Proof of Chebyshev Type for Maximal Sum of Random Variables II

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Chebyshev type for maximal sum of random variables II.  Let $\{X_n\}$ be independent random variables with finite means and suppose that there exists an $A$ such that $$\forall\,n,\,|X_n-\mathscr{E}(X_n)|\leq A<\infty,$$Then let $S_n=\sum_{j=1}^nX_j$, we have for every $\varepsilon>0$, $$\mathscr{P}\left\{\underset{1\leq j\leq n}{\max}|S_j|\leq\varepsilon\right\}\leq\frac{(2A+4\varepsilon)^2}{\sigma^2(S_n)}.$$
See List of Inequalities.

$\bullet$ Proof.

Let $M_0=\Omega$ and $$M_k=\left\{\underset{1\leq j\leq k}{\max}|S_j|\leq\varepsilon\right\},\;1\leq k\leq n.$$We may suppose $\mathscr{P}\{M_n\}>0$, otherwise, the conclusion is trivial.  Furthermore, let $S'_0=0$ and for $k\geq1$, $$\begin{array}{rl}\Delta_k&=M_{k-1}\setminus M_k,\;k=1,\ldots,n;\\ X'_k&=X_k-\mathscr{E}(X_k);\\ S'_k&=\sum_{j=1}^kX'_j=S_k-\sum_{j=1}^k\mathscr{E}(X_j).\end{array}$$Define numbers $a_k$, $0\leq k\leq n$, as $$\begin{array}{rl}a_k&=\frac{1}{\mathscr{P}\{M_k\}}\int_{M_k}S'_k\,d\mathscr{P}=\frac{1}{\mathscr{P}\{M_k\}}\int_{M_k}[S_k-\mathscr{E}(S_k)]\,d\mathscr{P}\\ & =\frac{1}{\mathscr{P}\{M_k\}}\int_{M_k}S_k\,d\mathscr{P}-\frac{1}{\mathscr{P}\{M_k\}}\mathscr{E}(S_k)\cdot\mathscr{P}\{M_k\}\end{array}$$so that $$\int_{M_k}(S'_k-a_k)\,d\mathscr{P}=0.$$Write$$\begin{array}{rl} & \int_{M_{k+1}}(S'_{k+1}-a_{k+1})^2\,d\mathscr{P} \\ = & \int_{M_k}(S'_{k+1}-a_{k+1})^2\,d\mathscr{P}-\int_{\Delta_{k+1}}(S'_{k+1}-a_{k+1})^2\,d\mathscr{P}\overset{define}{=}I_1-I_2.\end{array}$$Using the definition of $M_k$ and the assumption $|X_n\mathscr{E}(X_n)|\leq A<\infty$, we have $$\begin{array}{rl}|S'_k-a_k|
& =\left|S_k-\mathscr{E}(S_k)-\frac{1}{\mathscr{P}\{M_k\}}\int_{M_k}[S_k-\mathscr{E}(S_k)]\,d\mathscr{P}\right|\\
& =\left|S_k-\frac{1}{\mathscr{P}\{M_k\}}\int_{M_k}S_k\,d\mathscr{P}\right|\\
& \leq|S_k|+\frac{1}{\mathscr{P}\{M_k\}}\int_{M_k}|S_k|\,d\mathscr{P}\\
& \leq|S_k|+\frac{\varepsilon}{\mathscr{P}\{M_k\}}\int_{M_k}\,d\mathscr{P}\quad\left(\because\,\mbox{def. of }M_k\right)\\
& =|S_k|+\varepsilon.\end{array}$$and $$\begin{array}{rl}|a_k-a_{k+1}|
& =\left|\frac{1}{\mathscr{P}\{M_k\}}\int_{M_k}S'_k\,d\mathscr{P}-\frac{1}{\mathscr{P}\{M_{k+1}\}}\int_{M_{k+1}}S'_{k+1}\,d\mathscr{P}\right| \\
& =\left|\frac{1}{\mathscr{P}\{M_k\}}\int_{M_k}S'_k\,d\mathscr{P}-\frac{1}{\mathscr{P}\{M_{k+1}\}}\int_{M_{k+1}}S'_k\,d\mathscr{P}-\frac{1}{\mathscr{P}\{M_{k+1}\}}\int_{M_{k+1}}X'_{k+1}\,d\mathscr{P}\right| \\
&\leq 2\varepsilon+\frac{1}{\mathscr{P}\{M_{k+1}\}}\int_{M_{k+1}}|X'_{k+1}|\,d\mathscr{P}\quad\left(\because\,\mbox{def. of }M_k,\,M_{k+1}\right)\\
&\leq 2\varepsilon+A.\end{array}$$Now, let us focus on $I_1$ and $I_2$.  Since $|S_k|\leq\varepsilon$ on $\Delta_{k+1}$, $$\begin{array}{rl}I_2=\int_{\Delta_{k+1}}(S'_{k+1}-a_{k+1})^2\,d\mathscr{P}
& =\int_{\Delta_{k+1}}(S'_k+X'_{k+1}-a_k+a_k-a_{k+1})^2\,d\mathscr{P}\\
