Let $Z$ be a nonnegative random variable satisfying the following concentration inequality for all $t\ge 0$:
$$\mathbb{P}[Z\ge t]\le C e^{-\frac{t^2}{2(1+t)}}$$
(for some $C\ge 1$)
Then:
$$\mathbb{E}[Z]\le 2(\sqrt{\pi}+\sqrt{\log C})+4(1+\log C)$$
This is basically exercise 2.8 from High-Dimensional Statistics: A Non-Asymptotic Viewpoint. I tried to use:
$$\mathbb{E}[Z]=\int_0^\infty \mathbb{P}[Z\ge t]\, dt\le C\int_0^\infty e^{-\frac{t^2}{2(1+t)}}\, dt$$
But I don't know how to upper bound the term in the integral (also, I don't see how I would get $\log C$ or $\sqrt{\log C}$ terms from this expression). Is there another way to get $\mathbb{E}[Z]$ from $\mathbb{P}[Z\ge t]$?
Best Answer
Thanks to the hint of @Saad in comments, I am now able to finish the proof.
First I separate the summand in the integral into two parts (the first one will be relevant when $t$ is small, the second one when $t$ is large):
$$\exp\left(-\frac{t^2}{2(1+t)}\right)\le \exp\left(-\frac{t^2}{4\max(1,t)}\right)\le e^{-t^2/4}+e^{-t/4}$$
Now I can use the hint in the following way:
$$\mathbb{E} Z=\int_0^\infty\mathbb{P}[Z\ge t]\le \int_0^\infty \min\left(C \exp\left(-\frac{t^2}{2(1+t)}\right),1\right)\,dt\le A+B$$
with:
$$A=\int_0^\infty \min(C e^{-t^2/4},1)\,dt$$ $$B=\int_0^\infty \min(C e^{-t/4},1)\,dt$$
For $B$ the crossing point of the two regimes is for $t=4 \log C$, hence:
$$B\le 4\log C+\int_{4\log C}^\infty C e^{-t/4}\,dt=4(\log C+1)$$
For $A$ it happens at $t=2\sqrt{\log C}$, so that:
$$A\le 2\sqrt{\log C}+\int_{2\sqrt{\log C}}^\infty Ce^{-t^2/4}\,dt$$
And the integral can be upper bounded as follows:
$$\int_{2\sqrt{\log C}}^\infty Ce^{-t^2/4}\,dt=\sqrt{2} C\int_{\sqrt{2\log C}}^\infty e^{-t^2/2}\,dt\le 2\sqrt{\pi}$$
where I used the standard inequality:
$$\int_x^\infty e^{-t^2/2}\, dt\le \sqrt{2\pi} e^{-x^2/2}$$