How to show ergodicity on this probability measure.

Question

I am looking at a way of describing an infinite checkerboard where in each tile a random constant matrix of size $d \times d$ is given.

Step 1 : introduction

Let $z$ a random vector with uniform distribution in $[-\frac{1}{2},\frac{1}{2}]^d$ and $(a_k)_{k \in \mathbb{Z}^d}$ a family of independent, identically distributed random matrices in $\Omega_0:=\mathbb{R}^{d^2}$ (i.e in more usual probabilistic terms, if we note $(\bar{\Omega},A,\mu)$ a probability space, it is a family $(X_k)_{k \in \mathbb{Z}^d}$ of independent, identically distributed random variables from $\bar{\Omega}$ to $\mathbb{R}^{d^2}$).

Our goal is to define a random variable $a \in \Omega:=\{ a : \mathbb{R}^d \rightarrow \mathbb{R}^{d^2}, \quad \text{a is measurable} \}$ such that $a(x)$ is worth a random matrix on each tile of the checkerboard.

Step 2 : description of one tile

We introduced the following space :

$$(\Omega_0,\mathcal{F}_0,\mathbb{P}_0)$$ where $\mathcal{F}_0$ is the Borel $\sigma$-algebra on $\Omega_0$, and $\mathbb{P}_0$ describes the distribution on a single tile i.e for any geometry paver $[\alpha_1,\beta_1] \times...\times [\alpha_{d^2},\beta_{d²}] \subset \mathbb{R}^{d^2}$ :

$$\mathbb{P}_0([\alpha_1,\beta_1] \times...\times [\alpha_{d^2},\beta_{d²}]) \text{ gives the probability chance that $a_k$ } \in [\alpha_1,\beta_1] \times...\times [\alpha_{d^2},\beta_{d²}].$$

Step 3 : description of the checkerboard

We then introduce the following product probability space :

$$(\Omega',\mathcal{F}',,\mathbb{P}')=(\Omega_0^{\mathbb{Z^d}} \times \Box, \ \mathcal{F}_0^{\otimes\mathbb{Z}^d} \times \mathcal{B}(\Box), \ \mathbb{P}_0^{\otimes\mathbb{Z}^d} \otimes \lambda)$$ where $\lambda$ is the Lebesgue measure on $\Box=[-\frac{1}{2},\frac{1}{2}]^d$ and $\mathcal{B}(\Box)$ is a the Borel tribe on $\Box \subset \mathbb{R}^d$.

We also introduce the following map :

$$\pi : \Omega' \rightarrow \Omega, \quad \pi((a_k)_{k \in \mathbb{Z}^d},z):= \sum_{k \in \mathbb{Z}^d} \mathbb{1}_{k+z+\Box}(\cdot) a_k$$

with $\Omega$ define in the introduction. Finaly, we can equip $\Omega$ with a canonic tribe $\mathcal{F}$ and a probability $\mathbb{P}$ defines as the push-forward of $\mathbb{P}'$ under $\pi$ i.e :

$$\mathbb{P}(B)= \mathbb{P}'(\pi^{-1}(B)), \quad \forall B \in \mathcal{F}.$$

Step 4 : interpretation

The function $f : x \mapsto \sum_{k \in \mathbb{Z}^d} \mathbb{1}_{k+z+\Box}(x) a_k$ describe the checkerboard where :

• $f$ is worth the constant matrix $a_k$ on the tile number $k$
• the vector $z$ describe the center of the checkerboard. If $z=0$, then the first tile is $[-\frac{1}{2},\frac{1}{2}]$, centered in $0$.

We have define a probability $\mathbb{P}$ that describes the chance that $a : x \mapsto \sum_{k \in \mathbb{Z}^d} \mathbb{1}_{k+z+\Box}(x) a_k \in B$, for any set $B$ of $\mathcal{F}$, so we have define a law for the random variable $a$ presented in the introduction.

Step 5 : Question time

As I am quite new to probability, I'm not sure to completely understand the above construction.

1. First of all, do you have any remarks or thoughts that could help understand this probability model ?

2. I would like to show that the measure $\mathbb{P}$ is ergodic and stationary. I have succeed to prove the stationarity i.e :

for all $z \in \mathbb{R}^d$ and random real variables $f$ in $L^1(\Omega,\mathbb{P})$, we have :

$$\mathbb{E}[f \circ \tau_z]=\mathbb{E}[f]$$ where $\mathbb{E}$ is the expectation on $(\Omega,\mathcal{F},\mathbb{P})$ and $\tau_z$ is the shift operator defined by $\tau_z(a)=a(\cdot +z)$ from $\Omega$ to $\Omega$.

However, I am strugling to prove the ergodicity, defined by :

for any measurable set $E \subset \Omega$ such as $\tau_zE=E$ for all $z \in \mathbb{R}^d$, then $\mathbb{P}(E)=0 \text{ or } 1$.

I know that there exist another caracterization of ergodicity using Birkhoff's ergodic theorem but I would rather not use it for now.

Update 1

I changed my mind about using Birkoff's theorem, so now it would be enough to prove that for any random variables $f \in L^1(\Omega,\mathbb{P})$ we have :

$$\underset{R \rightarrow +\infty}{\lim} \frac{1}{|R\Box|} \int_{R \Box} f(\tau_z a) \mathrm{d}z = \mathbb{E}[f], \quad \mathbb{P}-a.e \ \ a \in \Omega$$ and it will give me the ergodicity I desire.

Update 2

I've proposed a solution using Kolmogoroff's law inspired from sand piles problem but I'm unsure if it is correct or not. If anyone wants to give me his opinion on this solution I'll be happy to hear it.

Answer 1

I think I might have found a beginning of an answer for the ergodicity of my checkboard problem, however I think it still need some verification.

The idea is to use the law 0-1 of Kolmogoroff.

Let $E \in \mathcal{F}$ measurable. We construct the following tribes :

$$\mathcal{F}_n = \sigma(\{ a(x) \ | \ |x| \geq n \}), \quad \forall n>0$$ which verifies $\mathcal{F}_0 \supset \mathcal{F}_1 \supset \mathcal{F}_2 \dots$, so that we have $\bigcup_{i=0}^n \mathcal{F}_i = \mathcal{F}_n.$ We then pose the following asymptotic tribe $$\mathcal{F}_\infty= \bigcap_{n=0}^{+ \infty} \mathcal{F}_n.$$

Now, since we have $\tau_z E= E, \forall z \in \mathbb{R}^d$ because of the ergodicity hypothesis, I believe it gives us that $$E \in \mathcal{F}_n, \quad \forall n>0$$ since $E=\{a(\cdot) \ | \ a \in E \}= \{ a(\cdot + z) \ | \ a \in E \} $ for any vector $z \in \mathbb{R}^d$. This proves that $E \in \mathcal{F}_\infty$.

Then using Kolmogoroff's law it gives us that $\mathbb{P}(E)=0 \text{ or }1$.

However I feel quite uncomfortable with this prove when I construct the tribes $\mathcal{F}_n$ since I'm not really sure $\{ a(x) \ | \ |x|> n\}$ is a part of $\Omega$. There is probably a more meaningful way of writing this...