Kolmogorov like Maximal inequality with exponential expected value Let $X_i$ a countable collection of independent random variables with symmetric distribution i.e. $P(X_i\in A)=P(X_i \in -A)$ for all $i\geq 1$. If $\lambda\in\mathbb{R}$ is such that $E(e^{\lambda X_i})$ is finite for all $i$ I want to prove the following Kolmogorov like inequality
$$P\left(\max_{1\leq k\leq n} \left|\sum_{i=1}^{k}X_{k}\right|>t\right)\leq e^{-\lambda t}\prod_{i=1}^{n}E(e^{\lambda X_i })$$
Have you some hint for solution?

Using extrapolation I am able to prove your statement but only for $t$ large relative to $\lambda^{-1}$ and with the stronger condition of $E[e^{\lambda |X_i|}] < \infty$ (maybe you were missing an absolute value). Let us denote by $S$ the random variable 
$$
  S = \sum_{j = 1}^\infty X_j
$$
and by $\Sigma_n$ the $\sigma$-algebra generated by the random variables $X_1,...X_n$. Using the fact that each $X_j$ is of mean $0$, the conditional expectation $E_n: L^1(\Omega) \to L^1(\Omega; \Sigma_n)$ satisfies that
$$
  E_n(S) = \sum_{j = 1}^n X_j := S_n.
$$
By Doob's Theorem and Marcinkiewicz interpolation we have that
\begin{align}
  P \Big( \max_{1 \leq k \leq n} |E_k(S_n)| > t \Big) & \lesssim \frac{\|S_n\|_1}{t}\\
  P \Big( \max_{1 \leq k \leq n} |E_k(S_n)| > t \Big) & \lesssim \Big( \frac{p}{p-1}\Big) \frac{\|S_n\|^p_p}{t^p}.
\end{align}
Therefore 
\begin{align}
  (e^{t \lambda} -1) \, P \Big( \max_{1 \leq k \leq n} |E_k(S_n)| > t \Big)  
  & \lesssim \sum_{p=1}^\infty \frac{t^p \lambda^p}{p!} P \Big( \max_{1 \leq k \leq n} |E_k(S_n)| > t \Big)\\
  & \lesssim \sum_{p=1}^\infty \frac{t^p \lambda^p}{p!} \frac{E [|S_n|^p ]}{t^p}\\
  & = E \Big[ \sum_{p=1}^\infty \frac{\lambda^p |S_n|^p }{p!} \Big]\\
  & = E \Big[ \mathrm{exp}\big(\lambda |S_n|\big) - 1 \Big]
   \leq E \Big[ \mathrm{exp}\big(\lambda |S_n|\big)  \Big].
\end{align}
But now, using independence, we obtain that
$$
  E \Big[ \mathrm{exp}\big(\lambda |S_n|\big)  \Big] \leq \prod_{k = 1}^n E \big[e^{\lambda |X_k|}\big].
$$
In sum, we have
$$
 P \Big( \sup_{1 \leq k \leq n} \Big| \sum_{k=1}^n X_k \Big| > t \Big)  \lesssim \frac{1}{e^{t \lambda} - 1} \prod_{k = 1}^n E \big[e^{\lambda |X_k|}\big].
$$