& =\int_{\Delta_{k+1}}\left[(S'_k-a_k)+(a_k-a_{k+1})+X'_{k+1}\right]^2\,d\mathscr{P}\\
& \leq\int_{\Delta_{k+1}}\left(|S_k|+\varepsilon+2\varepsilon+A+|X'_{k+1}|\right)^2\,d\mathscr{P}\\ & \leq(4\varepsilon+2A)^2\mathscr{P}\{\Delta_{k+1}\}.\end{array}$$and $$\begin{array}{rl}I_1&=\int_{M_k}(S'_{k+1}-a_{k+1})^2\,d\mathscr{P}\\
& =\int_{M_k}\left(S'_k+X'_{k+1}-a_k+a_k-a_{k+1}\right)^2\,d\mathscr{P}\\
& =\int_{M_k}\left[(S'_k-a_k)^2+(a_k-a_{k+1})^2+{X'_{k+1}}^2 \right. \\
&\qquad \left. +2(S'_k-a_k)(a_k-a_{k+1})+2(S'_k-a_k)X'_{k+1}+2(a_k-a_{k+1})X'_{k+1}\right]\,d\mathscr{P}\\
& =\int_{M_k}\left[(S'_k-a_k)^2+(a_k-a_{k+1})^2+{X'_{k+1}}^2\right]\,d\mathscr{P}+2(a_k-a_{k+1})\int_{M_k}(S'_k-a_k)\,d\mathscr{P}\\
&\qquad +2\int_{M_k}(S'_k-a_k)\,d\mathscr{P}\int_{M_k}X'_{k+1}\,d\mathscr{P}+2(a_k-a_{k+1})\int_{\Omega}I\{M_k\}\,d\mathscr{P}\int_{\Omega}X'_{k+1}\,d\mathscr{P}\\
&\qquad\left\{\because\,\mbox{indep. between }X_j\mbox{'s},\,\mathscr{E}(X'_j)=0\mbox{ and }\int_{M_k}(S'_k-a_k)\,d\mathscr{P}=0\right\}\\
& =\int_{M_k}\left[(S'_k-a_k)^2+(a_k-a_{k+1})^2+{X'_{k+1}}^2\right]\,d\mathscr{P}\\
& \geq\int_{M_k}(S'_k-a_k)^2\,d\mathscr{P}+\int_{M_k}{X'_{k+1}}^2\,d\mathscr{P}\\
& =\int_{M_k}(S'_k-a_k)^2\,d\mathscr{P}+\int_{\Omega}I\{M_k\}\,d\mathscr{P}\int_{\Omega}{X'_{k+1}}^2\,d\mathscr{P}\;\left\{\because\,\mbox{indep. between }X_j\mbox{'s}\right\}\\
& =\int_{M_k}(S'_k-a_k)^2\,d\mathscr{P}+\mathscr{P}\{M_k\}\sigma^2(X_{k+1}). \end{array}$$Thus, $$I_1-I_2\geq\int_{M_k}(S'_k-a_k)^2\,d\mathscr{P}+\mathscr{P}\{M_k\}\sigma^2(X_{k+1})-(4\varepsilon+2A)^2\mathscr{P}\{\Delta_{k+1}\}.$$Substituting $I_1$ and $I_2$ and using $M_k\supset M_n$, we have $$\begin{array}{c}\int_{M_{k+1}}(S'_{k+1}-a_{k+1})^2\,d\mathscr{P}-\int_{M_k}(S'_k-a_k)^2\,d\mathscr{P}\\ \geq\mathscr{P}\{M_n\}\sigma^2(X_{k+1})-(4\varepsilon+2A)^2\mathscr{P}\{\Delta_{k+1}\}.\end{array}$$Summing over $k$, we have $$\begin{array}{rl}\int_{M_n}(S'_n-a_n)^2\,d\mathscr{P}& \geq\mathscr{P}\{M_n\}\sum_{j=1}^n\sigma^2(X_j)-(4\varepsilon+2A)^2\sum_{j=1}^n\mathscr{P}\{\Delta_j\}\\
& =\mathscr{P}\{M_n\}\sigma^2(S_n)-(4\varepsilon+2A)^2\mathscr{P}\{\Omega\setminus M_n\}. \end{array}$$Therefore, using $|S'_k-a_k|\leq|S_k|+\varepsilon$ again, the inequality turns to $$\begin{array}{rl}\mathscr{P}\{M_n\}\sigma^2(S_n)
& \leq\int_{M_n}(S'_n-a_n)^2\,d\mathscr{P}+(4\varepsilon+2A)^2\mathscr{P}\{\Omega\setminus M_n\}\\
& \leq\int_{M_n}(|S_n|+\varepsilon)^2\,d\mathscr{P}+(4\varepsilon+2A)^2\mathscr{P}\{\Omega\setminus M_n\}\\
& \leq4\varepsilon^2\mathscr{P}\{M_n\}+(4\varepsilon+2A)^2\mathscr{P}\{\Omega\setminus M_n\}\\
& \leq(4\varepsilon+2A)^2\mathscr{P}\{M_n\}+(4\varepsilon+2A)^2\mathscr{P}\{\Omega\setminus M_n\}\\
& \leq(4\varepsilon+2A)^2. \\ \end{array}$$Thus, $$\mathscr{P}\left\{\underset{1\leq j\leq n}{\max}|S_j|\leq\varepsilon\right\}\leq\frac{(2A+4\varepsilon)^2}{\sigma^2(S_n)}.$$

$\Box$