Doob martingale

A Doob martingale (named after Joseph L. Doob,^[1] also known as a Levy martingale) is a mathematical construction of a stochastic process which approximates a given random variable and has the martingale property with respect to the given filtration. It may be thought of as the evolving sequence of best approximations to the random variable based on information accumulated up to a certain time.

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 $Y$ be any random variable with $\mathbb {E} [|Y|]<\infty$ . Suppose $\left\{{\mathcal {F}}_{0},{\mathcal {F}}_{1},\dots \right\}$ is a filtration, i.e. ${\mathcal {F}}_{s}\subset {\mathcal {F}}_{t}$ when $s<t$ . Define

Z_{t}=\mathbb {E} [Y\mid {\mathcal {F}}_{t}],

then $\left\{Z_{0},Z_{1},\dots \right\}$ is a martingale,^[2] namely Doob martingale, with respect to filtration $\left\{{\mathcal {F}}_{0},{\mathcal {F}}_{1},\dots \right\}$ .

To see this, note that

$\mathbb {E} [|Z_{t}|]=\mathbb {E} [|\mathbb {E} [Y\mid {\mathcal {F}}_{t}]|]\leq \mathbb {E} [\mathbb {E} [|Y|\mid {\mathcal {F}}_{t}]]=\mathbb {E} [|Y|]<\infty$ ;
$\mathbb {E} [Z_{t}\mid {\mathcal {F}}_{t-1}]=\mathbb {E} [\mathbb {E} [Y\mid {\mathcal {F}}_{t}]\mid {\mathcal {F}}_{t-1}]=\mathbb {E} [Y\mid {\mathcal {F}}_{t-1}]=Z_{t-1}$ as ${\mathcal {F}}_{t-1}\subset {\mathcal {F}}_{t}$ .

In particular, for any sequence of random variables $\left\{X_{1},X_{2},\dots ,X_{n}\right\}$ on probability space $(\Omega ,{\mathcal {F}},{\text{P}})$ and function $f$ such that $\mathbb {E} [|f(X_{1},X_{2},\dots ,X_{n})|]<\infty$ , one could choose

Y:=f(X_{1},X_{2},\dots ,X_{n})

and filtration $\left\{{\mathcal {F}}_{0},{\mathcal {F}}_{1},\dots \right\}$ such that

{\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. $\sigma$ -algebra generated by $X_{1},X_{2},\dots ,X_{t}$ . Then, by definition of Doob martingale, process $\left\{Z_{0},Z_{1},\dots \right\}$ where

{\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 $Z_{n}=f(X_{1},X_{2},\dots ,X_{n})$ . This martingale can be used to prove McDiarmid's inequality.

McDiarmid's inequality

Statement^[1]

Consider independent random variables $X_{1},X_{2},\dots X_{n}$ on probability space $(\Omega ,{\mathcal {F}},{\text{P}})$ where $X_{i}\in {\mathcal {X}}_{i}$ for all $i$ and a mapping $f:{\mathcal {X}}_{1}\times {\mathcal {X}}_{2}\times \cdots \times {\mathcal {X}}_{n}\rightarrow \mathbb {R}$ . Assume there exist constant $c_{1},c_{2},\dots ,c_{n}$ such that for all $i$ ,

{\underset {x_{1},\cdots ,x_{i-1},x_{i},x_{i}',x_{i+1},\cdots ,x_{n}}{\sup }}|f(x_{1},\dots ,x_{i-1},x_{i},x_{i+1},\cdots ,x_{n})-f(x_{1},\dots ,x_{i-1},x_{i}',x_{i+1},\cdots ,x_{n})|\leq c_{i}.

