A maximal Hoeffding's inequality?

A maximal Hoeffding's inequality?

Let $X_1, \cdots, X_n$ be real-valued independent random variables
satisfying $|X_k|\le 1$ and $\mathbb EX_k=0$. Hoeffding's inequality tells
us that for any $k=1,\cdots, n$ and $t>0$, $$\mathbb P\Big( \Big |
\frac{X_1+\cdots+ X_k}{\sqrt{k}} \Big | \ge t \Big) \le 2 e^{-t^2/2}.$$
My question is whether there exists a similar bound for the maximum over
$k$. More precisely:
Question: Do there exist absolute constants $C>0$ and $A>0$ so that
$$\mathbb P\Big( \max_{1\le k\le n} \Big | \frac{X_1+\cdots+
X_k}{\sqrt{k}} \Big | \ge t \Big) \le C e^{-t^2/A}$$ holds for all $t>0$?
If not what can we say about the left hand side?