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

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