(In other words, changing the value of the $i$ th coordinate $x_{i}$ changes the value of $f$ by at most $c_{i}$ .) Then, for any $\epsilon >0$ ,

{\text{P}}(f(X_{1},X_{2},\cdots ,X_{n})-\mathbb {E} [f(X_{1},X_{2},\cdots ,X_{n})]\geq \epsilon )\leq \exp \left(-{\frac {2\epsilon ^{2}}{\sum _{i=1}^{n}c_{i}^{2}}}\right),

{\text{P}}(f(X_{1},X_{2},\cdots ,X_{n})-\mathbb {E} [f(X_{1},X_{2},\cdots ,X_{n})]\leq -\epsilon )\leq \exp \left(-{\frac {2\epsilon ^{2}}{\sum _{i=1}^{n}c_{i}^{2}}}\right),

and

{\text{P}}(|f(X_{1},X_{2},\cdots ,X_{n})-\mathbb {E} [f(X_{1},X_{2},\cdots ,X_{n})]|\geq \epsilon )\leq 2\exp \left(-{\frac {2\epsilon ^{2}}{\sum _{i=1}^{n}c_{i}^{2}}}\right).

Proof

Pick any $x_{1}',x_{2}',\cdots ,x_{n}'$ such that the value of $f(x_{1}',x_{2}',\cdots ,x_{n}')$ is bounded, then, for any $x_{1},x_{2},\cdots ,x_{n}$ , by triangle inequality,

{\begin{aligned}&|f(x_{1},x_{2},\cdots ,x_{n})-f(x_{1}',x_{2}',\cdots ,x_{n}')|\\\leq &|f(x_{1},x_{2},\cdots ,x_{n})-f(x_{1}',x_{2},\cdots ,x_{n})|\\&+\sum _{i=1}^{n-1}|f(x_{1}',\cdots ,x_{i}',x_{i+1},\cdots ,x_{n})-f(x_{1}',x_{2}',\cdots ,x_{i}',x_{i+1}',x_{i+2},\cdots ,x_{n})|\\\leq &\sum _{i=1}^{n}c_{i},\end{aligned}}

thus $f$ is bounded.

Define $Z_{i}:=\mathbb {E} [f(X_{1},X_{2},\cdots ,X_{n})\mid X_{1},X_{2},\cdots ,X_{i}]$ for all $i\geq 1$ and $Z_{0}:=\mathbb {E} [f(X_{1},X_{2},\cdots ,X_{n})]$ . Note that $Z_{n}=f(X_{1},X_{2},\cdots ,X_{n})$ . Since $f$ is bounded, by the definition of Doob martingale, $\left\{Z_{i}\right\}$ forms a martingale. Now define ${\begin{aligned}U_{i}&={\underset {x\in {\mathcal {X}}_{i}}{\sup }}\mathbb {E} [f(X_{1},\cdots ,X_{n})\mid X_{1},\cdots ,X_{i-1},x]-\mathbb {E} [f(X_{1},\cdots ,X_{n})\mid X_{1},\cdots ,X_{i-1}],\\L_{i}&={\underset {x\in {\mathcal {X}}_{i}}{\inf }}\mathbb {E} [f(X_{1},\cdots ,X_{n})\mid X_{1},\cdots ,X_{i-1},x]-\mathbb {E} [f(X_{1},\cdots ,X_{n})\mid X_{1},\cdots ,X_{i-1}].\end{aligned}}$

Note that $L_{i}\leq Z_{i}-Z_{i-1}\leq U_{i}$ and $U_{i},L_{i}$ are both ${\mathcal {F}}_{i-1}$ -measurable. In addition,

