Doob martingale

When analyzing sums, random walks, or other additive functions of independent random variables, one can often apply the central limit theorem, law of large numbers, Chernoff's inequality, Chebyshev's inequality or similar tools. When analyzing similar objects where the differences are not independent, the main tools are martingales and Azuma's inequality.

Definition

Let ${\displaystyle Y}$ be any random variable with ${\displaystyle \mathbb{E}[|Y|]<\mathrm{\infty }}$. Suppose ${\displaystyle \left\{{\mathcal{F}}_0,{\mathcal{F}}_1,\dots \right\}}$ is a filtration, i.e. ${\displaystyle {\mathcal{F}}_s\subset {\mathcal{F}}_t}$ when ${\displaystyle s<t}$. Define

${\displaystyle Z_t=\mathbb{E}[Y\mid {\mathcal{F}}_t],}$

then ${\displaystyle \left\{Z_0,Z_1,\dots \right\}}$ is a martingale, namely Doob martingale, with respect to filtration ${\displaystyle \left\{{\mathcal{F}}_0,{\mathcal{F}}_1,\dots \right\}}$.

To see this, note that

• ${\displaystyle \mathbb{E}[|Z_t|]=\mathbb{E}[|\mathbb{E}[Y\mid {\mathcal{F}}_t]|]\le \mathbb{E}[\mathbb{E}[|Y|\mid {\mathcal{F}}_t]]=\mathbb{E}[|Y|]<\mathrm{\infty }}$;
• ${\displaystyle \mathbb{E}[Z_t\mid {\mathcal{F}}_{t-1}]=\mathbb{E}[\mathbb{E}[Y\mid {\mathcal{F}}_t]\mid {\mathcal{F}}_{t-1}]=\mathbb{E}[Y\mid {\mathcal{F}}_{t-1}]=Z_{t-1}}$ as ${\displaystyle {\mathcal{F}}_{t-1}\subset {\mathcal{F}}_t}$.

In particular, for any sequence of random variables ${\displaystyle \left\{X_1,X_2,\dots ,X_n\right\}}$ on probability space ${\displaystyle (\mathrm{\Omega },\mathcal{F},\text{P})}$ and function ${\displaystyle f}$ such that ${\displaystyle \mathbb{E}[|f(X_1,X_2,\dots ,X_n)|]<\mathrm{\infty }}$, one could choose

${\displaystyle Y:=f(X_1,X_2,\dots ,X_n)}$

and filtration ${\displaystyle \left\{{\mathcal{F}}_0,{\mathcal{F}}_1,\dots \right\}}$ such that

${\displaystyle \begin{array}{cc}{\mathcal{F}}_0& :=\left\{\varphi ,\mathrm{\Omega }\right\},\\ {\mathcal{F}}_t& :=\sigma (X_1,X_2,\dots ,X_t),\mathrm{\forall }t\ge 1,\end{array}}$

i.e. ${\displaystyle \sigma }$-algebra generated by ${\displaystyle X_1,X_2,\dots ,X_t}$. Then, by definition of Doob martingale, process ${\displaystyle \left\{Z_0,Z_1,\dots \right\}}$ where

${\displaystyle \begin{array}{cc}{Z}_0& :=\mathbb{E}[f(X_1,X_2,\dots ,X_n)\mid {\mathcal{F}}_0]=\mathbb{E}[f(X_1,X_2,\dots ,X_n)],\\ Z_t& :=\mathbb{E}[f(X_1,X_2,\dots ,X_n)\mid {\mathcal{F}}_t]=\mathbb{E}[f(X_1,X_2,\dots ,X_n)\mid X_1,X_2,\dots ,X_t],\mathrm{\forall }t\ge 1\end{array}}$

forms a Doob martingale. Note that ${\displaystyle Z_n=f(X_1,X_2,\dots ,X_n)}$. This martingale can be used to prove McDiarmid's inequality.