R´ENYI FUNCTION FOR MULTIFRACTAL RANDOM FIELDS NIKOLAI N. LEONENKO AND NARN-RUEIH SHIEH Abstract.

NIKOLAI N. LEONENKO AND NARN-RUEIH SHIEH

Abstract. This paper presents the basic scheme and the log-normal, log-gamma and log- negative-inverted-gamma scenarios to establish the R´enyi function for infinite products of ho- mogeneous isotropic random fields on Rⁿ; in particular for random fields on the on the sphere in R³. The motivation of this paper is the test of (non-)Gaussianity on the Cosmic Microwave Background Radiation (CMBR) in Cosmology. In the presentation, we need to employ spherical harmonics for some concrete computations.

1. Introduction

Multifractal models have been used in many applications in hydrodynamic turbulence, fi- nance, genomics, computer network traffic, etc; see Kolmogorov (1941,1962), Kahane (1985,1987), Gupta and Waymire (1993), Novikov (1994), Frisch (1995), Mandelbrot (1974), and Falconer (1997). Harte (2001) and Riedi (2003) contain an extensive bibliography of the subject. There are many ways to construct random multifractal measures such as via the binomial cascade, branching processes and other stochastic processes; see Kahane (1985,1987), Gupta and Waymire (1993), Frisch (1995), Taylor (1995), Molchan (1996), Falconer (1997), Jaffard (1999), Schmitt and Marsan (2001), Barral and Mandelbrot (2002), Shieh and Taylor (2002), Riedi (2003), Bacry and Muzy (2003), Schmitt (2003), Barndorf-Nilsen and Shmigel (2004), M¨orters and Shieh (2002,2004,2008). In these works of such Multifracal Analysis, the R´enyi function, which is also termed as the deterministic partition function or the moment-scaling function, plays a central role, as we may see in the seminal works of Mandelbrot (1974) and Kahane (1987). In a recent work, Mannersalo et al (2002) discuss R´enyi functions for random measures induced from infinite products of stationary processes. Their work has been much examined for several classes of exponential ( geometric ) processes by Anh, Leonenko, and Shieh (2008a,b, 2009a,b, 2010);

see also the results related to the topics in Shieh (2009) and Matsui and Shieh (2009,2012), and a simulation paper by Anh, Leonenko, Shieh and Taufer (2010).

The purpose of this work is to present a basic scheme and some important scenarios for the R´enyi function of the infinite products of measurable homogeneous and isotropic random fields, which may show the multifractality of such fields. Our scheme is novel in two aspects; the first one is that the process (one-parameter) case is not easily seen to have the field (multi-parameter) corresponding, and the second one is that our scheme may include the model needed for study on the problems in Cosmic Microwave Background Radiation (CMBR); see a review paper by Marinucci (2004), a recent monograph by Marinucci and Peccati (2011), and the references therein. Moreover, our scenarios include several ones which are of intrinsic interest in previous mathematics and physics literatures.

The paper is organized as follows. In Section 2, we list some preliminaries and present the basic scheme. In Section 3, we provide some important scenarios. All proofs are given in the Section 4. In the concluding Section 5 we summarize our results and remark the possible statistical calibration of our model based on the dataset released by The Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP).

Key words and phrases. R´enyi function; infinite products; random fields; multifractality; singularity spectrum;

log-normal scenario; log-gamma scenario; log-negative-inverted- gamma scenario.

1

Acknowledgment. This work is initiated while both authors visited Queensland University of Technology, and is continued while N.-R. Shieh visited York University, Canada. The support and the hospitality is acknowledged. N. N. Leonenko is supported by the Australian Research Council grant DP0559807 and the EPSRC grant RCMT119. The authors thanks to the editor and the referee for their valuable comments which make the present version of the paper much more edged.

2. Infinite products of random fields: multifractal schemes

In this paper we consider a measurable homogeneous and isotropic random field (HIRF, for brevity) on the n-dimensional Euclidean space Rⁿ, n ≥ 2 (the case n = 1, i.e. the process case, is already in literatures), that is

Λ = {Λ(x) = Λω(x), x ∈ Rⁿ, ω ∈ Ω} ,

is a measurable separable mean-square continuous random field on a complete probability space (Ω, z, P ) , such that

EΛ(x) = m = const, VarΛ(x) < ∞, Cov(Λ(x), Λ(y)) = RΛ(kx − yk),

where R_Λ(kx − yk), (x, y) ∈ Rⁿ× Rⁿ, is a continuous positive definite kernel, which depends on the Euclidean distance r = rxy = kx − yk only.

The Schoenberg theorem implies that a function R_Λ(kx − yk) is the covariance function of a mean-square continuous homogeneous isotropic random field Λ(x), x ∈ Rⁿ, if and only if there exists a finite measure G on the measurable space (R¹+, B(R¹+)) such that

(2.1) R_Λ(r) =

∞

Z

0

Y_n(ur) G(du),

To define Y_n, let, for ν > −¹₂, Jν(z) =

∞

X

m=0

(−1)^m(z

2)^2m+ν[m! Γ(m + ν + 1)]⁻¹, z > 0 be the Bessel function of the first kind of order ν, and we define

Yn(z) = 2^(n−2)/2Γ(n

2) J_(n−2)/2(z) z^(2−n)/2, z ≥ 0, n ≥ 2.

See, for example, Leonenko (1999), for more details and proofs on the above facts.

The 2-dimensional and the 3-dimensional fields are physically important and the above dis- plays are then more explicit; thus, we list them in the below.

For n = 2 formula (2.1) is

R_Λ(r) =

∞

Z

0

J₀(ur) G(du).

