Ergodicity and Gaussianity for Spherical Random Fields

TL;DR

This paper explores the connection between ergodicity and Gaussianity in isotropic spherical random fields, showing they are often equivalent in the high-frequency limit, with implications for CMB data analysis.

Contribution

It demonstrates the equivalence of ergodicity and asymptotic Gaussianity for high-frequency spherical fields under broad conditions.

Findings

Sample angular power spectrum converges to the true value if and only if the field is asymptotically Gaussian.

Results provide insights into the role of Cosmic Variance in CMB analysis.

Establishes conditions linking ergodicity and Gaussianity in spherical random fields.

Abstract

We investigate the relationship between ergodicity and asymptotic Gaussianity of isotropic spherical random fields, in the high-resolution (or high-frequency) limit. In particular, our results suggest that under a wide variety of circumstances the two conditions are equivalent, i.e. the sample angular power spectrum may converge to the population value if and only if the underlying field is asymptotically Gaussian, in the high frequency sense. These findings may shed some light on the role of Cosmic Variance in Cosmic Microwave Background (CMB) radiation data analysis.

Equations

$$T(x)=\sum _{l=0}^{\infty }\sum _{m=-l}^l{a}_{lm}{Y}_{lm}(x)\text{ , }x\in S^2\text{ .}$$

$$\left\{Y_{lm}:l\ge 0,m=-l,...,l\right\}$$

$$T_l\left(x\right):=\sum _{m=-l}^l{a}_{lm}{Y}_{lm}\left(x\right)\text{, }x\in S^2\text{, }l\ge 0\text{,}$$

$${\hat{C}}_l=\frac{1}{2l+1}\sum _{m=-l}^l|a_{lm}{|}^2\text{ , }l=0,1,2,...$$

$$ET{\left(x\right)}^2=\sum _{l\ge 0}\frac{(2l+1)}{4\pi }{C}_l<\infty \text{,}$$

$$\sum _{l\ge 0}E{\hat{C}}_l=\sum _{l\ge 0}{C}_l<\infty \text{.}$$

$$P\left\{\underset{l\to \infty }{\lim }\sup {\hat{C}}_l>\epsilon \right\}\le \underset{l\to \infty }{\lim }\sum _{\ell \ge l}P\left\{{\hat{C}}_{\ell }\ge \epsilon \right\}\le \underset{l\to \infty }{\lim }\frac{1}{\epsilon }\sum _{\ell \ge l}{C}_{\ell }=0,$$

$${\tilde{C}}_l=\frac{1}{2l+1}\sum _{m=-l}^l\frac{|a_{lm}{|}^2}{C_l}=\frac{{\hat{C}}_l}{C_l},l\ge 0\text{.}$$

$$\underset{l\to \infty }{\lim }E{\left\{{\tilde{C}}_l-1\right\}}^2=\underset{l\to \infty }{\lim }E{\left\{\frac{{\hat{C}}_l}{C_l}-1\right\}}^2=0\text{.}$$

$${\tilde{C}}_l=\frac{1}{2l+1}\sum _{m=-l}^l\frac{|a_{lm}{|}^2}{C_l}\to 1,$$

$$E{\left\{{\tilde{C}}_l-1\right\}}^2$$

$$\sqrt{\frac{2l+1}{2}}\left\{\frac{{\hat{C}}_l}{C_l}-1\right\}=\sqrt{\frac{2l+1}{2}}\left\{{\tilde{C}}_l-1\right\},$$

$$d_{TV}\left(X,Y\right)=\underset{A}{\sup }\left|P\left(X\in A\right)-P\left(Y\in A\right)\right|\text{,}$$

$$d_{TV}\left(\sqrt{\frac{2l+1}{2}}\left\{\frac{{\hat{C}}_l}{C_l}-1\right\},N(0,1)\right)\le \sqrt{\frac{8}{2l+1}}\text{,}$$

$$\sqrt{\frac{2l+1}{2}}\left\{{\tilde{C}}_l-1\right\}\stackrel{law}{\to }N\left(0,1\right).$$

$$\sqrt{\frac{2l+1}{2}}\left\{\frac{{\hat{C}}_l}{C_l}-1\right\}$$

$$cu{m}_4\left\{\frac{\sqrt{2}}{\sqrt{(2l+1)}}\left[\sum _{m=1}^{2l+1}\frac{(x_{lm}^2-1)}{2}\right]\right\}=\frac{12}{2l+1}\text{,}$$

$$d_{TV}(F_q,N(0,1))\le 2\sqrt{\frac{q-1}{3q}}\sqrt{cu{m}_4\left(F_q\right)}\text{ .}$$

$$\frac{T_l(x)}{\sqrt{Var\left\{T_l(x)\right\}}}\stackrel{law}{\to }N(0,1)\text{ , as }l\to \infty \text{, }$$

