ThmDex – An index of mathematical definitions, results, and conjectures.

Result R4583 on D2858: Random variable right tail
Subresult of R3262: Dilations of unsigned tail probability converge to zero

Formulation 0

Let $X \in \mathsf{Random}[0, \infty)$ be a D5452: Random unsigned real number such that

(i)	\begin{equation} \mathbb{E} X < \infty \end{equation}

Let $\lambda \in (0, \infty)$ be a D5407: Positive real number.

Then \begin{equation} \mathbb{P}(X \geq \lambda) = o(\lambda^{-1}) \end{equation}

Proofs

Proof 0

Let $X \in \mathsf{Random}[0, \infty)$ be a D5452: Random unsigned real number such that

(i)	\begin{equation} \mathbb{E} X < \infty \end{equation}

Let $\lambda \in (0, \infty)$ be a D5407: Positive real number.

This result is a particular case of R3262: Dilations of unsigned tail probability converge to zero. $\square$