Using the Markov inequality Let $x_k$ is some stochastic process as $k$ progresses. I have a basic question of using Markov inequality in this scenario. For example if I have some constraint
$$
Pr\{|x_k|\le \alpha\}\ge \beta
$$
How can we use Markov inequality to obtain expression in terms of expectation of $|x_k|$, $E\{|x_k|\}$?
Thank You.
 A: Let us use some intuition first. Your constraint says that $|x|$ is smaller than $\alpha$ with a probability of at least $\beta$. Can it give us any information about the expected value of $|x|$? Unfortunately, no. On the one hand, nothing prevents the case that $\mathsf E|x| = 0$ since the case $x\equiv 0$ always satisfy your constraints. On the other hand, the expectation can be as big as possible. Indeed, let us consider
$$
  y = 
\begin{cases}
0,&\text{ with probability } \beta
\\
n,&\text{ with probability }1-\beta.
\end{cases}
$$
Clearly,
$
  \mathsf Ey = n(1-\beta)
$ can be any real number if $\beta<1$. As a result, unless you have a trivial case $\beta = 1$ in which case $\mathsf E|x|\leq \alpha$, you constraint does not provide any guarantees for the expectation.
The trick of Markov-inequality-like-bounds is to write the expectation as an integral, and then restrict your attention to the region that appears in the probability. But since we work with a non-negative random variable, only the bounds of the shape
$$
  \mathsf P\{|x|\geq a\}\geq b \tag{1}
$$
are useful. Or alternatively, bounds of the shape
$$
  \mathsf P\{|x|\leq c\}\leq d 
$$
since you can restate them as
$$
  \mathsf P\{|x|\geq c\}\geq 1-d.
$$
Provided the constraint of the shape $(1)$ we denote
$$
  A = \{\omega\in \Omega:|x(\omega)|\geq a\}
$$
so that we know that $\mathsf P(A)\geq b$. Moreover, $|x(\omega)|\geq a$ for any $\omega\in A$. As a result:
$$
  \mathsf E|x| = \int_\Omega |x(\omega)|\mathsf P(\mathrm d\omega)\geq\int_A |x(\omega)|\mathsf P(\mathrm d\omega) \geq \int_A a\mathsf P(\mathrm d\omega) \geq a\cdot b.
$$
Bottom line: the constraints that you have provide no guarantees on the expected value. For such guarantees you need something like $(1)$.