For n = 3 formula (2.1) is

R_Λ(r) =

∞

Z

0

sin ur

ur G(du).

We begin with the following conditions:

A⁰. Let a given random field Λ = {Λ(x), x ∈ Rⁿ} be a HIRF such that EΛ(x) = 1, VarΛ(x) = σ_Λ² < ∞, Λ(x) > 0, x ∈ Rⁿ, Cov(Λ(x), Λ(y)) = R_Λ(kx − yk) = σ²_Λρ_Λ(kx − yk), ρ(0) = 1,

and Λ⁽ⁱ⁾(x), x ∈ Rⁿ, i = 0, 1, 2, . . . be a sequence of independent fields re-scaled from Λ by Λ⁽ⁱ⁾(x)= Λ^{d (i)}(bⁱx),

where b > 1 is a scaling factor, and= denotes equality in finite-dimensional distributions. That^d bⁱx denotes the product of a vector x by a scalar bⁱ.

For the convenience, we will call the field Λ in the above condition A⁰ to be a mother field;

referring the sense that Λ generates our scheme presented in the main Theorem 2.1 in the below.

We denote the n-dimensional (n ≥ 2) unit ball as

Bⁿ= {x ∈ Rⁿ: kxk ≤ 1},

and we will use the spherical coordinates for x: x = (r, e) = re, r > 0, e ∈ Sⁿ⁻¹; the latter one denotes the unit spherical surface in Rⁿ.

Now, we define a sequence of finite-product fields on Bⁿ, k = 1, 2, . . ., by

(2.2) Λ_k(x) =

k

Y

i=0

Λ⁽ⁱ⁾(xbⁱ), and the random measure on Borel σ-algebra B of a unit ball Bⁿ

µ_k(B) = Z

y∈B

Λ_k(y)dy, k = 0, 1, 2, . . . , B ∈ B;

using the spherical coordinates, we may write the above as, µ_k(B) =

Z Z

y=(r,e)∈B

rⁿ⁻¹Λ_k(re)drde, k = 1, 2, . . . .

In the above, we have used the normalized Lebesque measure dy on the ball Bⁿ and the normalized uniform spherical measure de on Sⁿ⁻¹ (normalized so that the total measure is 1).

We denote

µk→^D µ, k → ∞,

the weak convergence of the measures µ_k to some measure µ, that is Z

Bⁿ

f (y)µ_k(dy) → Z

Bⁿ

f (y)µ(dy), k → ∞, for all continuous functions f (y), y ∈ Bⁿ.

For a random measure µ, the R´enyi function of µ is a deterministic function defined as

T (q) = lim inf

m→∞

log₂EP

lµ

 B_l^(m)

q

log₂   B^(m)_l

where {B_l^(m), l, m}, l = 0, 1, . . . , 2^m− 1 and m = 1, 2, . . . , denotes the mesh formed by the m-th level dyadic decomposition of the unit ball Bⁿ based on the spherical coordinates (note that this mesh is different from the rectangle-type decomposition; we thanks to the referee to remind this). The notation | · | denotes the normalized Lebesgue measure on Bⁿ= {x ∈ Rⁿ: kxk ≤ 1};

in the above and henceforth, the log_a denotes the logarithm in the base a > 0.

Remark 2.1. R´enyi function plays a central role in multifractal analysis, since the multifractal formalism in the theory of random cascades can be understood in the sense that the Legendre transform of the R´enyi function

LT (z) =min

q (qz − T (q))

would be the dimension spectrum (in the sense of Hausdorff dimension) of the following sets (indexed by α)

F_α =n

x ∈ Bⁿ: lim

m→∞log₂µ

B^(m)_l (x) / log₂

  B^(m)_l

  = αo

,

where B_l^(m)(x) is the sequence in the mesh {B_l^(m), l, m} that contain x; it shrinks to x as m → ∞.

Our basic scheme can be summarized in the following statement.

Theorem 2.1. Suppose that the condition A⁰holds.

(i) Assume that the correlation function ρΛ(kx − yk) = ρ(r) of a mother field Λ satisfies the following condition:

(2.3) |ρ_Λ(r)| ≤ Ce^−γr, r > 0,

for some positive C and γ. Then, when the scaling factor b : b > ⁿ

q

1 + σ_Λ², on Bⁿ the measures

µk→^D µ, k → ∞,

the random measure µ is non-degenerate, it has the finite second moment: Eµ²(Bⁿ) < ∞, and it has the stochastic scaling-invariant (or say self-similar) property:

µ(dy) = b⁻ⁿΛ(y)ˆµ(bdy),

where the measure ˆµ(dy) is independent of Λ and has the same distribution as µ(dy).

(ii) Assume that for some range

q ∈ Q = [q−, q+],

both E^qΛ(0) < ∞ and Eµ^q(Bⁿ) < ∞, B ∈ B, then, the R´enyi function T (q) of µ is given by

(2.5) T (q) = q − 1 − 1

nlog_bEΛ^q(x) , q ∈ Q,

Remark 2.2. Our scheme, suppose n being reduced to be 1, is complementary to that in Man- nersalo et al (2002, p. 894); see the remark below the proof of Theorem 2.1. The scheme may be traced back to the “multiplicative chaos” theory of Kahane (1985, 1987), which lays a math- ematical foundation on Mandelbrot’s cascades; confer to the pioneering paper of Mandelbrot (1974) for the details of the physical theory.

Remark 2.3. The range Q in Theorem 2.1 at least contains [1,2]; while to determine the full range of validity for a given scenario is mathematically challenging, even in the classical cascades case; see Kahane (1985,1987).

The proof of Theorem 2.1 will be given in the Section 4.

