Inequalities

Markov

For $\lambda > 0$,

$$P(|X| > \lambda) \le \frac{E|X|}{\lambda}$$

Chebyshev

If $X \in L^2(P)$ and $\lambda > 0$,

$$P(|X - E(X)| \ge \lambda) \le \frac{\mathrm{Var}(X)}{\lambda^2}$$

Cauchy-Schwarz

If $X, Y \in L^2$, then

$$E|XY| \le (E X^2)^{1/2} (E Y^2)^{1/2}$$

Hölder

If $p, q > 1$ with $\frac{1}{p} + \frac{1}{q} = 1$, $X \in L^p$ and $Y \in L^q$, then

$$E|XY| \le \|X\|_p \|Y\|_q$$

Minkowski

If $1 \le p < \infty$ and $X, Y \in L^p$, then

$$\|X + Y\|_p \le \|X\|_p + \|Y\|_p$$

Jensen

If $v : \mathbb{R} \to \mathbb{R}$ is convex, $E|X| < \infty$ and $E|v(X)| < \infty$, then

$$v(EX) \le E[v(X)]$$

If $v$ is concave the inequality is reversed.

Norm Monotonicity

If $0 < \alpha \le \beta$ and $E|X|^\beta < \infty$, then

$$\|X\|_\alpha \le \|X\|_\beta$$

Consequently, if $1 \le q \le p$ and $X_n \xrightarrow{L^p} X$, then $X_n \xrightarrow{L^q} X$.

With $L^\infty = \{X : \|X\|_\infty < \infty\}$ and

$$\|X\|_\infty = \sup\{x : P(|X| > x) > 0\}$$

the smallest number $|X|$ stays below with probability 1, i.e. the largest value $|X|$ can take ignoring probability 0 events. For $1 < p < q < \infty$,

$$\|X\|_1 \le \|X\|_p \le \|X\|_q \le \|X\|_\infty$$