ThmDex – An index of mathematical definitions, results, and conjectures.
▼ Set of symbols
▼ Alphabet
▼ Deduction system
▼ Theory
▼ Zermelo-Fraenkel set theory
▼ Set
▼ Subset
▼ Power set
▼ Hyperpower set sequence
▼ Hyperpower set
▼ Hypersubset
▼ Subset algebra
▼ Subset structure
▼ Measurable space
▼ Measurable map
▼ Random variable
▼ Class of random variables
▼ Collection of random variables
▼ Set of random variables
▼ Random collection
▼ Identically distributed random collection
▼ Stationary random collection
Definition D5254
Stationary random sequence
Formulation 1
A D1721: Random collection $X : \mathbb{N} \to \mathsf{Random}(\Omega \to \Xi)$ is stationary if and only if \begin{equation} \forall \, M \in \mathbb{N} : \forall \, n_1, \ldots, n_M \in \mathbb{N} : \forall \, k \in \mathbb{N} : (X_{n_1}, \ldots, X_{n_M}) \overset{d}{=} (X_{n_1 + k}, \ldots, X_{n_M + k}) \end{equation}
Results
▶ R3921: Identically distributed random collection need not be stationary