Motivated by CMBR problems (see Section 1), we consider the case when the the underlying random field is a 3-dimensional spherical one. Let the spherical surface in R³(as a 2-dimensional manifold) with a given radius r > 0 be

s2(r) = s(r) =x ∈ R³ : kxk = r ⊂ R³ while the Lebesgue measure (the area element on the sphere)

σr(du) = σr(dθ.dϕ) = r²sin θdθdϕ , (θ, ϕ) ∈ s(1) , r = kxk > 0 . A spherical random field, denoted by

T = {T (r, θ, ϕ) : 0 ≤ θ ≤ π, 0 ≤ ϕ ≤ 2π, r > 0} or T = n

T (x) , x ∈ s(r)e o

,

is a stochastic function defined on the sphere s(r). We consider a real-valued spherical field T with the mean m, and finite second moments such that it is continuous in the mean-square

sense and its (continuous) covariance function depends on the geodesic (or angular) distance between two points on the sphere. Under these conditions, the the isotropic random field on the sphere s₂(r) can be expanded in mean-square sense as a Laplace series

(2.1) T (r, θ, ϕ) = m +

∞

X

l=0 l

X

m=−l

Y_l^m(θ, ϕ)alm(r) , where Y_l^m(θ, ϕ) represent the spherical harmonics, i.e.

(2.2) Y_l^m(θ, ϕ) = clmexp(imϕ)P_l^m(cos θ) , − l ≤ m ≤ l , l = 0, 1, ..., where

clm= (−1)^m 2l + 1 4π

(l − m)!

(l + m)!

1/2

, − l ≤ m ≤ l , and P_l^m(cos θ) denotes the associated Legendre polynomial of degree l, m, i.e.

(2.3) P_l^m(x) = (−1)^m(1 − x²)^m/2 d^m

dx^mP_l(x) , where

P_l(x) = 1 2^ll!

d^l

dx^l(x²− 1)^l

is the Legendre polynomial. The spherical harmonics have the following properties Z π

0

Z 2π 0

Y_l^m(θ, ϕ)Y_l^{m0 0}(θ, ϕ) sin θdθdϕ = δ^l_l⁰δ_m^m0 , (2.4)

Y_l^m(θ, ϕ) = (−1)^mY_l^−m(θ, ϕ) , Y_l^m(π − θ, ϕ + π) = (−1)^lY_l^m(θ, ϕ) ,

where δ_l^l0 represent the Kronecker delta. The random coefficients in the Laplace series (2.1) can be obtained through inversion arguments in the form of mean-square stochastic integrals

a^m_l (r) = Z π

0

Z 2π 0

T (r, θ, ϕ)Y_l^m(θ, ϕ)r²sin θdθdϕ

= Z

s(1)

T (ru)Ye _l^m(u)σ1(du) , u = x

kxk ∈ s(1) , r = kxk , (2.5)

see, for example, Leonenko (1999), for more details.

The field T (r, θ, ϕ) = eT (x), x ∈ R³, is said to be isotropic in R³, if E eT (x)² < ∞, and its first and second order moments are invariant with respect to the group of rotations on the sphere, i.e.

E eT (x) = E eT (gx), E eT (x) eT (y) = E eT (gx) eT (gy) ,

for every g ∈ SO(3), the group of rotations in R³. The restriction of an isotropic random field T (x), x ∈ Re ³ on a sphere s(r) is an isotropic random field on the sphere. This is equivalent to saying that the covariance function ET (r, θ, ϕ)T (r, θ⁰, ϕ⁰) depends only on the angular distance θ = θP Q between the points P = (r, θ, ϕ) and Q = (r, θ⁰, ϕ⁰). The restriction of the isotropic field on the sphere is spherical field on s(r) if and only if

(2.6) Ea^m_l (r)a^m_l0⁰(r) = δ^l_l⁰δ_m^m0Cl(r) , − l ≤ m ≤ l , − l⁰ ≤ m⁰≤ l⁰ , or

(2.7) E|a^m_l (r)|² = C_l(r) , m = 0, ±1, ..., ±l .

The functional series {C₁(r), C₂(r), ..., C_l(r), ...} is called the angular power spectrum of the isotropic random field T (r, θ, ϕ). From (2.1), (2.5) and (2.6) we deduce that

(2.8) Cov(T (r, θ, ϕ)T (r, θ⁰, ϕ⁰)) = 1 4π

∞

X

l=1

(2l + 1)Cl(r)Pl(cos θ) , where

∞

X

l=1

(2l + 1)C_l(r) < ∞ ,

for every fixed r > 0. If T (r, θ, ϕ) is an isotropic Gaussian field on the sphere s(r), then the coefficients a^m_l (r) are complex-valued independent Gaussian random processes with

Ea^m_l (r) = 0 , Ea^m_l (r)Ea^m_l0⁰(r) = δ^m_m⁰δ^l_l⁰C_l(r).

A random field eT (x), x ∈ R³, with E eT (x)² < ∞, is called homogenous (in the wide sense) if its first two moments are invariant with respect to the Abelian group of shifts in R³. The restriction of the HIRF eT (x), x ∈ R³on the sphere s(r) is an spherical field on s(r) if and only if

(2.9) Ea^m_l (r)a^m_l0⁰(r) = δ^l_l⁰δ_m^m0C_l(r, s) with

(2.10) Cl(r, s) = 2π²

Z ∞ 0

J_l+1 2(µr) (µr)^1/2

J_l+1 2(µs)

(µs)^1/2 G(dµ) , l = 1, 2, ..., where G is a finite measure on the Borel sets of [0, ∞) such that

σ² = Var n

T (0)e o

= Z ∞

0

G(dµ) < ∞ , and J_ν(z) again is the Bessel function of the first kind of order ν.

