TITLE: Proof of Pinelis (1992) - Banach space inequalities
QUESTION [5 upvotes]: I am reading Pinelis "An approach to inequalities for the distributions of infinite -dimensional martingales" and cannot follow his proof of Theorem 3:
Let $(f_n)$ be a martingale in a separable Banach space $(\mathcal{X},||~||)$, $\mathcal{X} = L^p, p \geq 2$ and $f^*=\sup\{||f_n||\}$.
Theorem 3 says

\begin{align} P(f^*>r) \leq 2\exp\big(-r^2/2(p-1)\big), \quad r \geq 0
\end{align}

and Pinelis writes 

One can compare the last inequality with
  \begin{align}
P(f^*>r) \leq (r+1)\exp(-r^2/2)
\end{align}
  in Kallenberg ans Sztencel (1991) proved for $\mathcal{X} = L^2$.

Does he refer to
\begin{align}
P(f^*>r) \leq \frac{1+r}{1+rc}\exp\big(-\frac{r}{2c} \ln(1+rc)\big), \quad r \geq 0
\end{align}
from Kallenberg and Sztencel (1991)? If so, I still cannot see the path he takes to prove his theorem.
Kallenberg and Sztencel (1991): Some dimension-free features of vector-valued martingales

REPLY [14 votes]: As written in my paper [1], the inequality 
$$P(f^*>r) \le 2\exp\big(-r^2/2(p-1)\big)
$$
in Theorem 3 in [1] 
for martingales in $\mathcal{X}=L^p$ can be compared with the inequality 
$$
P(f^*>r) \le C (r+1)\exp(-r^2/2)
$$
with an unidentified absolute constant $C>0$, given 
(as noted in the comment by Nate Eldredge) in Theorem 5.3 of [Kallenberg and Sztencel], proved for $\mathcal{X} = L^2$.
As for the inequality 
$$P(f^*>r) \le C \frac{1+r}{1+rc}\exp\big(-\frac{r}{2c} \ln(1+rc)\big) 
$$
with an absolute constant $C>0$, 
which is inequality (5.2) of [Kallenberg and Sztencel], it can be compared with the inequality in Theorem 2 of [1].  
More general results were given in [3]. 
Also, you wrote: "I still cannot see the path he takes to prove his theorem." Can you specify the steps that seem unclear?