$$\E (|M_n - M_{n-1}|^p \| \F_{n-1}) \leq L .$$ Then there is $c_p$ such that $\E |M_n|^p \leq c_p L n^{p/2}$.

[Proof:]{} Burkholder’s inequality (see Stout 1974, Theorem 3.3.6; Burkholder, 1966; Chow and Teicher, 1988, p. 396) tells us that for $p>1$, there is a constant $c_p$ for which $$\E |M_n|^p \leq c_p \E \left ( \sum_{k=1}^n (M_k - M_{k-1})^2 
   \right)^{p/2}.$$ (In fact, for $p \ge 2$ we may take $c_p = (p-1)^p$, cf. Burkholder, 1988, Theorem 3.1.) For any $Z_1 , \ldots , Z_n$, Hölder’s inequality gives $$\E |Z_1 + \cdots + Z_n|^{p/2} \leq n^{p/2} \max_{1 \leq k \leq n}
   \E |Z_k|^{p/2} .$$ Set $Z_k = (M_k - M_{k-1})^2$ and observe that $\E Z_k^{p/2} \leq 
\E ((\E (Z_k \| \F_{k-1})^{p/2}) \leq \E (\E |M_k - M_{k-1}|^p \| \F_{k-1}) 
\leq L$, so the conclusion of the lemma follows. $\Cox$

\[lem:tau\] Assume the notation and hypotheses of Theorem \[th:2\]. For $x > 0$, let $S_x = \inf \{k : M_k \geq x \}$ be the time to hit value $x$ or greater. Then there is a $C' = C' (b , p)$ such that $$\P (\tau > S_x) \leq {C' \over x^{p/2}} \, .$$

[Proof:]{} Fix $x \ge 1$ and bound 