channels and price movements of financial markets are generally heavy-tailed, they can serve good random-number generators for Monte Carlo simulations of such practical applications with power law. However, few studies have been made except the studies [1, 2, 8]. Here we study ergodic transformations on the real line R having Cauchy law and their mixing behavior. This paper is constructed as follows. In Section 2, we introduce ergodic transformations on the real line R and prove the basic properties. In Section 3, we introduce our theorems on preservation of Cauchy laws and prove the theorems. Finally, we conclude this paper.
2. Ergodic transformations on the real line R The following simple ergodic transformation T : R → R defined by T 0 := 0,
T x :=
1 1 (x − ) for x = 0 2 x
appears in many literatures [1, 7–10]. Consider more generic ergodic transformation Bα,β : R → R which is defined by 1 for x = 0. Bα,β (0) := 0, Bα,β (x) := αx − β x In 1857, G. Boole [4] obtained the Boole transformation B1,1 : R → R defined by B1,1 (0) := 0,
B1,1 (x) := x −
1 x
for x = 0
14 Nonlinear Theory and Its Applications, IEICE, vol. 7, no. 1, pp. 14–20
c IEICE 2016
DOI: 10.1588/nolta.7.14
which has the invariance property for an integrable function f (x) and the Boole transformation T = B1,1 (x) as follows: f (T x)dx = f (x)dx. R
R
This means the Boole transformation has the infinite invariant measure μ(dx) given by the Lebesgue measure. In 1973, Adler and Weiss proved the ergodicity of the Boole transformation [3] and later 1 1 on, the ergodicity of more generalized Boole transformation such as T x = (x − ) was shown in [1, 2 x 8]. In 1998, the present author [1] showed ergodicity and mixing property of such transformations on R given by the addition theorem of cot θ and tan θ according to the general method of constructing exactly solvable chaos [11]. Here, we introduce a family of an infinite number of ergodic transformations FK : R → R defined by the addition theorem of cot θ: FK (cot θ) := cot Kθ,
(1)
where K(≥ 2) is a positive integer. Note that the particular transformations are: F2 (x) =
1 1 (x − ), 2 x
F3 (x) =
x3 − 3x , 3x2 − 1
F4 (x) =
x4 − 6x2 + 1 , 4x3 − 4x
··· .
(2)
In the similar way, we introduce ergodic transformations IK : R → R defined by the addition theorem of tan θ: IK (tan θ) := tan Kθ, (3) where K(≥ 2) is a positive integer. Note that those counterpart transformations are: I2 (x) =
2x , 1 − x2
I3 (x) =
3x − x2 , 1 − 3x2
I4 (x) =
4x − 4x4 , 1 − 6x2 + x4
··· .
(4)
All the transformations FK , IK at K ≥ 2 have mixing property (thus ergodic) [1] with the positive Lyapunov exponents λ(= log K), which equals to its Kolmogorov-Sinai entropy hμ due to the Pesin identity with respect to the ergodic invariant measure μ given by the Cauchy density on R: μ(dx) = ρ(x)dx =
dx . π(1 + x2 )
(5)
The key to proving this fact is simple: all the transformations FK , IK at K ≥ 2 have the topological conjugacy relation with the piecewise linear map with the absolute value of the slope being K by a diffeomorphism. Thus, by the pointwise ergodic theorem of Birkhoff [5], we have the following theorem: Theorem (Basic Ergodicity) Consider a map T : R → R, which is given by FK or IK at K ≥ 2. For an integrable function f , the following relation holds for almost everywhere x ∈ R: N −1 dy 1 n . f (T x) = f (y) lim N →∞ N π(1 + y 2 ) R n=0
Here, T n x is the n times iteration of the mapping T of x. Furthermore, when we set f (x) = eikx with k ∈ R, we have the following relation regarding the Chaos-Fourier Transform computing the Fourier Transform by Chaotic Monte Carlo method [6] based on the pointwise ergodic theorem with chaotic maps with absolutely continuous invariant measure μ(dx) = ρ(x)dx. Theorem (Chaos Fourier Transform) Consider a map T : R → R, which is given by FK or IK at K ≥ 2. The following relation holds for almost everywhere x ∈ R:
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N −1 1 ikT n x e = e−|k| . N →∞ N n=0
lim
Proof With an integrable function f (x) = eikx with k ∈ R, the pointwise ergodic theorem states: N −1 1 ikT n x dy lim = e−|k| e = eiky μ(dy) = eiky 2) N →∞ N π(1 + y R R n=0
(6)
for almost everywhere x. The last equality holds because of the well-known mathematical fact that 1 is the function of the characteristic function φ(k) of the Cauchy density function ρ(x) = π(1 + x2 ) the exponential decay: dx = e−|k| . eikx π(1 + x2 ) R More generally, the time average of eikXn on R naturally computes the associated characteristic function φ(k) of the Fourier spectrum k for an ergodic invariant measure μ(dx) on R by this chaos Fourier transform method just utilizing the pointwise ergodic theorem of Birkhoff. This is the fundamental property of ergodic transformations on R. Recently, Steunding [7] found a remarkable relation between the Riemann hypothesis on the Riemann zeta function ζ(s) and the ergodic transformation 1 1 T x = F2 (x) = (x − ) by using this type of ergodic theorem: 2 2 Theorem (Steuding [7], 2012) For almost all x ∈ R and the ergodic transformation T x = − x1 ) on R, N −1 1 ρ 1 n 1 ; lim log ζ( + iT x) = log N →∞ N 2 2 1 − ρ n=0 Reρ> 12 in particular, the Riemann hypothesis is true if, and only if, one and thus either side vanishes, the left-hand side for almost all real x. 1 2 (x
