In studying for an upcoming qualifying exam, I have encountered the following problem:
Suppose $X_p \sim \text{Geo}(p)$, which is to say random viable $X_p$ has PMF $$ \mathbb{P}(X_p = n) = p(1-p)^{n-1} \quad \text{ for } \quad n \in \mathbb{N} $$ Determine the limit in probability of $X_p$ as $p \to 0$.
I think I have the intuitive solution: $X_p \to \infty$ as $p \to 0$. As the probability of success for any single trial becomes smaller and smaller, the number of trials needed to obtain one success becomes larger and larger. This intuitive idea is reinforced by the facts that as $p \to 0$
- $\qquad \mathbb{P}(X_p = n) = p(1-p)^{n-1} \to 0$
- $\qquad \mathbb{E}[X_p] = \frac{1}{p} \to \infty$
- $\qquad \text{Var}(X_p) = \frac{1-p}{p^2} \to \infty$
My real question comes down to how to prove this formally. I know that given a sequence of random variables $\{Z_n\}_{n \geq 1}$ we say $Z_n \xrightarrow{\text{P}} c$ (i.e. $Z_n$ converges in probability to c as $n \to \infty$) if for every $\varepsilon > 0$ we have that $\mathbb{P}(\vert Z_n - c \vert \geq \varepsilon) \to 0 $. In this case, if I take $c = \infty$ for the definition of convergence in probability for $X_p$ as $p \to 0$ the overall statement seems kind of meaningless. Is there a formal approach to this solution, or am I trapped to this hand-wavy argument?
$\newcommand{\pr}{\operatorname{Pr}}$$|Z_n-c|\ge\epsilon$ is capturing the idea of "being far away from $c$". The suitable generalisation for $c=\infty$ is this: $$\tag{$\ast$}\forall R>0:\quad\quad\lim_{n\to\infty}\Pr(|Z_n|\le R)=0$$
Since being bounded is the same thing as being far away from $\infty$ "". Something something something, topology of the compactification, ... this type of definition is found in other contexts.
It suffices to check $(\ast)$ for $R\in\Bbb N$. Then: $$\pr(|X_p|\le R)=\pr(X_p\le R)=\sum_{n=1}^Rp(1-p)^{n-1}=1-(1-p)^R$$Tends to zero as $p\to0^+$. So condition $(\ast)$ seems to hold and I think it is reasonable to say $(X_p)_{0<p<1}\to\infty$ as $p\to0^+$.
You should check with your instructors or classmates if a specific definition (that is different from mine) of convergence in probability to $\infty$ is expected.