Good day to All. 

Let $S_{1,n} = \sum_{i=1}^{n}\xi_{i}$, where $(\xi_{i})_{i \in \mathbb{N}}$ be independent RV with values in some Banach space.
 
On pages 79-80 in [this book][1] author provides an example that illustrates the fact that in the infinite dimensional spaces it may be not enough to have assumptions on the distribution of the individual summands $\xi_{i}$ in order to control the deviations of $\lVert {S_{1,n}}\rVert $, where $\lVert \cdot \rVert $ is the norm of corresponding Banach space. Author says:

\begin{quote}
The following example serves
to show the relative independence of restrictions on concentration of the sum
and those concerning the distributions of individual summands.
\end{quote}

In his example 3.0.1 let the space $c_0$ (with the norm $\|x\|_{\infty} = \sup_{n \in \mathbb{N}} |x^{(n)}|$) be the space of real-valued convergent to zero sequences and we consider an array of i.i.d. real-valued $\xi_{j}^{i} = \frac{1}{2}\delta_{-1}+\frac{1}{2}\delta_{1}$. 

Consider the sequence $\xi_{j} = (\xi_{j}^{i} = \frac{\xi_{j}^{i}}{\ln\ln(10 i)}, i \in \mathbb{N} )$ which are centered, in $c_{0}$ and a.s. bounded, i.e. $\|\xi_j\|_{\infty} = \frac{1}{\ln \ln 10}$. Consider RV's $\eta_{n}^{i} = \sum_{j=1}^{n}\xi_{j}^{i}$ and notice that they are independent. One easily sees that $\mathbb{P}(\eta_{n}^{i}=n) = 2^{-n}$ and therefore it implies: 
\begin{align*}
\mathbb{P}(\max_{i \leq 2^n}\eta_{n}^{i}<n) = \prod_{k=1}^{2^n}(1-\mathbb{P}(\eta_{n}^{i}=n)) \leq \exp\paren{-2^{n} 2^{-n}} = e^{-1},
\end{align*}
where we used classical $1-x \leq \exp(-x)$ for $x\geq 0$. Thefore, for all $n$ we have: 
\begin{align*}
P\left(\|S_{1,n}\| \geq \frac{n}{\ln \ln (10 2^n)}\right) \geq P\left(\max_{i \leq 2^n}\eta_{n}^{i} \geq n\right) \geq 1-e^{-1}. 
\end{align*}

Thus, the norm of sum scales (with constant probability) like $\frac{n}{\ln (n)}$, whereas $\|\xi\|_{\infty} = \frac{1}{\ln \ln (10)}$.

My question: can't we also consider $l^{\infty} := \{x=(x^{(n)}) : \sup_{n\in \mathbb N}| x^{(n)}| \lt \infty\}$ (which is Banach space with respect to the norm  $\|x\|_{\infty} = \sup_{n \in \mathbb N} |x^{(n)}|$) and just the sequences $\xi_{j} = (\xi_{j}^{j}, j \in \mathbb{N})$ (without additional log weight ) and $\xi_{j}^{i} = \frac{1}{2}\delta_{-1}+\frac{1}{2}\delta_{1}$ i.i.d in $i,j$ as in example 3.0.1 to illustrate the same fact? 

  [1]: https://www.springer.com/de/book/9783540603115