Independent random variables with zero mean: inequalities Let $X_1,\ldots,X_n,\ldots$ independent random variables with zero mean, define:
$$S_n=X_1+X_2+\ldots+X_n$$
and
$$M_n=\text{max}\{\vert S_k \vert : k \in \{1,\ldots,n\} \} $$
Prove that:
$$\mathbb{E}(S_n^2 I_{A_k}) > c^2 \mathbb{P}(A_k)$$
where $A_k=\{M_{k-1}\leq c < M_k\}$ and $c>0$. Also prove that:
$$ \mathbb{P}( \text{max}\{\vert S_k \vert ; k \in \{1,\ldots,n\}\} >c) \leq \dfrac{\mathbb{E}(S_n^2)}{c^2}, \quad c>0$$
Honestly I have tried using inequalities and integrals (for the first part) and have not achieved much. Any help would be great, thank you very much.
 A: Both problems are essentially consequences of the Kolmogorov's inequality . First, let's to make the notation a bit simpler, for that you can read the reference easier.
Let $X_{1},X_{2},\ldots,X_{n}$ be independent random variables with mean $\mathbb{E}[X]=0$ and suppose that $\forall k\in \mathbb{N}$ we have that $\mathbb{V}ar[X_{k}]<\infty$. Also, let's to make  $$S_{n}:=X_{1}+X_{2}+\cdots+X_{n},$$
$$M_{n}:=\max_{1\leq k\leq n}|S_{k}|$$
So, in the first part you need to prove that $$\mathbb{E}[S_{n}^{2}\mathbb{I}\{A_{k}\}]>c^{2}\mathbb{P}[A_{k}]$$
Sketch prove: First, note that since that the mean is $0$, so  $ \displaystyle \mathbb{E}[S_{n}^{2}]=\sum_{k=1}^{n}\mathbb{V}ar[X_{k}]$.
Now,
\begin{eqnarray*}
\mathbb{E}[S_{n}^{2}\mathbb{I}\{A_{k}\}]&=&\mathbb{E}[(S_{k}^{2}+2S_{k}(S_{n}-S_{k})+(S_{n}-S_{k})^{2})\mathbb{I}\{A_{k}\}]\\
&\geq &\mathbb{E}[((S_{k}^{2}+2S_{k}(S_{n}-S_{k}))\mathbb{I}\{A_{k}\}]\\
&=&\mathbb{E}[S_{k}^{2}\mathbb{I}\{A_{k}\}], \quad S_{n}-S_{k}\perp S_{k}\mathbb{I}\{A_{k}\},\\
&>&^{2}\mathbb{E}[\mathbb{I}\{A_{k}\}]\\
&=&c^{2}\mathbb{P}[A_{k}]
\end{eqnarray*}
For the second part, you need to prove that $$\mathbb{P}\left[\max_{1\leq k\leq k} |S_{k}|>c \right]\geq \frac{1}{c^{2}}\mathbb{E}\left[\max_{1\leq k \leq n}|S_{k}| \right]^{2}$$
for that you can use the first part and the Chebyshev's inequality.
Remark 1: The Kolmogorov's inequality says that: If $X_{1},X_{2},\ldots,X_{n}$ be independent random variables with mean $0$ and asumming that $\mathbb{V}ar[X_{k}]<\infty$ for all $k\in \mathbb{N}$. Then, we have that for all $c>0$, $$\mathbb{P}\left[ \max_{1\leq k\leq n} |S_{k}|>c \right]\leq \frac{1}{c^{2}}\sum_{k=1}^{n}\mathbb{V}ar[X_{k}].$$
Remark 2: Also, note that $\mathbb{V}ar[X]=\mathbb{E}[X^{2}]-[\mathbb{E}{X}]^{2}$.
