# Inequality relating expected value and tail probability

I am currently working through Scornet2015 - Consistency of Random Forests.

I'm having trouble understanding a specific inequality that is used in the proofs without further explanation. I am assuming it is something rather general and not immediately related to the topic at hand.

As far as I can see, the inequality boils down to the following (I'll quote the full statements below).

$$\mathbb{E}[X] \leq \xi + u \mathbb{P}[X > \xi]$$ where $$u$$ such that $$X \leq u$$. Why is that? I've looked at various basic tools such as Markov or Chebyshev inequalities as well as inequalities for tail probabilities but none seems to apply here.

## First application

This considers (something like) the estimation error of a truncated estimate.

\begin{aligned} & \left.\mathbb{E}\left[\sup _{\substack{f \in \mathcal{F}_n(\Theta) \\ \|f\|_{\infty} \leq \beta_n}} \mid \frac{1}{a_n} \sum_{i=1}^{a_n}\left[f\left(\mathbf{X}_i\right)-Y_{i, L}\right]^2-\mathbb{E}\left[f(\mathbf{X})-Y_L\right]^2\right]\right] \\ & \quad \leq \xi+2\left(\beta_n+L\right)^2 \mathbb{P}\left[\sup _{\substack{f \in \mathcal{F}_n(\Theta) \\ \|f\|_{\infty} \leq \beta_n}}\left|\frac{1}{a_n} \sum_{i=1}^{a_n}\left[f\left(\mathbf{X}_i\right)-Y_{i, L}\right]^2-\mathbb{E}\left[f(\mathbf{X})-Y_L\right]^2\right|>\xi\right] \end{aligned}

Earlier, it is established that

$$\sup _{\substack{f \in \mathcal{F}_n(\Theta) \\\|f\|_{\infty} \leq \beta_n}}\left|\frac{1}{a_n} \sum_{i \in \mathcal{I}_{n, \Theta}}\left[f\left(\mathbf{X}_i\right)-Y_{i, L}\right]^2-\mathbb{E}\left[f(\mathbf{X})-Y_L\right]^2\right| \leq 2\left(\beta_n+L\right)^2$$

## Second application

This considers the variation of the estimate $$m$$ in cells $$A_{n}(\mathbf{X}, \Theta)$$ of the random forest. The variation is defined as $$\Delta(m, A)=\sup _{\mathbf{x}, \mathbf{x}^{\prime} \in A}\left|m(\mathbf{x})-m\left(\mathbf{x}^{\prime}\right)\right|$$ and thus upper-bounded by the supremum norm $$\|f\|_{\infty}:=\sup _{x \in[0,1]}|f(x)|$$.

\begin{aligned} \mathbb{E}\left[\Delta\left(m, A_n(\mathbf{X}, \Theta)\right)\right]^2 & \leq \xi^2+4\|m\|_{\infty}^2 \mathbb{P}\left[\Delta\left(m, A_n(\mathbf{X}, \Theta)\right)>\xi\right] \end{aligned}

• In the applications you describe it is assumed that $X \lt u$, not that $\mathbb E[X] \lt u$. Deriving the inequality from there is straightforward Mar 22, 2023 at 10:45
• Thank you for your comment, I have edited the question. Mar 22, 2023 at 12:32

Let $$\Omega$$ be the sample space and $$X: \Omega \to \mathbb{R}$$ a random variable. Assume $$X < u$$, i.e. $$X(\omega) < u$$ for all $$\omega \in \Omega$$.
\begin{align} \mathbb{E}[X] &= \sum_{\omega \in \Omega} \mathbb{P}(\omega) X(\omega) \\ &= \sum_{\substack{\omega \in \Omega \\ X(\omega) \leq \xi}} \mathbb{P}(\omega ) \underbrace{X(\omega)}_{\leq \xi} + \sum_{\substack{\omega \in \Omega \\ X(\omega) > \xi}} \mathbb{P}(\omega) \underbrace{X(\omega)}_{\leq u} \\ & \leq \xi ~ \underbrace{ \mathbb{P}(X \leq \xi) }_{\leq 1} + u ~ \mathbb{P}(X > \xi) \\ & \leq \xi + u ~\mathbb{P}(X > \xi) \end{align}
• on a second thought, the first equality might not be completely correct, maybe you also have to sum over the range of $X$. Mar 31, 2023 at 8:00