Proof of Pinelis (1992) - Banach space inequalities

Question

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

Answer 1

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?