$$cum\left\{a_{l_1{m}_1},a_{l_2{m}_2},a_{l_3{m}_3},a_{l_4{m}_4}\right\}$$

$$=$$

$$C_{l_{1}m_{1}l_{2}m_{2}}^{LM}=(-1)^{l_{1}-l_{2}+m_{3}}\sqrt{2L+1}\left(\begin{array}[]{ccc}l_{1}&l_{2}&L\\ m_{1}&m_{2}&M\end{array}\right)\text{ .}$$

$$T_l(\vartheta ,\phi )\stackrel{law}{=}{T}_l(\overline{N})=\sum _{lm}{a}_{lm}{Y}_{lm}(\overline{N})\stackrel{law}{=}{a}_{l0}\sqrt{\frac{2l+1}{4\pi }}\text{ ,}$$

$$H_q(T(x)):=T_q(x)=\sum _{l=0}^{\infty }{T}_{l;q}(x)\text{, }x\in S^2\text{, }q\ge 2\text{ ,}$$

$$T_{l;q}(x)=\sum _{m=-l}^l{a}_{lm;q}{Y}_{lm}\left(x\right)$$

$$C_{l;q}=q!\sum _{l_1,...,l_q=0}^{\infty }{C}_{l_1}\cdot \cdot \cdot C_{l_q}\frac{4\pi }{2l+1}\left\{\prod _{i=1}^q\frac{2l_i+1}{4\pi }\right\}\sum _{L_1...L_{q-2}}{\left\{C_{l_1,0;...;l_{q}0}^{L_1,L_2,...,L_{q-2},l;0}\right\}}^2\text{,}$$

$$C_{l_1,m_1;...;l_p{m}_p}^{{\lambda }_1,{\lambda }_2,...,{\lambda }_{p-1};\mu }:=\sum _{{\mu }_1=-{\lambda }_1}^{{\lambda }_1}...\sum _{{\mu }_{p-2}=-{\lambda }_{p-2}}^{{\lambda }_{p-2}}{C}_{l_1,m_1,l_2,m_2}^{{\lambda }_1,{\mu }_1}{C}_{{\lambda }_1,{\mu }_1;l_3,m_3}^{{\lambda }_2,{\mu }_2}\cdot \cdot \cdot C_{{\lambda }_{p-2},{\mu }_{p-2};l_p,m_p}^{{\lambda }_{p-1},\mu }$$

$$w_{1l}(L):={\left(C_{l0l0}^{L0}\right)}^2\text{ and }{w}_{2l}(L)=\frac{(2L+1)}{(2l+1{)}^2}\text{,}$$

$$\sum _{L=0}^{2l}{w}_{1l}(L)=\sum _{L=0}^{2l}{w}_{2l}(L)=1\text{ .}$$

$$\underset{l\to \infty }{\lim }\sum _{L=0}^{2l}{w}_{1l}(L)\frac{{\mathcal{T}}_{ll}^{ll}(L;q)}{C_{l;q}^2}=0.$$

$$\underset{l\to \infty }{\lim }\sum _{L=0}^{2l}{w}_{2l}(L)\frac{{\mathcal{T}}_{ll}^{ll}(L;q)}{C_{l;q}^2}=0.$$

Full text

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Keywords and Phrases: Spherical Random Fields, High-Frequency Gaussianity, High-Frequency Ergodicity, Gaussian Subordination, Clebsch-Gordan Coefficients, Cosmic Variance

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PACS: 02.50-r, 98.70-Vc, 98.80-k

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AMS Classification: 60G60; 60G15, 60K40, 62M15, 42C10

1 Introduction and background

1.1 Overview

The usual framework for proving asymptotic results in probability (for instance, central limit theorems or laws of large numbers) lies within the so-called large sample paradigm, according to which more and more (independent or weakly dependent) random variables are generated, and the limiting behaviour of some functionals of these variables (e.g., averages or empirical moments) is studied.

Physical applications, however, are prompting the development of a stochastic asymptotic theory of a rather different nature, where the indefinite repetition of a single experience is no longer available, and one relies instead on observations of the same (fixed) phenomenon with higher and higher degrees of resolution.

One crucial instance of this situation appears when dealing with the statistical analysis of random fields indexed by compact manifolds, the quintessential example being provided by the case of the sphere $S^{2}$. Indeed, we are especially concerned with issues arising from the analysis of the Cosmic Microwave Background (CMB) radiation, a theme which is currently at the core of physical and cosmological research, see for instance [12, 20] for textbook references, and [38, 39, 21] for further discussions around the latest experimental data.

