Doob martingale

A Doob martingale (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

A Doob martingale (named after Joseph L. Doob)^[1] is a generic construction that is always a martingale. Specifically, consider any set of random variables

\vec{X}=X_1, X_2, ..., X_n

taking values in a set $A$ for which we are interested in the function $f:A^n \to \Bbb{R}$ and define:

B_i=E_{X_{i+1},X_{i+2},...,X_{n}}[f(\vec{X})|X_{1},X_{2},...X_{i}]

where the above expectation is itself a random quantity since the expectation is only taken over

X_{i+1},X_{i+2},...,X_{n},

and

X_{1},X_{2},...X_{i}

are treated as random variables. It is possible to show that $B_i$ is always a martingale regardless of the properties of $X_i$ .

The sequence ${B_i}$ is the Doob martigale for f.^[2]

Application

Thus if one can bound the differences

|B_{i+1}-B_i|

one can apply Azuma's inequality and show that with high probability $f(\vec{X})$ is concentrated around its expected value

E[f(\vec{X})]=B_0.

McDiarmid's inequality

One common way of bounding the differences and applying Azuma's inequality to a Doob martingale is called McDiarmid's inequality.^[3]

Suppose $X_1, X_2, \dots, X_n$ are independent and assume that $f$ satisfies

\sup_{x_1,x_2,\dots,x_n, \hat x_i} |f(x_1,x_2,\dots,x_n) - f(x_1,x_2,\dots,x_{i-1},\hat x_i, x_{i+1}, \dots, x_n)| \le c_i \qquad \text{for} \quad 1 \le i \le n \; .

(In other words, replacing the $i$ -th coordinate $x_i$ by some other value changes the value of $f$ by at most $c_i$ .)

It follows that

|B_{i+1}-B_i| \le c_i

and therefore Azuma's inequality yields the following McDiarmid inequalities for any $\varepsilon > 0$ :

\Pr \left\{ f(X_1, X_2, \dots, X_n) - E[f(X_1, X_2, \dots, X_n)] \ge \varepsilon \right\} \le \exp \left( - \frac{2 \varepsilon^2}{\sum_{i=1}^n c_i^2} \right)

and

\Pr \left\{ E[f(X_1, X_2, \dots, X_n)] - f(X_1, X_2, \dots, X_n) \ge \varepsilon \right\} \le \exp \left( - \frac{2 \varepsilon^2}{\sum_{i=1}^n c_i^2} \right)

and

\Pr \left\{ |E[f(X_1, X_2, \dots, X_n)] - f(X_1, X_2, \dots, X_n)| \ge \varepsilon \right\} \le 2 \exp \left( - \frac{2 \varepsilon^2}{\sum_{i=1}^n c_i^2} \right). \;

Notes

↑ J. L. Doob, Transactions of the American Mathematical Society 47 (1940), 486, Theorem 3.7
↑ Anupam Gupta (2011) http://www.cs.cmu.edu/~avrim/Randalgs11/lectures/lect0321.pdf Lecture notes
↑ McDiarmid, Colin (1989). "On the Method of Bounded Differences". Surveys in Combinatorics 141: 148–188.

References

McDiarmid, Colin (1989). "On the Method of Bounded Differences". Surveys in Combinatorics 141: 148–188.