The covariance function Cov n

T (x), ee T (y) o

of a mean-square continuous HIRF eT (x), x ∈ R³, depends only on the Euclidean distance

ρ = |x − y| = q

ρ²₁+ ρ²₂− 2ρ₁ρ₂cos γ , cos γ = < x, y >

kxkkyk , x = (ρ₁, u₁) , y = (ρ₂, u₂) , and by the addition theorem for Bessel functions can be represented as

B(ρ) = Z ∞

0

sin(µρ) µρ G(dµ)

= 2π²

∞

X

l=1 l

X

m=−l

Y_l^m(u1)Y_l^m(u2) Z ∞

0

J_l+¹

2(µρ1) (µρ1)^1/2

J_l+¹

2(µρ2)

(µρ2)^1/2 G(dµ) .

By Karhunen’s Theorem, each mean-square continuous homogenous isotropic spherical random field with zero mean has a spectral representation

(2.11) T (x) = m +e

∞

X

l=1 l

X

m=−l

Y_l^m(θ, ϕ)a^m_l (r), with

(2.12) a^m_l (r) = π√

2 Z ∞

0

J_l+¹

2(µr)

p(µr) Z_l^m(dµ) ,

where Z_l^m, −l ≤ m ≤ l, l = 1, 2, .., are the family of complex-valued random measures on Borel sets of [0, ∞) such that

(2.13) EZ_l^m(A) = 0 , EZ_l^m(A)Z_l^m0⁰(B) = δ_l^l0δ^m_m⁰G(A ∩ B) If there exists an isotropic spectral density g(µ) ≥ 0 such that

(2.14) G(dµ)

dµ = ks(1)kµ²g(µ) , µ²g(µ) ∈ L₁([0, ∞)) , where ks(1)k = 4π(the area of the unit sphere). Then (2.11) holds with (2.15) a^m_l (r) = (2π)^3/2

Z ∞ 0

√µJ_l+¹

2(µr)p

g(µ)W_l^m(dµ) , where

EW_l^m(A)W_l^m0 ⁰(B) = δ^l_l⁰δ_m^m0|A ∩ B| ,

that is W_l^m, −l ≤ m ≤ l, l = 1, 2, ... is a family of white noise random measures (Gaussian is field is Gaussian). The restriction of an HIRF eT (x), x ∈ R³ on a sphere s(r) is an isotropic random field on the sphere. In this particular case, the covariance function of this isotropic field T on s(r) is representable in the form (2.8) with the angular power spectrum

(2.16) C_l(r) = 2π²

Z ∞ 0

J²

l+¹₂(µr)

µr G(dµ) , l = 1, 2, ...

or

(2.17) C_l(r) = (2π)³

Z ∞ 0

J_l+² 1 2

(µr)

µr µ²g(µ)dµ,

whenever (2.14) is satisfied. For the correlation function of HIRF eT (x), x ∈ R³of the form ρ(r) = e^−ar, r ≥ 0, the isotropic spectral density g(µ) = a(µ²+ a²)⁻³²/(2π³²).

We introduce the following conditions for the spherical random fields on R³: A⁰⁰. Let the mother field Λ = n ˜Λ(x), x ∈ s2(1)

o

, be isotropic random field and the sphere s2(1) =x ∈ R³ : kxk = 1 ⊂ R³ such that

E ˜Λ(x) = 1, Var ˜Λ(x) = σ²_Λ< ∞, Λ(x) > 0, x ∈ s₂(1), Cov(Λ(θ, ϕ), Λ(θ⁰, ϕ⁰)) = 1

4π

∞

X

l=1

(2l + 1)C_lP_l(cos θ) ,

∞

X

l=1

(2l + 1)C_l< ∞ ,

and ˜Λ⁽ⁱ⁾(x), x ∈ s2(1), i = 0, 1, 2, . . . be a sequence of independent fields on x = (1, θ, ϕ) ∈ s2(1) such that

Λ˜⁽ⁱ⁾(x)= ˜^d Λ⁽ⁱ⁾(bⁱx),

where b > 1 is a scaling factor, and we interpret bⁱx by bⁱx := (1, bⁱ×

π θ, bⁱ ×

2πϕ) ∈ s2(1), where the modulus algebra is used accordingly.

Define the finite product fields on s2(1) Λ˜_k(x) =

k

Y

i=0

Λ˜⁽ⁱ⁾(bⁱx), and the random measure on Borel σ-algebra B of s₂(1)

(2.18) µ_k(B) = Z

B

Λ_k(y)dy, k = 0, 1, 2, . . . , B ∈ B, in the 3-dimensional spherical coordinate, the above is then in the form

µ_k(B) = Z Z

x=(1,θ,ϕ)

Λ_k(1, θ, ϕ) sin θdθdϕ, k = 1, 2, . . . . We denote

µ_k→^D µ, k → ∞,

the weak convergence of the measures µ_k to some non-degenerate measure µ, that is Z

s2(1)

f (y)µ_k(dy) → Z

s2(1)

f (y)µ(dy), k → ∞, for all continuous functions f (y), y ∈ s2(1).

The R´enyi function of µ on s₂(1) is now defined as

T (q) = lim inf

m→∞

log₂EP

lµ S^(m)_l q

log₂   S_l^(m)

where {S_l^(m), l, m}, l = 0, 1, . . . , 2^m− 1, is the mesh formed by the m-th level dyadic decompo- sition of the spherical surface s₂(1).

We then reformulate the R´enyi function T (q) in our basic scheme Theorem 2.1 in the fol- lowing spherical form. It is a direct consequence of Theorem 2.1 (the s₂(1) is identified as a 2-dimensional surface) and thus the proof is omitted; we refer it as a theorem since its formu- lation is of individual interest.