Here, ρ is the non-trivial zeros of ζ(s). The key of proving the above theorem is based on the fact 1 dx 1 . that T x = F2 (x) = (x − ) is ergodic with respect to the invariant measure μ(dx) = 2 x π(1 + x2 ) Thus, we obtain the similar theorem related to an infinite number of ergodic transformations on R by using the fact that T x = FK (x) or T x = IK (x) on R with K ≥ 2 is ergodic with respect to the dx as follows. invariant measure μ(dx) = π(1 + x2 ) Theorem (Riemann Hytpothesis and Ergodic Transformations on R) For almost all x ∈ R and the ergodic transformations T x = FK (x) or T x = IK (x) on R with K ≥ 2, N −1 1 ρ 1 1 ; log ζ( + iT n x) = log lim N →∞ N 2 2 1 − ρ 1 n=0 Reρ> 2 in particular, the Riemann hypothesis is true if, and only if, one and thus either side vanishes, the left-hand side for almost all real x. That is an interesting relation between the Riemann hypothesis and ergodic transformations on R.
3. Preservation of Cauchy laws In this section, we further investigate about how the Cauchy distribution ρλ (x)dx with the scale parameter λ(> 0) is affected by a change of variable by the ergodic transformations on R. Let us consider the density function of the Cauchy law with the scale parameter λ(> 0) :
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ρλ (x) =
π(λ2
λ . + x2 )
Here, we have the following theorem. Theorem 1 (The Cauchy Invariance for T x = F2 (x)) Assume that a variable X ∈ R obeys the Cauchy law with the density ρλ (x)dx for λ > 0. Then, the probability density function of the 1 1 variable Y = T X = F2 (X) obeys the Cauchy law with the scale parameter λ = (λ + ); namely, 2 λ the following relation holds: ρλ → ρλ = ρ 12 (λ+ λ1 )
x → T x = F2 (x) =
when
1 1 (x − ). 2 x
Thus, after n times iterations of T , we have the semi-group property of the scale parameter of the Cauchy Law: ρλ → ρλn = ρGn (λ) when x → T n x = F2n (x), where G is defined by G(λ) =
1 1 (λ + ). 2 λ
Note that λ∞ = lim λn = lim Gn (λ) = 1 n→∞
n→∞
for any λ(> 0) because λ∞ is the unique fixed point of the equation λ = G(λ) for λ(> 0) and the map G corresponds to one iteration process of the Newton method solving the equation λ2 − 1 = 0 with the quadratic convergence speed [10]. This is consistent with the fact the ergodic transformation 1 as T x = F2 (x) has the standard Cauchy law with the absolutely continuos density ρ1 (x) = π(1 + x2 ) the invariant measure and due to the ergodic and mixing property, the limiting distribution converges to the invariant measure μ(dx) = ρ1 (x)dx for almost all distributions of the initial value x. Proof. (Proof of Theorem 1) For an arbitrary y = 0, there are always two solutions x1 , x2 ∈ R satisfying the equation y = F2 (x) =
1 1 (x − ). 2 x
We denote them as x1 and x2 respectively. Then, we have the relation x1 + x2 = 2y,
x1 x2 = −1.
(7)
Let us denote the density function of a variable y = T x = F2 (x) by pY (y). The probability pY (y)|dy| equals the sum of the probabilities ρλ (x1 )|dx|x=x1 and ρλ (x2 )|dx|x=x2 . Thus, we have the following probability preservation relation: pY (y)|dy| = ρλ (x1 )|dx|x=x1 + ρλ (x2 )|dx|x=x2 . Remark that
dy dF2 (x) 1 + x2j 2(1 + y 2 ) = = = , dx dx 2 xj 1 + x2j x=xj x=xj
for j = 1
(8)
or 2,
we obtain pY (y) by the relation pY (y)
=
2 1 ρλ (xj )(1 + x2j ) 2(1 + y 2 ) j=1
1 4(λ2 + 1)(y 2 + 1) = 2(1 + y 2 ) π((1 + λ2 )2 + 4y 2 λ2 )
=
1 λ(1 + x21 )(λ2 + x22 ) + λ(1 + x22 )(λ2 + x21 ) 2(1 + y 2 ) π(x21 x22 + λ2 (x21 + x22 ) + λ4 ) 2
λ +1 1 2λ = = ρ 12 (λ+ λ1 ) (y) = ρG(λ) (y). 2 +1 2 π ( λ2λ ) + y2
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1 1 Thus, we can say that the generalized Boole transformation T x = F2 (x) = (x − ) preserves 2 x 1 1 the Cauchy law with the scale parameter λ → G(λ) = (λ + ). 2 λ Here, we define GK as an addition theorem of coth θ satisfying GK (coth θ) = coth Kθ. Theorem 2 (The Cauchy Invariance for T x = FK (x) at K ≥ 2) Assume that a variable X ∈ R obeys the Cauchy law with the density ρλ (x)dx for λ > 0. Then, the probability density function of the variable Y = T X = FK (X) at K ≥ 2 obeys the Cauchy law with the scale parameter λ = GK (λ); namely, the following relation holds: ρλ → ρλ = ρGK (λ)
when
x → T x = FK (x).