{\begin{aligned}U_{i}-L_{i}&={\underset {x_{u}\in {\mathcal {X}}_{i},x_{l}\in {\mathcal {X}}_{i}}{\sup }}\mathbb {E} [f(X_{1},\cdots ,X_{n})\mid X_{1},\cdots ,X_{i-1},x_{u}]-\mathbb {E} [f(X_{1},\cdots ,X_{n})\mid X_{1},\cdots ,X_{i-1},x_{l}]\\&={\underset {x_{u}\in {\mathcal {X}}_{i},x_{l}\in {\mathcal {X}}_{i}}{\sup }}\int _{{\mathcal {X}}_{i+1}\times \cdots \times {\mathcal {X}}_{n}}f(X_{1},\cdots ,X_{i-1},x_{u},x_{i+1},\cdots ,x_{n}){\text{d}}{\text{P}}_{X_{i+1},\cdots ,X_{n}\mid X_{1},\cdots ,X_{t-1},x_{u}}(x_{i+1},\cdots ,x_{n})\\&\quad -\int _{{\mathcal {X}}_{i+1}\times \cdots \times {\mathcal {X}}_{n}}f(X_{1},\cdots ,X_{i-1},x_{l},x_{i+1},\cdots ,x_{n}){\text{d}}{\text{P}}_{X_{i+1},\cdots ,X_{n}\mid X_{1},\cdots ,X_{t-1},x_{l}}(x_{i+1},\cdots ,x_{n})\\&={\underset {x_{u}\in {\mathcal {X}}_{i},x_{l}\in {\mathcal {X}}_{i}}{\sup }}\int _{{\mathcal {X}}_{i+1}\times \cdots \times {\mathcal {X}}_{n}}f(X_{1},\cdots ,X_{i-1},x_{u},x_{i+1},\cdots ,x_{n}){\text{d}}{\text{P}}_{X_{i+1},\cdots ,X_{n}}(x_{i+1},\cdots ,x_{n})\\&\quad -\int _{{\mathcal {X}}_{i+1}\times \cdots \times {\mathcal {X}}_{n}}f(X_{1},\cdots ,X_{i-1},x_{l},x_{i+1},\cdots ,x_{n}){\text{d}}{\text{P}}_{X_{i+1},\cdots ,X_{n}}(x_{i+1},\cdots ,x_{n})\\&={\underset {x_{u}\in {\mathcal {X}}_{i},x_{l}\in {\mathcal {X}}_{i}}{\sup }}\int _{{\mathcal {X}}_{i+1}\times \cdots \times {\mathcal {X}}_{n}}f(X_{1},\cdots ,X_{i-1},x_{u},x_{i+1},\cdots ,x_{n})\\&\quad -f(X_{1},\cdots ,X_{i-1},x_{l},x_{i+1},\cdots ,x_{n})\ {\text{d}}{\text{P}}_{X_{i+1},\cdots ,X_{n}}(x_{i+1},\cdots ,x_{n})\\&\leq {\underset {x_{u}\in {\mathcal {X}}_{i},x_{l}\in {\mathcal {X}}_{i}}{\sup }}\int _{{\mathcal {X}}_{i+1}\times \cdots \times {\mathcal {X}}_{n}}c_{i}\ {\text{d}}{\text{P}}_{X_{i+1},\cdots ,X_{n}}(x_{i+1},\cdots ,x_{n})\\&\leq c_{i}\end{aligned}}

where the third equality holds due to the independence of $X_{1},X_{2},\cdots ,X_{n}$ . Then, applying the general form of Azuma's inequality to $\left\{Z_{i}\right\}$ , we have

{\text{P}}(f(X_{1},\cdots ,X_{n})-\mathbb {E} [f(X_{1},\cdots ,X_{n})]\geq \epsilon )={\text{P}}(Z_{n}-Z_{0}\geq \epsilon )\leq \exp \left(-{\frac {2\epsilon ^{2}}{\sum _{i=1}^{n}c_{i}^{2}}}\right).

One-sided bound from the other direction is obtained by applying Azuma's inequality to $\left\{-Z_{i}\right\}$ and two-sided bound follows from union bound. $\square$

References

^ ^a ^b Doob, J. L. (1940). "Regularity properties of certain families of chance variables" (PDF). Transactions of the American Mathematical Society. 47 (3): 455–486. doi:10.2307/1989964. JSTOR 1989964.
^ Doob, J. L. (1953). Stochastic processes. 101. New York: Wiley. p. 293.

[Doob-1] Doob, J. L. (1940). "Regularity properties of certain families of chance variables" (PDF). Transactions of the American Mathematical Society. 47 (3): 455–486. doi:10.2307/1989964. JSTOR 1989964.

[2] Doob, J. L. (1953). Stochastic processes. 101. New York: Wiley. p. 293.

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