Theorem 2.2. Suppose that the condition A⁰⁰ holds and the isotropic random field is the re- striction of the HIRF Λ(x), x ∈ R³ with correlation function ρΛ(kx − yk) = ρ(r) on the sphere s₂(1). We assume the similar assumptions as those in Theorem 2.1.

The R´enyi function T (q) of the limit measure µ on s₂(1) is given by T (q) = q − 1 −1

2log_bEΛ^q(t) , q ∈ Q.

3. Some important scenarios

In this section we consider several scenarios for multifractal random fields on the n-dimensional Euclidean space Rⁿ, n ≥ 2. When n is reduced to be 1, they are consistent with the previous known results of the exponential OU-type stationary processes studied in a series of papers by Anh, Leonenko, and Shieh (2008a,b, 2009a,b, 2010).

Firstly, we provide the the lognormal scenario for multifractal products of HIRF’s, which follow the prominent lognormal hypothesis of Kolmogorov (1962) in turbulent cascades. In fact, this lognormal scenario has its origin in Kahane (1985,1987). We reformulate Theorem 2.1 for this special case in order to have a precise scaling law for the moments.

B⁰. Consider a mother field of the form Λ(x) = exp



Y (x) −1 2σ_Y²

 ,

where Y (x), x ∈ Rⁿ is a zero-mean Gaussian, measurable, separable random field with covari- ance function

R_Y(r) = σ_Y²ρ_Y(r), ρ_Y(0) = 1.

Let us consider the following more specific case. The Gaussian solution of the stochastic Laplace or stochastic Helmholtz equation studied in Kelbert, Leonenko and Ruiz-Medina (2005);

the equation is in the from:

α²I − ∆
ν/2

Y (x) = ε(x), x ∈ Rⁿ, n ≥ 2, ν > 0,

where ε(x), x ∈ Rⁿ, is white noise with variance σ², ∆ is the Laplacian and I is identity operator. The authors show that the homogeneous isotropic solution to this equation has the covariance function R_Y(r), which belongs to Mat´ern class, as one can see Stein (2000, pp.

49-51), and takes the following form:

R_Y(r) = π^n/2

α^2ν−n2^ν−n/2−1Γ(ν) σ²

(2π)ⁿK_ν−n/2(rα)(rα)^ν−n/2, r > 0, α > 0, ν > n/2, where

K_λ(x) = 1 2

Z ∞ 0

u^λ−1e^−12x(u+1u)du, x > 0,

is the modified Bessel function of the third kind or McDonald function with index λ.For λ = r + 1/2, where r is an nonnegative integer, the Bessel function Kλ(x) has the closed form

K_r+1/2(x) =r π 2xe^−x

r

X

k=0

(r + k)!

(r − k)!k!(2x)^−k. Thus, for ν −ⁿ₂ = ¹₂, (up to constant)

R_Y(r) = const1

αe^−αr, r ≥ 0,

and condition (2.3) holds; for instance, ν = ³₂ for n = 2, and ν = 2 for n = 3.

Under condition B⁰, we have the following moment generating functions:

M (ζ) = E exp

 ζ



Y (x) −1 2σ²_Y



= e^{12σ2Y(ζ2−ζ)}, ζ ∈ R¹, M (ζ1, ζ2; kx1− x₂k) = E exp

 ζ1



Y (x1) −1 2σ²_X

 + ζ2



Y (x2) −1 2σ_Y²



= exp 1

2σ_Y² ζ₁²− ζ₁+ ζ₂²− ζ₂ + ζ₁ζ₂R_Y(kx₁− x₂k)



, ζ₁, ζ₂ ∈ R¹, It turns out that, in this case,

M (1) = 1; M_θ(2) = e^σY2; σ_Λ² = e^σ2Y − 1;

Cov(Λ(x₁), Λ(x₂)) = M (1, 1; kx₁− x₂k) − 1 = e^{RY(kx1−x2k)}− 1 and

log_bEΛ(x)^q= (q²− q)σ_Y²

2 log b , q > 0.

Using Theorem 2.1, we obtain

Theorem 3.1. Suppose that condition B⁰ holds with the correlation function 0 < |ρ_Y(r)| ≤ Ce^−γr, r > 0,

for some positive C and γ.

Then, for any b > exp

σ2Y

n



, the measures

µ_k→^D µ, k → ∞,

and the R´enyi function of measure µ is given by T (q) = q



1 + σ_Y² 2n log b



− q²

 σ_Y² 2n log b



− 1, q ∈ [1, 2].

Moreover, for n = 3, if Y (x) in the condition B⁰, is a spherical isotropic random field ob- tained as restriction of the HIRF Y (x), x ∈ R³ with correlation function ρ(kx − yk) = ρ_Y(r) on the sphere s2(1), then the random measures (2.18) generated by the spherical fields Λ(x) = expY (x) −¹₂σ²_Y , x ∈ s₂(1), converge weakly to the random measure µ, with the R´enyi func- tion

T (q) = q



1 + σ_Y² 4 log b



− q²

 σ_Y² 4 log b



− 1, q ∈ [1, 2].

In the theory of turbulence cascades, the log-gamma scenario is known as an alternative to the lognormal scenario; see Saito (1992) for details and discussions. Now, we propose a homogeneous isotropic random field version of the log-gamma scenario. We will use a specific construction of the gamma-correlated random field Z(x), x ∈ Rⁿ, as that in Leonenko (1999) and the references therein). However, for our present purpose, we will extend the construction in there to allow two parameters β > 0 and λ > 0 (in the usual literatures the parameter λ = 1).