1 1 When K = 2, λ = G2 (λ) = (λ + ). Thus, Theorem 2 for K = 2 corresponds to Theorem 1. Thus, 2 λ Theorem 2 can be regarded as the generalization of Theorem 1. Proof. (Proof of Theorem 2) Consider a change of variable as Y = FK (X). For a given Y = y ∈ R, there are K solutions satisfying the equation y = FK (x). We denote the solutions by x1 , x2 , · · · , xK . By noting that FK can be seen as K-th multiplication formula of cot θ, we can write the solutions explicitly as π xj = − cot(φ + j K ), j = 1, · · · , K for y = − cot Kφ (0 ≤ Kφ ≤ π). Denote the density function of a variable y = T x = FK (x) by pY (y). Then we have the following probability preservation relation: pY (y)|dy| = ρλ (x1 )|dx|x=x1 + ρλ (x2 )|dx|x=x2 + · · · + ρλ (xK )|dx|x=xK . It is easy to check that the relation holds: dy dFK (x) dφ K K(1 + y 2 ) d π (− cot Kφ) = = = . sin2 (φ + j ) = dx dx 2 dφ dx K 1 + x2j sin (Kφ) x=xj x=xj Thus, pY (y) satisfies the relation pY (y) =
K K λ 1 1 2 . ρ (x )(1 + x ) = λ j j π 2 2 2 K(1 + y ) j=1 Kπ(1 + y ) j=1 1 + (λ − 1) sin2 (φ + j K )
By noting that the following identity K j=1
1+
(λ2
KGK (λ)(1 + cot2 (Kφ)) λ = 2 π G2K (λ) + cot2 (Kφ) − 1) sin (φ + j K )
always holds, we obtain pY (y) as pY (y) =
K 1 1 λ KGK (λ)(1 + y 2 ) = ρλ (y), = 2 π Kπ(1 + y 2 ) j=1 1 + (λ2 − 1) sin (φ + j K ) Kπ(1 + y 2 ) (G2K (λ) + y 2 )
where λ = GK (λ). After n-th iteration of the ergodic transformation T x = FK (x), the density function obeys the semi1 group property as ρλn = ρGnK (λ) approaching the standard Cauchy density ρ1 (x) = of the π(1 + x2 ) unique invariant invariant measure μ(dx) on the real line R as the limiting distribution for n → ∞. Letac [9] showed that generalized Boole transformations including T x = F2 (x) on the real line R preserve Cauchy laws. While Theorem 1 is considered as a special case of the theorem of Letac, Theorem 2 generalizing Theorem 1 covers wider ergodic transformations FK at K ≥ 2 (not necessary
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the generalized Boole transformations) preserving Cauchy laws as compared to the existing articles on preservation of Cauchy laws such as [8, 9]. Thus, Theorem 2 states a new preservation law of Cauchy laws. Furthermore, we can proceed further as we treat ergodic transformations IK at K ≥ 2 satisfying the functional relation tan Kθ = IK (tan θ). Here we consider rational mappings JK at K ≥ 2 which satisfy the relation tanh Kθ = JK (tanh θ). We have the following theorem. Theorem 3 (The Cauchy Invariance for T x = IK (x) at K ≥ 2) Assume that a variable X ∈ R obeys the Cauchy law with the density ρλ (x)dx for λ > 0. Then, the probability density function of the variable Y = T X = IK (X) at K ≥ 2 obeys the Cauchy law with the scale parameter λ = JK (λ); namely, the following relation holds: ρλ → ρλ” = ρJK (λ)
when
x → T x = IK (x).
Proof. (Proof of Theorem 3) 1 where X obeys the Cauchy law with the scale parameter λ, obeys We note that distribution of X 1 the Cauchy law with the scale parameter . Thus, by Theorem 2, λ λ =
1 = JK (λ). GK ( λ1 )
4. Conclusion An infinite number of ergodic transformations FK and IK given by the addition theorems of cot θ and 1 1 tan θ at K ≥ 2 are proven to preserve Cauchy laws. The case that F2 (x) = (x − ) corresponds 2 x to the well-known case having the standard Cauchy law as the ergodic invariant measure. The scale parameter λ dynamics is also obtained by mappings GK and JK given by the addition theorems of coth θ and tanh θ respectively. Thus, explicit mixing behavior of the probability distribution of FK and IK towards the limiting standard Cauchy law with the density ρ1 (x) is systematically obtained by this scale parameter dynamics GK and JK respectively.
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