Doob's Maximal Inequality/Discrete Time

Theorem
Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \mathop \ge 0}, \Pr}$ be a filtered probability space.

Let $\sequence {X_n}_{n \mathop \ge 0}$ be a non-negative $\sequence {\FF_n}_{n \mathop \ge 0}$-submartingale.

Let:


 * $\ds X_n^\ast = \max_{0 \mathop \le k \mathop \le n} X_k$

where $\max$ is the pointwise maximum.

Let $\lambda > 0$.

Then:


 * $\lambda \map \Pr {X_n^\ast \ge \lambda} \le \expect {X_n}$

Proof
Let:


 * $\map T \omega = \inf \set {k \ge 0 : \map {X_k} \omega \ge \lambda} \wedge n$

for each $\omega \in \Omega$.

From:
 * Least Time at which Discrete-Time Adapted Stochastic Process equals or exceeds Real Number is Stopping Time
 * Constant Function is Stopping Time
 * Pointwise Minimum of Stopping Times is Stopping Time

we have:


 * $T$ is a stopping time with respect to $\sequence {\FF_n}_{n \mathop \ge 0}$.

Further:


 * $T \le n$

Note that $X_T \ge \lambda$ :


 * $X_k \ge \lambda$ for some $0 \le k \le n$.

This is equivalent to:


 * $\ds \sup_{0 \mathop \le k \mathop \le n} X_k = X_n^\ast \ge \lambda$

Let $\FF_T$ be the stopped $\sigma$-algebra associated with $T$.

By Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time: Discrete Time: Submartingale, we have:


 * $\expect {X_n \mid \FF_T} \ge X_T$ almost surely.

From Adapted Stochastic Process at Stopping Time is Measurable with respect to Stopped Sigma-Algebra:


 * $X_T$ is $\FF_T$-measurable.

From Conditional Expectation of Measurable Random Variable, we have:


 * $X_T = \expect {X_T \mid \FF_T}$ almost surely.

So, by Conditional Expectation is Linear we have:


 * $\expect {X_n - X_T \mid \FF_T} \ge 0$ almost surely.

So from Condition for Conditional Expectation to be Almost Surely Non-Negative, we have:


 * $\expect {X_n \cdot \chi_A} \ge \expect {X_T \cdot \chi_A}$

for all $A \in \FF_T$.

We can now calculate:

Multiplying through $\lambda > 0$ allows us to conclude.