Accordingly, there exists a HIRF Z(x), x ∈ Rⁿ, with given marginal density

(3.1) p (u) = λ^β

Γ (β)u^β−1e^−λu, u > 0, λ > 0, β > 0 and fixed correlation function γ = ρZ(τ ).

The bivariate density function of Z(x) can be constructed via the bilinear expansion p (u, w; γ) = p(u)p(w)

"

1 +

∞

X

k=1

γ^ke_k(u)e_k(w)

#

=

(3.2) = λ²p₀(uλ, wλ; γ) , , 0 < γ < 1, where

e_k(u) = e^(β)_k (u) = L^(β−1)_k (u)

 k!Γ(β) Γ(β + k)

1/2

, k = 0, 1, 2, . . . ,

where L^(β)_k (u) are the generalized Laguerre polynomials of index β for k ≥ 0, defined as L^(β)_k (u) = (k!)⁻¹u^−βe^u d^k

du^k{e^−uu^β+k}.

One can show that

e^(β)₀ (u) ≡ 1, e^(β)₁ (u) = (β − u) β^−1/2, ...

It is known that

(3.3) p₀(u, w; γ) = uw γ

^β−1₂ exp



−u + w 1 − γ

 I_β−1

 2

√u · w · γ 1 − γ

 1

Γ (β) (1 − γ), where

I_ν(z) =

∞

X

k=0

z 2

2k+ν 1

k! Γ(k + ν + 1).

being the modified Bessel function of the first kind of order ν. In our case γ = ρZ(τ ) . The existence of gamma-correlated HIRF with marginal density (3.1), bivariate density (3.3) and correlation function γ = ρZ(τ ) , in which ρZ(τ ) is a continuous positive-definite function with ρ_Z(0) = 1 can be proven by using the maximum entropy principle, see Joe (1997) or Dozzi and Leonenko (2011).

Let Z(x), x ∈ Rⁿ, be homogeneous isotropic random field with one dimensional densities (3.1) and two-dimensional densities of the form (3.3), and let γ = γ(kx − yk) be a continuous non-negative definite kernel on Rⁿ× Rⁿ. Then becomes

(3.4) Ee^(β)_k (Z(x)) = 0, E e^(β)_m (Z(x)) e^(β)_k (Z(y)) = δ^k_m γ^m(kx − yk).

As we shall see below this field can be given constructive for every α = p/2, where p ≥ 1 is integer. So the class of Gamma correlated random fields is not empty. To see this we consider the so-called χ -squared random field:

(3.5) Z_p(x) = 1

2(η₁²(x) + · · · + η_p²(x)), x ∈ Rⁿ, p ≥ 1,

where Y1(x), . . . , Yp(x) are independent copies of homogeneous Gaussian field Y (x), with EY (x) = 0, EY²(x) = 1, Cov(Y (0), Y (x)) = R(kxk), x ∈ Rⁿ

Let the random variables (X1, Y1), . . . , (Xp, Yp) be i.i.d. with common standard normal bivariate distribution with correlation coefficient ρ. Then it can be shown that the r.v. ¹₂(X₁², Y₁²) has the characteristic function

(3.6) ϕβ(t, s, γ) = [1 − it − is − t s (1 − γ)]^−β.

with γ = ρ² and β = ¹₂. Consequently, the function (3.6) is the characteristic function of

1

2(X₁²+ · · · + X_p², Y₁²+ · · · + Y_p²) with β = p/2. In what follows, the Gamma correlated random field Z(x), x ∈ Rⁿ may be realized for β = p/2, γ(kxk) = R²(kxk), as χ-squared random field (3.5). Note that

(3.7) EZ_p(x) = p

2, VarZ_p(x) = p

2, Cov(Z_p(0), Z_p(x)) = p

2R²(kxk).

The one-dimensional and two-dimensional densities of χ-squared random field ξ_p(x) are given by (3.1) and (3.3) respectively with β = p/2. We obtain

Ee^(p/2)_k (Z_p(x))) e^(p/2)_m (Z_p(x)) = δ_m^k R^2m(kx − yk).

Note that the positive definiteness is generally not sufficient for a given function to be the covariance function of a χ-squared random field (3.5), since the covariance function of the χ-squared random field (3.5) must be also nonnegative as it follows from (3.7).

B⁰⁰. Consider a mother field of the form

(3.8) Λ(x) = exp {Z(x) − c_Z} , c_Z = log 1 1 −¹_λ
β,

where Z(x), x ∈ Rⁿ is a gamma-correlated HIRF with marginal density (3.1), bivariate density (3.3) and correlation function ρ_Z(r).

The covariance function of the general gamma-correlated random field then takes the form R_Z(r) = β

λ²ρ_Z(r) .

Under condition B⁰⁰, we obtain the following moment generating function (3.9) M (ζ) = E exp {ζ (Z (x) − c_Z)} = e^−cZζ



1 −_λ^ζβ, ζ < λ, λ > 1, β ≥ 1.

and the bivariate moment generating function is

M (ζ1, ζ2; kx1− x₂k) = E exp {ζ₁(Z(x1) − cZ) + ζ2(Z(x2) − cZ)}

= e^{−cZ(ζ1+ζ2)}/

 1 −ζ1

λ − ζ2

λ +ζ1ζ2

λ² (1 − ρZ(τ ))

β

, ζ1 < λ, ζ2 < λ, λ > 1,

Thus, the correlation function of the mother process (3.8) takes the form (3.10) ρΛ(r) =

"

e^−2cZ

1 −_λ² +_λ²2 (1 − ρZ(τ ))β − 1

#, "

e^−2cZ

1 − _λ¹β − 1

# , where c_Z is defined in (3.8). It turns out that, in this case,

log_bEΛ (x)^q= 1 log b

"

−q log 1

1 −_λ¹
β − β log 1 − q

λ



# , and we can formulate the following

Theorem 3.2. Suppose that condition B⁰⁰ holds and λ > 2, and the correlation function 0 < ρ_Z(r) ≤ Ce^−γr, r > 0,

for some positive C and γ.

