0
$\begingroup$

The general question is if we are given $\mathrm{E} X$ and $\mathrm{Var} X$ for some random variable $X$, can we deduce some information about $|X|$, such as bounds on $\mathrm{E} |X|$ and $\mathrm{Var} |X|$? Note that Chebyshev's and Markov's inequality only provide information about $\mathrm{P} (|X| \geq a)$.

For example, let $X_t, t \geq 0$ be a stochastic process and $a, b, c, d>0$. Suppose we know $ \mathrm{E}(X_t)=-a e^{-b t} \!\ $ and $ \mathrm{Var} (X_t ) = c\left( 1 - e^{-d t} \right). $

Can we deduce some information about $\mathrm{E} |X_t|$ and $\mathrm{Var} |X_t|$, such as bounds on them?

Does $\mathrm{E} |X_t|$ also non-decrease as $t \to \infty$, as $\mathrm{E} X_t$ does? Does $\mathrm{Var} |X_t|$ also non-decrease as $t \to \infty$, as $\mathrm{Var} X_t$ does?

When $X_t$ is normally distributed, $X_t$ is folded normally distributed.Analyzing monotonicity of $\mathrm{E} |X_t|$ and $\mathrm{Var} |X_t|$ is still hard for me to do.

Thanks!

1 Answers 1

1

I think you need more information about the distribution of $X_t$ to obtain non-trivial bounds. Without any assumptions, I can only find the following simple bounds.

Since $\left|\cdot\right|$ is convex, we have $\mathbb{E}\left[\left|X_t\right|\right]\geq\left|\mathbb{E}\left[X_t\right]\right|= ae^{-bt}.$ Furthermore, we can write $\mathbb{E}\left[X_t\right]^2+\mathrm{var}\left[X_t\right]=\mathbb{E}\left[X_t^2\right]=\mathbb{E}\left[\left|X_t\right|\right]^2+\mathrm{var}\left[\left|X_t\right|\right].$ Therefore, $\mathrm{var}\left[\left|X_t\right|\right]\leq \mathrm{var}\left[X_t\right]=c\left(1-e^{-dt}\right).$