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

Result R3581 on D1719: Expectation
Subresult of R3571: Random Lebesgue P-norm inherits finiteness from higher exponents

Absolute moment inherits finiteness from greater exponents for random complex number

Formulation 1

Let $Z \in \text{Random}(\mathbb{C})$ be a D4877: Random complex number such that

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

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

Formulation 2

Let $Z \in \text{Random}(\mathbb{C})$ be a D4877: Random complex 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} \|Z\|^p \in [0, \infty) \end{equation}

Then \begin{equation} \mathbb{E} |Z|^q \in [0, \infty) \end{equation}

Subresults

▶	R5046: Absolute moment inherits finiteness from greater exponents for random real number

Proofs

Proof 0

Let $Z \in \text{Random}(\mathbb{C})$ be a D4877: Random complex number such that

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

Since $\mathbb{E} |Z|^p$ is finite, then so is its exponentiation $(\mathbb{E} |Z|^p)^{1 / p}$. Since $p > q$, result R3571: Random Lebesgue P-norm inherits finiteness from higher exponents now states that $(\mathbb{E} |Z|^q)^{1 / q}$ is also finite. Exponentiating this finite quantity by $q$, we find that $\mathbb{E} |Z|^q < \infty$. $\square$