Then, for any

(β, λ) ∈ L_β,λ: b >

( 1 +

1 λ²

1 −_λ² )^β_n

∩ {λ > 2}, the measures

µ_k→^D µ, k → ∞, and the R´enyi function of µ is given by

T (q) = q 1 + 1

n log blog 1 1 −¹_λ
β

! +

 β

n log b

 log

 1 − q

λ



− 1, where

q ∈ Q = {0 < q < λ, λ > 2} ∩ [1, 2] ∩ L_β,λ. B⁰⁰⁰. Consider a mother field of the form

(3.11) Λ(x) = exp {U (x) − c_U} , x ∈ Rⁿ, where

U (x) = − 1

Z(x), x ∈ Rⁿ,

and Z(x), x ∈ Rⁿ is a gamma-correlated HIRF with marginal density (3.1), bivariate density (3.3) and correlation function ρ_Z(r).

Note that the field U (x) = −_Z(x)¹ , x ∈ Rⁿ has the negative inverted gamma marginal density p(u) = λ^β

Γ(β)(−u)^−β−1e^λ/u, u < 0, whose moment generating function is

(3.12) M (ζ) = Ee^{ζU (x)}= 2λ^β Γ(β)

 ζ λ

β/2

K_β 2√

λp ζ

, ζ > 0,

where Kλ(x) is modified Bessel function of the third kind or McDonald function We have then

(3.13) c_U = − log Γ(β)

2λ^β/2K_β(2√ λ). and

(3.14) log_bEΛ(x)^q = 1 log b

"

−qc_U + log2λ^β/2

Γ(β) + log{q^β2K_β

 2p

qλ

 }

#

, q > 0.

Theorem 3.3. Suppose that condition B⁰⁰⁰ holds with the correlation function 0 < ρZ(r) ≤ Ce^−γr, r > 0,

for some positive C and γ. Then, for any

(β, λ) ∈ L_β,λ: b >







Γ(β)2^β2−1K_β 2√

2λ λ^β/2h

K_β 2√

λi

2







1 n

,

the measures

µ_k→^D µ, k → ∞, and the R´enyi function of measure µ is given by

T (q) = q(1 +cUlog_Γ(β)^2λβ

n log b ) − 1

n log blogn

q^β2K_β 2p

qλo

− (1 + log^2λ_Γ(β)^β/2

n log b ), q ∈ Q = [1, 2] ∩ L_β,λ. 4. Proofs

Proof of Theorem 2.1.

For each Borel B ⊂ Bⁿ, define Y_k:=R

BΛ_k(y)dy. Then it is seen that Y_k, k = 0, 1, 2, . . . , is a martingale. The assumption (i) asserts it converges both in the mean square and almost surely.

Indeed, we have

E(Y_k− Y_k−1)²= Z Z

(y,y⁰)∈B×B

E(Λ_k(y)Λ_k(y⁰))E([1 − Λ_k(y)][1 − Λ_k(y⁰)])dydy⁰, which is equal to

X

k

E(Yk− Y_k−1)² =X

k

Z Z

(y,y⁰)

"_k−1 Y

i=0

1 + σ²ρ bⁱ y − y⁰

 σ²ρ

 b^k

y − y⁰



# dy dy⁰. We claim that, under the exponential delay assumption on ρ, for b > √ⁿ

1 + σ² the above sum is finite.

Write ky − y⁰k = r, and assume the condition (i) that |ρ(r)| ≤ Ce^−γr. For each k ≥ 1, we have the following estimates:

^k−1Y

i=0

|1 + σ²ρ bⁱ

y − y⁰
|

|σ²ρ

 b^k

y − y⁰



|

≤ 

1 + σ²

k

× σ²ρ

 b^k

y − y⁰



≤ 

1 + σ²k

×

σ²Ce^−γbkr .

Using the spherical coordinates to integrate over y, y⁰ together with a change of variable γb^kr = s, we have

X

k

E(Y_k− Y_k−1)² ≤ const ×X

k



1 + σ²k

×R∞

s=0e^−ssⁿ⁻¹ds (γb^k)ⁿ

 , the above sum is finite when bⁿ> 1 + σ². This asserts the claim.

By martingale L² convergence, we see that Y_k converges in mean square to a limit.

Allowing B to vary over the meshes of all dyadic decompositions of Bⁿ in the spherical coordinate, we obtain the limiting measure µ. The measure µ is non-degenerate and Eµ²(Bⁿ) <

∞.

The scaling-invariant property of µ is a consequence of the construction of the µ_k, indeed µk(dy) = Λk(y)dy=^d

Λ(y)Λ(by) · · · Λ(b^ky)dy=^d

Λ(y)Λ(y⁰)Λ(by⁰) · · · Λ(b^k−1y⁰)b⁻ⁿdy⁰, by = y⁰,

where= means the equality in the distribution. Thus the scaling invariance of µ is a consequence^d of letting k → ∞; whenever it is permitted to take the limit, as we have shown above.

By the assumption (ii), both EΛ^q(0) < ∞ and Eµ^q(Bⁿ) < ∞. Assume for simplicity that b = 2 in the following. The scaling-invariant property of µ asserts that

Eµ^q(dy) = 2^−nqEΛ^q(0)Eµ^q(2ⁿdy).

