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

Result R5046 on D1719: Expectation
Subresult of R3581: Absolute moment inherits finiteness from greater exponents for random complex number

Absolute moment inherits finiteness from greater exponents for random real number

Formulation 0

Let $X \in \text{Random}(\mathbb{R})$ be a D3161: Random real number such that

(i)	$1 \leq q < p < \infty$ are each a D5407: Positive real number
(i)	\begin{equation} \mathbb{E} \|X\|^p < \infty \end{equation}

Then \begin{equation} \mathbb{E} |X|^q < \infty \end{equation}

Proofs

Proof 0

Let $X \in \text{Random}(\mathbb{R})$ be a D3161: Random real number such that

(i)	$1 \leq q < p < \infty$ are each a D5407: Positive real number
(i)	\begin{equation} \mathbb{E} \|X\|^p < \infty \end{equation}