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
Random process
Random real collection process
Random Euclidean real process
Random complex process
Random real process
Random rational process
Random integer process
Random natural number process
Definition D3745
Renewal process
Formulation 0
Let $X_1, X_2, X_3, \ldots \in \text{Random}[0, \infty)$ each be a D5452: Random unsigned real number such that
(i) $X_1, X_2, X_3, \ldots$ is an D3358: I.I.D. random collection
(ii) \begin{equation} \mathbb{E} |X_1| < \infty \end{equation}
A D6140: Random natural number process $N : [0, \infty) \to \text{Random}(\mathbb{N})$ is a revewal process with respect to $X_1, X_2, X_3, \ldots$ if and only if \begin{equation} \forall \, t \in [0, \infty) : N_t \overset{d}{=} \max \left\{ M \in \mathbb{N} : \sum_{m = 1}^M X_m \leq t \right\} \end{equation}
Subdefinitions
Real poisson process
Children
Real poisson process