I am reading Pinelis "[An approach to inequalities for the distributions of infinite -dimensional martingales][1]" 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][2]


  [1]: http://link.springer.com/chapter/10.1007/978-1-4612-0367-4_9#page-1
  [2]: http://link.springer.com/article/10.1007/BF01212560#page-1