Summing ∆ over the class Dm of all members in the m-th level dyadic decomposition, and making backward recursion, we have

X

∆∈Dm

Eµ^q(∆) = Eµ^q(Bⁿ)2^nm2^−nmq(EΛ^q(0))^m,

then, R´enyi function has the claimed expression with b = 2; we can use base b for any b > 1, by using decompositions based on b⁻¹-adic base. 

Remark 4.1. The above proof on the martingale L² convergence, suppose n being reduced to be 1, is complementary to that in Mannersalo et al (2002, pp. 892-3); in which the authors allow the weaker correlation decay ( power decay), yet the positivity of the correlation is essential there, and moreover the arguments there are restricted to the 1-parameter (the process) case.

Proof of Theorem 3.1 We consider the random field

ηq(x) = Λ^q(x) = Gq ˜Y (x)

 , where

Gq(u) = exp n

quσX −q 2σ_X²

o ,

and the Gaussian field ˜Y (x) = Y (x)/σ_X.has the covariance function ρ_Y(kxk).

Let H_k(u), k = 0, 1, 2, ..., be Hermite polynomials with the leading coefficient equal to one, which form a complete system of orthogonal polynomials in the Hilbert space L₂(R, ϕ(u)du), where

ϕ(u) = 1

√2πe^−u22 , u ∈ R.

Then G_q(u) has the expansion

G_q(u) =

∞

X

k=0

(C_k(q)/k!) H_k(u),

Ck(q) = Z

R

Hk(u)

√

2π e^{quσX−q2σ2y−u22} du,

∞

X

t=0

C_k²(q) k! = 1

√2π Z

R

e²(^qσX−^q₂σ_y²)^−u22 du < ∞.

Taking into account the well-known formula E



Hk ˜Y (x))



Hj ˜Y (y)



= k!δ_k^jR^k(kx − yk), k, j > 1,

where δ^j_k is the Kronecker symbol, we obtain R_Λ(t, s) = Cov (Λ^q(x), Λ^q(y)) =

∞

X

k=1

C_k²(q) k!
ρ^k_Y(kx − yk), and

Ce^−γrC₁²(q) ≤ R_Λ(r) 6 Ce^−γr

∞

X

k>1

C_k²(q) k!.

This completes the proof.  Proof of Theorem 3.2

Similar to the proof of Theorem 3.1, we can show that the process Z(x) has continuous sample paths. We now consider the process Λ(x) as a non-linear transformation of a gamma-correlated process Z(x), that is,

Λ(t) = G_q(Z(x)) , G_q(u) = exp {qu − qc_Z} . Let {ek(u)}^∞_k=0 be generalized Laguerre polynomials; then, for λ > 2,

Gq(u) ∈ L2((0, ∞) , p(u)du) , q ∈ Q = Q = {0 < q < λ, λ > 2}, and we have the expansion

Gq(u) =

∞

X

k=0

C_k(q)e_k(u), C_k(q) = Z ∞

0

e_k(u)e^−qcXe^{−qu(λ−2)}u^β−1du, where

C_k(q) = e^−qcXλ^β Γ(β)

Z ∞ 0

e^{−qu(λ−1)}u^β−1e_k(u)du, k = 0, 1, . . . and

Ce^−γrC₁²(q) ≤ σ²_Λρ_Λ(τ ) ≤ Ce^−γr

∞

X

k=1

C_k²(q).

This completes the proof.  Proof of Theorem 3.3

The proof again follows the main steps of that of Theorem 3.1, with necessary modifications.

We only need to note that, in this case,

Λ(x) = G (Z (x)) , G(u) = e^−u1−cU, where Z(x) is a gamma-correlated process defined in A⁰⁰. Then

EG²(Z(x)) = λ^β Γ(β)

Z ∞ 0

e^−2x−2cXx^β−1e^−λxdx

= λ^β Γ(β)e^−2cX

Z ∞ 0

e⁻

√ 2λ



√x 2/λ+

√

2/λ x



x^β−1dx

= e^−2cXα^β2^β/2 Γ(β) K_β

√

2λ

< ∞ for all β > 0, λ > 0.

We have again

Ce^−γrC₁²(q) ≤ σ²_ΛρΛ(τ ) ≤ Ce^−γr

∞

X

k=1

C_k²(q),

where now

Ck(q) = e^−qcXλ^β Γ(β)

Z ∞ 0

e^{−q(1u−cU)}u^β−1ek(u)du.

This completes the proof. 

5. Conclusion and Perspective

In this paper, we address the R´enyi function, which is a central core of Multifractal Analy- sis, for the infinite-products generated by several random fields on the multi-dimensional, 3- dimensional in particular, sphere. The scenarios for the random fields presented in this paper are log-normal, log-gamma, and log-negative-inverted-gamma.

The main motivation of this paper is to provide scenarios which may test the possible non- Gaussianity for the statistical distribution of Cosmic Microwave Background Radiation(CMBR);

a challenging issue of cosmology presented for example in the review paper by Marinucci (2004) and the monograph by Marinucci and Peccati (2011, Chapter 1, §1.2).

There are statistical calibrations which are not able to present in this paper. The Nine- Year Wilkinson Microwave Anisotropy Probe (WMAP) data released at December 20, 2012 ( arXiv:1212.5226 ) could be an extremely valuable database for future statistical testing of the models.

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School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4AG, UK. E-mail:

LeonenkoN@Cardiff.ac.uk

Department of Mathematics, Chinese University of Hong Kong, Shatin N.T. China. On leave from National Taiwan University. E-mail: shiehnr@ntu.edu.tw URL: http://www.math.ntu.edu.tw/~shiehnr/