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

Formulation 0

Let $G_n \in \text{Geometric}(\theta_n)$ be a D4001: Geometric random positive integer for each $n \in \mathbb{N}$ such that

(i)	\begin{equation} \lim_{n \to \infty} \theta_n n = 1 \end{equation}

Let $a \in (0, \infty)$ be a D5407: Positive real number.

Then \begin{equation} \lim_{n \to \infty} \mathbb{P} \left( \frac{G_n}{n} \leq a \right) = 1 - e^{-a} \end{equation}

Proofs

Proof 0

Let $G_n \in \text{Geometric}(\theta_n)$ be a D4001: Geometric random positive integer for each $n \in \mathbb{N}$ such that

(i)	\begin{equation} \lim_{n \to \infty} \theta_n n = 1 \end{equation}

Let $a \in (0, \infty)$ be a D5407: Positive real number.