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Empirical phase measures and Fourier order statistics

Definition

Let $n \geq 1$ and let $θ_{1}, \dots, θ_{n}$ be real representatives of phases, where a phase is an angle modulo $2 π$ . Put $z_{j} = e^{i θ_{j}}$ and $S^{1} = {z \in C : ∣ z ∣ = 1}$ , with its Borel sets, the sigma-algebra generated by its open subsets. The empirical phase measure is $ν_{n} = n^{- 1} \sum_{j = 1}^{n} δ_{z_{j}}$ : for a Borel set $A$ , $ν_{n} (A) = n^{- 1} # {j : z_{j} \in A}$ . Here $δ_{z} (A)$ is $1$ if $z \in A$ and $0$ otherwise. Repeated phases are counted with their multiplicities. This is a Borel probability measure: a countably additive nonnegative set function of total mass one.

For an integer $k$ , the $k$ th Fourier moment, also called a Fourier order statistic here, is $m_{k} (ν_{n}) = \int_{S^{1}} z^{k} d ν_{n} (z) = n^{- 1} \sum_{j = 1}^{n} e^{i k θ_{j}}$ . Negative powers use $z^{- 1} = \overline{z}$ on $S^{1}$ . In particular, $m_{1}$ is the mean phasor; its modulus is a scalar order parameter measuring the magnitude of that mean.

A continuum description may instead specify a Borel probability measure on the phase space, or a probability density with respect to a stated reference measure space $(X, A, λ)$ : a nonnegative measurable function $f : X \to [0, + \infty]$ with $\int_{X} f d λ = 1$ , assigning each $A \in A$ the mass $ν_{f} (A) = \int_{A} f d λ$ . Here $A$ is a sigma-algebra on $X$ and $λ$ is a measure on it; for phase densities, take $X = S^{1}$ and $A$ its Borel sigma-algebra. The resulting $ν_{f}$ is a probability measure. For probability measures $μ_{n}, μ$ on a metric space, weak convergence $μ_{n} \Rightarrow μ$ means that $\int f d μ_{n} \to \int f d μ$ for every bounded continuous real-valued test function $f$ .

For a frequency probability density $g$ on $R$ , symmetry about $a$ means $g (a + x) = g (a - x)$ for all $x$ ; unimodality about $a$ means $g$ is nondecreasing on $(- \infty, a]$ and nonincreasing on $[a, \infty)$ . These are additional assumptions when used, not consequences of the empirical measure. This definition asserts neither an interchange of the population-size limit $n \to \infty$ and a long-time limit $t \to \infty$ , nor the existence of a continuum phase transition, meaning a qualitative change of macroscopic behavior as a model parameter varies.

Facts & Assumptions

[F1]

The phasor $e^{i u}$ has modulus one, is unchanged by adding an integer multiple of $2 π$ to $u$ , and satisfies the exponential addition rule for real arguments (Phasors, trigonometric identities and periodic derivatives).

[F2]

On any measurable space, the empirical set function of finitely many points is their average of Dirac set functions and is a probability measure (Empirical measure of observed units, The empirical measure is a probability measure, The Dirac set function at a point, Probability measures and probability spaces).

[F3]

The nonnegative integral is the supremum of integrals of nonnegative simple measurable functions below the integrand; a simple integral is the finite sum of each value times the measure of its level set. Real integrals use positive and negative parts, and complex integrals use real and imaginary parts (The nonnegative Lebesgue integral, The integral of a nonnegative simple function, Integrable real and complex functions, and their integrals).

[F4]

Weak convergence of Borel probability measures on a metric space is convergence of their integrals against every bounded continuous real-valued function (Weak convergence of borel probability measures).

[F5]

On any measure space $(X, A, λ)$ , a nonnegative measurable function $f : X \to [0, + \infty]$ defines a measure by $ν_{f} (A) = \int_{A} f d λ$ for $A \in A$ (The indefinite integral of a nonnegative measurable function is a measure).

