\right)$, so that $\mu_i \le -a$ on $\{X_i>J\}$. We may recenter (see the proof of Corollary \[cor:1\] for details) to obtain $$\E\left( (X_{n+1}-X_n-\mu_n)^p \, | \, \F_n \right)
\ \le \ V'$$ for some $V' < \infty$.

But then, for $t > J$, we have $$\P( X_n \ge t)
= \sum_{k=0}^{n-1} \P(X_n \ge t, \ U=k)
\qquad\qquad\qquad$$ $$\le \sum_{k=0}^{n-1} \P(X_n - X_k \ge t-J, \ X_k \le J, \ X_i > J \ {\rm for} \
k<i<n)$$ $$\le \sum_{k=0}^{n-1} \P\bigg( (X_n - X_{n-1}-\mu_{n-1}) + \ldots 
+ (X_{k+1}-X_k-\mu_{k}) \ge t-J-V^{1/p}+a(n-k-1),$$ $$X_k \le J, \ X_i > J \ {\rm for} 
\ k<i<n \bigg)$$ \[since $\mu_i \le -a$ for $k<i<n$, and $\mu_k \le V^{1/p}$\] $$\le \sum_{k=0}^{n-1} \E\bigg( \Big| (X_n - X_{n-1}-\mu_{n-1}) + \ldots 
+ (X_{k+1}-X_k-\mu_{k}) \Big|^p \bigg) \Big( t-J-V^{1/p}+a(n-k-1) \Big)^{-p}$$ \[by Markov’s inequality\] $$\le \sum_{k=0}^{n-1} c_p V' (n-k)^{p/2} \Big( t-J-V^{1/p}+a(n-k-1) \Big)^{-p}$$ \[by Lemma \[lem:Lp\], which is a direct application of Burkholder’s inequality\] $$\le \sum_{\ell=0}^{\infty} c_p V' (\ell+1)^{p/2}
\Big( t-J-V^{1/p}+a\ell \Big)^{-p} \, .$$

It then follows that $$\E(X_n) \ = \
\int_0^\infty dt \, \P(X_n \ge t)
\qquad\qquad\qquad$$ $$\le \
(J+V^{1/p}+1) \ + \ \int_{J+V^{1/p}+1}^\infty dt \, 
\sum_{\ell=0}^{\infty} c_p V' (\ell+1)^{p/2}
\left( t-J-V^{1/p}+a\ell \right)^{-p}$$ This integral-of-sum does not depend on $n$. Furthermore, for 