\leq 
   K t^{r-p}$$ with $K = K(V,p,r)$. But for each $k$, $$X_N^+ \one_{U = k} \leq Y^{(k)}_{N -  k} \one_{\sigma^{(k)} > N - k}$$ and it follows that $$\E \left( (X_N^+)^r \one_{U = k} \right) \leq K (N - k)^{r-p}.$$ Now sum to get $$\E (X_N^+)^r \leq \sum_{k=0}^{N-1} K (N - k)^{r-p} \leq K \zeta (p-r) .$$ This completes the case $a = 1 , J = 0$.

For the general case, let $X_n' = (X_n - J) / a$. This process is covered by the analysis of the $a = 1 , J = 0$ case above, with $V / a^p$ in place of $V$. We conclude that $\E (X_n')^r \leq c (p , 1 , V / a^2 , 
0 , r)$, and hence that $\E X_n \leq c (p , a , V , J , r) := 
J + a^r c(p , 1 , V / a^p , 0 , r)$.

The proof of Corollary \[th:9\] from Corollary \[cor:better\] is virtually identical. $\Cox$

We now concern ourselves with the proof of Theorem \[th:2\]. We begin with two lemmas.

\[lem:Lp\] Let $\{ M_n \}$ be a martingale with $M_0 = 0$, and with increments bounded in $L^p$: 