[F6]

A sigma-algebra contains the ambient set and is closed under complements and countable unions; a measurable space is a set equipped with a sigma-algebra. A measure is a nonnegative countably additive set function vanishing at the empty set, and a measure space equips a measurable space with a measure. The Borel sigma-algebra is generated by the open sets (Sigma-algebras, Measurable spaces and measurable sets, Measures on sigma-algebras, Measure spaces, The Borel sigma-algebra of a topological space).

[F7]

Measurability between measurable spaces means that preimages of measurable sets are measurable. For an extended-real-valued function it is equivalently enough that every strict real superlevel set be measurable. A nonnegative simple measurable function has finite range, or equivalently is a finite nonnegative linear combination of indicators of pairwise disjoint measurable sets. Positive and negative parts of a measurable real function are measurable (A measurable function between measurable spaces, Extended-real-valued measurable functions, Nonnegative simple measurable functions, Closure properties of measurable functions used by the integral).

[F8]

A continuous map between topological spaces has Borel preimages of Borel sets (A continuous map has Borel preimages of Borel sets).

[F9]

For a nonnegative measurable function, integration over a measurable set is defined by $\int_{A} f d λ = \int f χ_{A} d λ$ (Integral over a measurable subset).

Justification

Given: An integer $n \geq 1$ , real phase representatives $θ_{1}, \dots, θ_{n}$ , and an integer $k$ . For the density construction, a stated measure space $(X, A, λ)$ and a nonnegative measurable $f$ with $\int_{X} f d λ = 1$ .

Goal: Verify the empirical probability measure and moment formulas, their independence of phase representatives, and the probability measure defined by a normalized density.

1.1 By [F1], every $z_{j}$ belongs to $S^{1}$ and is unchanged by changing $θ_{j}$ by an integer multiple of $2 π$ . Applying [F2] to the Borel measurable space $S^{1}$ gives the stated probability measure and set formula, including repeated points. Thus the measure is independent of representatives. [F1, F2, F6, given]

2.1 Let $E$ be the finite set of distinct $z_{j}$ and let $h : S^{1} \to [0, \infty)$ be bounded and Borel measurable. For each $a \in E$ , the singleton ${a}$ is Borel and has mass equal to its multiplicity divided by $n$ ; the complement of $E$ has mass zero. Every nonnegative simple function $s \leq h$ therefore has integral at most $\sum_{a \in E} h (a) ν_{n} ({a})$ . The simple function equal to $h (a)$ at each $a \in E$ and zero elsewhere attains this bound. It has finite range and measurable level sets by [F6] and [F7]. Hence [F3] gives $\int h d ν_{n} = n^{- 1} \sum_{j} h (z_{j})$ . For a bounded complex Borel function, its real and imaginary parts and its modulus are Borel measurable: their coordinate and modulus maps are continuous, and preimages compose, so [F7] and [F8] apply. The modulus and the positive and negative parts of each component are bounded nonnegative measurable functions by [F7], so their integrals are finite by the preceding evaluation formula. Applying this to the positive and negative parts of the real and imaginary parts of a bounded complex Borel function gives the same evaluation formula for that function. [F2, F3, F6, F7, F8, step 1.1, algebra]

3.1 The function $z \mapsto z^{k}$ is bounded and continuous on $S^{1}$ , including for negative $k$ , and $z_{j}^{k} = e^{i k θ_{j}}$ by [F1] and the addition rule. By [F8] it is Borel measurable, so step 2.1 gives the displayed moment formula. The weak-convergence wording is exactly [F4]; it makes no assertion that any such limit or interchange of limits exists. [F1, F4, F8, step 2.1, algebra]

4.1 For the stated reference measure space and nonnegative measurable $f$ , [F5] makes $ν_{f}$ a measure on $(X, A)$ . Its total mass is $ν_{f} (X) = \int_{X} f d λ = 1$ by the density normalization. Hence it is a probability measure by [F2]. This applies in particular to the Borel phase space $S^{1}$ with any stated reference measure $λ$ . [F2, F5, F6, F7, F9, given] ∎

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