Inequalities for Random Variable

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Let $X$ and $Y$ be random variables.

(1) [See Proof] H$\ddot{o}$lder's inequality.  Let $1<p<\infty$ and $\frac{1}{p}+\frac{1}{q}=1$. $$|\mathscr{E}(XY)|\leq \mathscr{E}|XY|\leq \left(\mathscr{E}|X|^p\right)^{\frac{1}{p}}\left(\mathscr{E}|Y|^q\right)^{\frac{1}{q}}.$$

(2) [See Proof] Minkowski's inequality.  Let $1<p<\infty$. $$\left(\mathscr{E}|X+Y|^p\right)^{\frac{1}{p}}\leq \left(\mathscr{E}|X|^p\right)^{\frac{1}{p}}+\left(\mathscr{E}|Y|^p\right)^{\frac{1}{p}}.$$

(3) [See Proof] Lyapunov's inequality.  For $0<s<t$, $$\left(\mathscr{E}|X|^s\right)^\frac{1}{s}\leq \left(\mathscr{E}|X|^t\right)^\frac{1}{t}.$$

(4) [See Proof] Jensen's inequality.  Let $\phi$ be a convex function.  Suppose $X$ and $\phi(X)$ are integrable. $$\phi(\mathscr{E}X)\leq \mathscr{E}[\phi(X)].$$

(5) [See Proof] Chebyshev's inequality.  Let $\phi$ be a strictly increasing function on $(0,\infty)$ and $\phi(u)=\phi(-u)$.  Suppose $\mathscr{E}[\phi(X)]<\infty$.  Then $\forall\,u>0$, $$\mathscr{P}\{|X|\geq u\}\leq\frac{\mathscr{E}[\phi(X)]}{\phi(u)}.$$

(6) [See Proof] If $X\geq0$ and $Y\geq0$, $p\geq0$, then $$\mathscr{E}\{(X+Y)^p\}\leq2^p\{\mathscr{E}(X^p)+\mathscr{E}(Y^p)\}.$$If $p>1$, the factor $2^p$ may be replaced by $2^{p-1}$. If $0\leq p\leq1$, it may be replaced by $1$.

(7) [See Proof] Cantelli's inequality.  Suppose $\sigma^2=\mbox{Var}(X)<\infty$.  Then for $a>0$, we have $$\mathscr{P}\{|X-\mathscr{E}(X)|>a\}\leq\frac{2\sigma^2}{a^2+\sigma^2}.$$

(8) [See Proof] Chebyshev type for maximal sum of random variables I.  Let $\{X_n\}$ be independent random variables such that $\mathscr{E}(X_n)=0$ and $\mathscr{E}(X_n^2)=\sigma^2(X_n)<\infty$ for all $n$, then let $S_n=\sum_{j=1}^nX_j$, we have for every $\varepsilon>0$, $$\mathscr{P}\left\{\underset{1\leq j\leq n}{\max}|S_j|>\varepsilon\right\}\leq\frac{\sigma^2(S_n)}{\varepsilon^2}.$$

(9) [See Proof] 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 $\varepsilon>0$, $$\mathscr{P}\left\{\underset{1\leq j\leq n}{\max}|S_j|\leq\varepsilon\right\}\leq\frac{(2A+4\varepsilon)^2}{\sigma^2(S_n)}.$$

(10) [See Proof] If $\mathscr{E}(X^2)=1$ and $\mathscr{E}|X|\geq a>0$, then $$\mathscr{P}\{|X|\geq\lambda a\}\geq(1-\lambda)^2a^2\mbox{  for }0\leq\lambda\leq1.$$