It is well-known that the CMB is a relic electromagnetic radiation providing a snapshot of the Universe at the so-called age of recombination, i.e. at the era when electrons in the primordial fluid arising from the Big Bang were captured by protons to form stable hydrogen atoms. Since the cross-section of hydrogen atoms is much smaller than for free electrons, after recombination photons can be viewed as diffusing freely across the Universe (to first order approximations). According to the latest experimental evidence, this has occurred some $3.7\times 10^{5}$ years after the Big Bang, i.e. 13.7 billion years from the current epoch. Several experiments have been devoted to collecting extremely refined observations of the CMB, the leading role being played by the currently ongoing NASA mission WMAP (launched in 2001, see http://map.gsfc.nasa.gov/) and the ESA mission Planck, which is just now starting to operate after the launch on May 14, 2009 (see http://www.sciops.esa.int/).

From a mathematical point of view, the CMB can be regarded as a single realization of an isotropic, zero-mean, finite variance spherical random field, for which the following spectral representation holds (see e.g. [1] or [22])

[TABLE]

Here, the collection

[TABLE]

stands for the usual triangular array of spherical harmonics, which are well-known to provide a complete orthonormal system for the $L^{2}(S^{2})$ space of square-integrable functions (with respect to Lebesgue measure) on the sphere – see [40, 43, 44]. In a loose sense, we can say that the frequency parameter $l$ is related to a characteristic angular scale, say $\vartheta_{l}$, according to the relationship $\vartheta_{l}\simeq\pi/l$. The (random) triangular array of spherical harmonic coefficients $\left\{a_{lm}:l\geq 0,\text{ \ }m=-l,...,l\right\}$ are such that $Ea_{lm}=0$ and $Ea_{lm}\overline{a}_{l^{\prime}m^{\prime}}=C_{l}\delta_{l}^{l^{\prime}}\delta_{m}^{m^{\prime}}$, the bar denoting complex-conjugation and $\delta_{a}^{b}$ indicating the Kronecker delta function. The non-negative sequence $\left\{C_{l}:l\geq 0\right\}$ (not depending on $m$ – see [29] as well as the forthcoming section) is the angular power spectrum of the spherical field (see for instance [2, 3]).

As recalled above, our work deals with asymptotic issues, where the expression “asymptotic” has to be understood in the high-resolution (or high-frequency) sense. This means that we focus on the behaviour of the Fourier components

[TABLE]

associated with a fixed spherical field, as the frequency $l$ grows larger and larger (plainly, each $T_{l}$ is the projection of the field $T$ into the orthogonal subspace of $L^{2}\left(S^{2}\right)$ spanned by the spherical harmonics $\left\{Y_{lm}:m=-l,...,l\right\}$). Note that this is the typical framework faced by experimentalists handling satellite missions as those mentioned above. Indeed, these missions are observing the same (unique) realization of our Universe on the so-called last scattering surface; more recent and more sophisticated experiments are then characterized by higher and higher frequencies (smaller and smaller scales) being observed. For instance, for the pioneering CMB mission COBE in 1989-1992 (which led to the Nobel Prize for Smoot and Mather in 2006) only frequencies in the order of a few dozens were recorded (i.e., scales of several degrees), a limit which was raised to few hundreds by WMAP (i.e., approximately a quarter of degree) and is expected to grow to a few thousands with Planck (i.e., a few arcminutes).

The principal goal of this paper is to enlighten some partial new connections between two high-resolution characterizations of spherical fields, that is, ergodicity and asymptotic Gaussianity. Roughly speaking (formal details are given in the forthcoming Sections 1.2 and 1.3), one says that the spherical field $T$ is ergodic if the empirical version of the power spectrum of $T$ (see formula (3) below) can be used as a consistent estimator of the sequence $\left\{C_{l}\right\}$ (at least for high values of $l$). On the other hand, we say that $T$ is asymptotically Gaussian, whenever suitably normalized versions of the frequency components of $T_{l}$ exhibit Gaussian fluctuations for high values of $l$. As discussed below, these two notions are tightly connected whenever one deals with fields having an isotropic (or, equivalently, rotationally-invariant) law.

**Remark. **For the rest of the paper, every random object is defined on a suitable (common) probability space $\left(\Omega,\mathcal{F},P\right).$

1.2 High-frequency ergodicity

In what follows, we shall consider a real-valued random field $T=\left\{T\left(x\right):x\in S^{2}\right\}$ indexed by the sphere $S^{2}$. The random field $T$ satisfies the following basic assumptions: (i) the law of $T$ is isotropic, that is, $T$ has the same law as $x\mapsto T\left(gx\right)$ for every rotation $g\in SO\left(3\right)$ (here, we select the canonical action of $SO\left(3\right)$ on $S^{2}$); (ii) $T$ is square-integrable and centered. Under assumptions (i)-(ii), the harmonic expansion (1) takes place, both in $L^{2}\left(P\right)$ (for fixed $x$) and in the product space $L^{2}\left(\Omega\times S^{2},P\otimes d\lambda\right)$