Doob martingale

Let ${\displaystyle Y}$ be any random variable with ${\displaystyle \mathbb {E} [|Y|]<\infty }$. 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,[2] namely Doob martingale, with respect to filtration ${\displaystyle \left\{{\mathcal {F}}_{0},{\mathcal {F}}_{1},\dots \right\}}$.

To see this, note that

• ${\displaystyle \mathbb {E} [|Z_{t}|]=\mathbb {E} [|\mathbb {E} [Y\mid {\mathcal {F}}_{t}]|]\leq \mathbb {E} [\mathbb {E} [|Y|\mid {\mathcal {F}}_{t}]]=\mathbb {E} [|Y|]<\infty }$;
• ${\displaystyle \mathbb {E} [Z_{t}\mid {\mathcal {F}}_{t-1}]=\mathbb {E} [\mathbb {E} [Y\mid {\mathcal {F}}_{t}]\mid {\mathcal {F}}_{t-1}]=\mathbb {E} [Y\mid {\mathcal {F}}_{t-1}]=Z_{t-1}}$ 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 ${\displaystyle (\Omega ,{\mathcal {F}},{\text{P}})}$ and function ${\displaystyle f}$ such that ${\displaystyle \mathbb {E} [|f(X_{1},X_{2},\dots ,X_{n})|]<\infty }$, 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

${\displaystyle {\begin{aligned}{\mathcal {F}}_{0}&:=\left\{\phi ,\Omega \right\},\\{\mathcal {F}}_{t}&:=\sigma (X_{1},X_{2},\dots ,X_{t}),\forall t\geq 1,\end{aligned}}}$

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

${\displaystyle {\begin{aligned}Z_{0}&:=\mathbb {E} [f(X_{1},X_{2},\dots ,X_{n})\mid {\mathcal {F}}_{0}]=\mathbb {E} [f(X_{1},X_{2},\dots ,X_{n})],\\Z_{t}&:=\mathbb {E} [f(X_{1},X_{2},\dots ,X_{n})\mid {\mathcal {F}}_{t}]=\mathbb {E} [f(X_{1},X_{2},\dots ,X_{n})\mid X_{1},X_{2},\dots ,X_{t}],\forall t\geq 1\end{aligned}}}$

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.

The Doob martingale was introduced by Joseph L. Doob in 1940 to establish concentration inequalities such as McDiarmid's inequality, which applies to functions that satisfy a bounded differences property (defined below) when they are evaluated on random independent function arguments.

A function ${\displaystyle f:{\mathcal {X}}_{1}\times {\mathcal {X}}_{2}\times \cdots \times {\mathcal {X}}_{n}\rightarrow \mathbb {R} }$ satisfies the bounded differences property if substituting the value of the ${\displaystyle i}$th coordinate ${\displaystyle x_{i}}$ changes the value of ${\displaystyle f}$ by at most ${\displaystyle c_{i}}$. More formally, if there are constants ${\displaystyle c_{1},c_{2},\dots ,c_{n}}$ such that for all ${\displaystyle i\in [n]}$, and all ${\displaystyle x_{1}\in {\mathcal {X}}_{1},\,x_{2}\in {\mathcal {X}}_{2},\,\ldots ,\,x_{n}\in {\mathcal {X}}_{n}}$,

${\displaystyle \sup _{x_{i}'\in {\mathcal {X}}_{i}}\left|f(x_{1},\dots ,x_{i-1},x_{i},x_{i+1},\ldots ,x_{n})-f(x_{1},\dots ,x_{i-1},x_{i}',x_{i+1},\ldots ,x_{n})\right|\leq c_{i}.}$