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}
Formulation 1
Let $X \in \text{Random}(\mathbb{R})$ be a D3161: Random real number such that
(i) $p, q \in [1, \infty)$ are each a D5407: Positive real number
(ii) \begin{equation} p > q \end{equation}
(iii) \begin{equation} \mathbb{E} |X|^p \in [0, \infty) \end{equation}
Then \begin{equation} \mathbb{E} |X|^q \in [0, \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}