# Doob’s Inequality

Doob’s ${{\cal L}^p}$ Inequality

Let ${(S_i)_{i \in {\mathbb N}}}$ be a (discrete) non-negative sub-martingale, bounded in ${{\cal L}^p}$ i.e. with ${\sup_{i \in {\mathbb N}} \mathop{\mathbb E} S^p_i < \infty}$ for some fixed ${p > 1}$. Define ${q > 1}$ by ${p^{-1} + q^{-1} = 1}$, ${S^* := \sup_{i \in {\mathbb N}} S_i}$ and ${S^*_n := \sup_{i \leq n} S_i}$.

Then ${\mathop{\mathbb E} ({S_n^*}^p) \leq q^p\mathop{\mathbb E} S_n^p}$. Further, ${{\lVert S^*\rVert}_p \leq q\ {\rm sup_{i \in N}}{\lVert S_i\rVert}_p}$.

Proof: Fix ${x > 0}$ and define the stopping time ${\tau = n \wedge \inf_{i \leq n}\{S_i > x\}}$. Then ${\{S^*_n \geq x\} = \{S_\tau \geq x\}}$ which is ${{\cal F}_\tau}$-measurable and thus by The Stopping Time Lemma since ${\tau \leq n}$

$\displaystyle x{\mathbb P}\{S^*_n \geq x\} \leq \mathop{\mathbb E} S_\tau\{S_\tau \geq x\} \leq \mathop{\mathbb E} S_n\{S^*_n \geq x\}$

Also by Fubini

$\displaystyle \int_0^\infty px^{p-1}\mathop{\mathbb P}\{X \geq x\}\,dx =$

$\displaystyle \int_0^\infty \int_\Omega px^{p-1}1\{X \geq x\}\,d\mu\,dx =$

$\displaystyle \int_\Omega \int_0^\infty px^{p-1}1\{X \geq x\}\,dx\,d\mu =$

$\displaystyle \int_\Omega \int_0^X px^{p-1}\,dx\,d\mu =$

$\displaystyle \int_\Omega X^pd\mu\ = \mathop{\mathbb E} X^p$

Using both of these facts and Fubini again

$\displaystyle \mathop{\mathbb E} (S^*_n)^p = \int_0^\infty px^{p-1}\mathop{\mathbb P}\{S^*_n \geq x\}\,dx \leq$

$\displaystyle \int_0^\infty px^{p-2}\mathop{\mathbb E} S_n\{S^*_n \geq x\}\,dx =$

$\displaystyle \mathop{\mathbb E} \left( S_n \int_0^\infty px^{p-2}\{S^*_n \geq x\}\,dx\right) =$

$\displaystyle \mathop{\mathbb E} \left( S_n \left[\frac{p}{p-1}x^{p-1}\right]_0^{S_n^*}\right) =$

$\displaystyle \mathop{\mathbb E} \left( S_n \frac{1}{1 - \frac{1}{p}}{S_n^*}^{p-1}\right) =$

$\displaystyle q\mathop{\mathbb E} \left( S_n {S_n^*}^{p-1}\right)$

Hölder’s inequality states that for ${f \in {\cal L}^p, g \in {\cal L}^q}$ then ${\mu \left| fg\right| \leq {\lVert f \rVert}_p {\lVert g \rVert}_q}$. Now ${({S^*_n}^{p-1})^q = {S^*_n}^p}$ which is in ${{\cal L}^1}$ hence ${{S^*}^{p-1} \in {\cal L}^q}$. Thus

$\displaystyle \mathop{\mathbb E} S_n{S^*_n}^{p-1} \leq {\left(\mathop{\mathbb E} S_n^p\right)}^{\frac{1}{p}}{\left(\mathop{\mathbb E} {S^*_n}^p\right)}^{\frac{1}{q}}$

Hence from above

$\displaystyle {\left(\mathop{\mathbb E} {S^*_n}^p\right)}^p \leq q^p \left(\mathop{\mathbb E} S_n{S^*_n}^{p-1}\right)^p \leq q^p\left(\mathop{\mathbb E} S_n^p\right){\left( \mathop{\mathbb E} {S^*_n}^p\right)}^{p-1}$

If ${\mathop{\mathbb E} S^*_n \neq 0}$ then

$\displaystyle \mathop{\mathbb E} {S^*_n}^p \leq q^p\left(\mathop{\mathbb E} S_n^p\right)$

If ${\mathop{\mathbb E} S^*_n = 0}$ then the inequality holds trivially.

Finally,

$\displaystyle q{\lVert S_n\rVert}_p \leq q \sup_{i \in {\mathbb N}}{\lVert S_i\rVert}_p$

and now apply monotone convergence to obtain

$\displaystyle {\lVert S^*\rVert}_p \leq q \sup_{i \in {\mathbb N}}{\lVert S_i\rVert}_p$

$\Box$