#
Information Properties of a Random Variable Decomposition through Lattices^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Lattices, Wrapping and Quantization

#### 2.1. Lattices and Fundamental Domains

#### 2.2. Wrapping and Quantization

**Example**

**1.**

**Example**

**2.**

- $E\left[X\right]=E\left[{X}_{\pi}\right]+E\left[{X}_{\mathcal{Q}}\right]$;
- $Var\left[X\right]=Var\left[{X}_{\pi}\right]+Var\left[{X}_{\mathcal{Q}}\right]+Cov[{X}_{\pi},{X}_{\mathcal{Q}}]+Cov[{X}_{\mathcal{Q}},{X}_{\pi}]$,

## 3. Information Properties

#### 3.1. Information-Theoretic Measures

**Proposition**

**1.**

**Proof.**

**Lemma**

**1.**

**Proof.**

**Proposition**

**2.**

- 1.
- If $\mathcal{D}$ is connected, and p is continuous and Riemann-integrable, then ${lim}_{\alpha \to 0}I({X}_{\pi};{X}_{\mathcal{Q}})=0$.
- 2.
- If 0 is an interior point of $\mathcal{D}$, then ${lim}_{\alpha \to +\infty}I({X}_{\pi};{X}_{\mathcal{Q}})=0$.

**Proof.**

**Example**

**3.**

**Example**

**4.**

#### 3.2. Fisher Information

**Proposition**

**3**

**.**Let X be a random variable distributed according to a distribution parametrized by θ, and $G\left(\theta \right)$ its information matrix. The following hold.

- 1.
**Monotonicity:**if $F:\mathcal{X}\to \mathcal{Y}$ is a measurable function (i.e., a statistic) and ${G}_{F}\left(\theta \right)$ is the information matrix of $F\left(X\right)$, then ${G}_{F}\left(\theta \right)\u2aafG\left(\theta \right)$, with equality if, and only if, F is a sufficient statistic for θ.- 2.
**Additivity:**if $X,Y$ are independent random variables, then the joint information matrix satisfies ${G}_{(X,Y)}\left(\theta \right)={G}_{X}\left(\theta \right)+{G}_{Y}\left(\theta \right)$.

**Example**

**5.**

## 4. A Generalization to Topological Groups

**Theorem**

**1.**

**Example**

**6.**

**Example**

**7.**

**Example**

**8.**

**Example**

**9.**

**Example**

**10.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Example of zero-mean Gaussian distributions and their corresponding wrapped, quantized and product distributions, with $\Lambda =\mathbb{Z}$ and $\mathcal{D}=[-\frac{1}{2},\frac{1}{2}[$ for different variances: ${\sigma}^{2}=0.25$ (blue), ${\sigma}^{2}=1$ (orange), ${\sigma}^{2}=4$ (green).

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**MDPI and ACS Style**

Meneghetti, F.C.C.; Miyamoto, H.K.; Costa, S.I.R. Information Properties of a Random Variable Decomposition through Lattices. *Phys. Sci. Forum* **2022**, *5*, 19.
https://doi.org/10.3390/psf2022005019

**AMA Style**

Meneghetti FCC, Miyamoto HK, Costa SIR. Information Properties of a Random Variable Decomposition through Lattices. *Physical Sciences Forum*. 2022; 5(1):19.
https://doi.org/10.3390/psf2022005019

**Chicago/Turabian Style**

Meneghetti, Fábio C. C., Henrique K. Miyamoto, and Sueli I. R. Costa. 2022. "Information Properties of a Random Variable Decomposition through Lattices" *Physical Sciences Forum* 5, no. 1: 19.
https://doi.org/10.3390/psf2022005019