Dyadic Martingales

Setting for this page

Throughout this section we will restrict our attention to the probability space $(\Omega ,\Sigma ,\mathbb{P})=([0,1],\mathcal{B}([0,1]),\lambda ),$ where $\mathcal{B}([0,1])$ is the Borel sigma-algebra and $\lambda$ is the Lebesgue measure. For $k\in \mathbb{N}$, define ${\mathcal{D}}_k:=\left\{\left[\frac{i}{2^k},\frac{i+1}{2^k}\right):i=0,\dots ,2^k-1\right\},$ and ${\mathcal{F}}_k:=\sigma ({\mathcal{D}}_k).$ The sets in ${\mathcal{D}}_k$ are called the dyadic intervals of $[0,1]$ at scale $k$. They have the properties that the cubes in ${\mathcal{D}}_k$ partition $[0,1]$ and each dyadic interval in ${\mathcal{D}}_k$ uniquely contains two dyadic “children” in ${\mathcal{D}}_{k+1}$.

Proposition

For $f\in L^2([0,1))$, define $M_k(f):=\mathbb{E}(f\mid {\mathcal{F}}_k)$ for $k\in \mathbb{N}$.

1. $(M_k(f){)}_{k\in \mathbb{N}}$ is a martingale.
2. For any $k\in \mathbb{N}$, $M_k(f)$ can be expressed as $M_k(f)={\sum }_{i=0}^{2^k-1}{1}_{\left[\frac{i}{2^k},\frac{i+1}{2^k}\right)}{2}^k{\int }_{\frac{i}{2^k}}^{\frac{i+1}{2^k}}f(s)ds.$
3. For any $t\in [0,1)$, $M_k(f)(t)=\frac{1}{|I_t|}{\int }_{I_t}f(s)ds,$ where $I_t$ is the unique interval in ${\mathcal{D}}_k$ that contains $t$.
4. There exists $M_{\infty }(f)\in L^2([0,1))$ such that $M_k(f)\to M_{\infty }(f)$ almost surely and in $L^2([0,1))$ as $k\to \infty$.
5. For any $g\in C([0,1))$, $M_{\infty }(g)=g$.

Lebesgue's Differentiation Theorem

For any $f\in L^2([0,1))$, $M_{\infty }(f)=f.$

Haar Function

Let $H:\mathbb{R}\to \mathbb{R}$ be the function $H(t):=\{\begin{array}{ll}1& \text{if }t\in [0,1/2)\\ -1& \text{if }t\in [1/2,1)\\ 0& \text{otherwise.}\end{array}$ The Haar functions are defined through $H_0(t):=1\forall t\in [0,1],$ and for any $k\in \mathbb{N}$ and $i=0,\dots ,2^k-1,$ $H_{2^k+i}(t):=2^{k/2}H(2^{k}t-i)\forall t\in [0,1].$

Lemma

For any $f\in L^2([0,1))$, the following statements are true.

1. For any $I=I_i^k=[i/2^k,(i+1)/2^k)$, where $k\in \mathbb{N}$ and $i\le 2^k-1$, $1_I\cdot (\mathbb{E}(f\mid {\mathcal{F}}_{k+1})-\mathbb{E}(f\mid {\mathcal{F}}_k))=\left({\int }_0^1{H}_{2^k+i}(s)f(s)ds\right)H_{2^k+i}=⟨H_{2^k+i},f⟩H_{2^k+i}.$
2. For any $k\in \mathbb{N}$, $\mathbb{E}(f\mid {\mathcal{F}}_k)={\sum }_{n=0}^{2^k-1}⟨f,H_n⟩H_n.$

Haar basis

$(H_n{)}_{n\in \mathbb{N}}$ is an orthonormal basis for the Hilbert space $L^2([0,1))$. It is called the Haar basis.