Can the ratio of the two smallest element of an iid sample converge to 1 if the support of $X$ is positive?

Question

We have: $\mathbb P(X \leq 0)=0$ and $\mathbb P (X \leq a)>0$ for any $a>0$.

Answer 1

The answer is no. Here is a counterexample. Lets $X_1, X_2, \dots, X_n$ random samples from an exponential distribution with parameter $\lambda$.the order statistics $X_{(1)}, X_{(2)}$ each have distribution \begin{align*} X_{(1)} &= \frac{Z_1}{\lambda n} \\ X_{(2)} &= \frac{1}{\lambda}\left(\frac{Z_1}{n}+\frac{Z_2}{n-1}\right) \end{align*} where the $Z_1, Z_2$ are iid standard exponential random variables (see Wikipedia). Therefore $$ \frac{X_{(2)}}{X_{(1)}} = \frac{Z_1}{Z_1+Z_2n/(n-1)} \rightarrow \frac{Z_1}{Z_1+Z_2} \neq0 $$

Answer 2

You're right for most cases, however this depends on the distribution as well.

If $X_1, \dots, X_n \sim \Bbb{P}_\theta$ where $Var_\theta(X_i) = 0$, this doesn't hold true, because $$\forall i,j \in \{1,\dots,n\}\quad\Bbb{P}(X_i = X_j) = 1 \rightarrow \Bbb{P}\left(\lim_{n \to \infty}\frac{X_i}{X_j}=1\right)=1$$

But for most cases, where $Var_\theta(X_i) \ne 0$, you can prove this indirectly:

Let's assume that $$\exists \lim_{n \to \infty}\frac{X^{(2)}}{X^{(1)}} := A \in \Bbb{R}^+_0 \cup \{+\infty\}$$ We can assume that $A \ge 0$, because $X^{(2)} \ge X^{(1)}$.

If

$$\lim_{n \to \infty}\frac{X^{(2)}}{X^{(1)}} = A \stackrel{\text{then }}{\longrightarrow}\lim_{n \to \infty}\Bbb{E}\left(\frac{X^{(3)}}{X^{(2)}}\right) = A$$

Since $\forall \varepsilon, N_1 \in \Bbb{R}^+ \exists, N_2 \in \Bbb{N}^+$ such that from a sample of $X_1,\dots,X_{N_1}$ $$A - \varepsilon < \frac{X^{(2)}}{X^{(1)}} < A + \varepsilon,$$ We can assure that from a sample of $X_1,\dots,X_{N_2}$, where $N_2$ is sufficiently large: $$A - \varepsilon < \frac{X^{(3)}}{X^{(2)}} < A + \varepsilon$$ Since as $n \to \infty$, $$\lim_{n \to \infty}\left(\frac{X^{(3)}}{X^{(2)}} - \frac{X^{(2)}}{X^{(1)}}\right) = 0$$

Now we'll have to prove $2$ cases separately:

Case $1$: $A = 1$

From our previous argument it follows that

$$\lim_{n \to \infty}\frac{X^{(2)}}{X^{(1)}} = \lim_{n \to \infty}\frac{X^{(3)}}{X^{(2)}} = \dots = \lim_{n \to \infty}\frac{X^{(n)}}{X^{(n-1)}} = 1$$

$$\Rightarrow \lim_{n \to \infty}\frac{X^{(n)}}{X^{(1)}} = 1$$

$$\Rightarrow \forall i \in \{1,\dots, n\} \quad \sup {Ran}_{\theta} X_i = \inf {Ran}_{\theta} X_i$$

$$\Rightarrow \forall i \in \{1,\dots, n\}\quad Var(X_i) = 0$$

Which is a contradiction, becuase in this case we assumed that $Var(X_i) \ne 0$.

Case $2$: $A \in \Bbb{R}^+_0 \backslash \{1\}$

Again, we get

$$\lim_{n \to \infty}\frac{X^{(2)}}{X^{(1)}} = \lim_{n \to \infty}\frac{X^{(3)}}{X^{(2)}} = \dots = \lim_{n \to \infty}\frac{X^{(n)}}{X^{(n-1)}} = A$$

Which is this case means:

$$\lim_{n \to \infty}\frac{X^{(2)}}{X^{(1)}} = \lim_{n \to \infty}\frac{1}{A}\frac{X^{(3)}}{X^{(1)}} = \dots =\lim_{n \to \infty}\frac{1}{A^n}\frac{X^{(n)}}{X^{(1)}}$$

$$\Rightarrow \lim_{n \to \infty}\frac{X^{(n)}}{X^{(1)}} = \lim_{n \to \infty}A^n\frac{X^{(2)}}{X^{(1)}} = \\ = 0, +\infty \text{ or does not exist (depending on the value of } A)$$

It can't be $0$, since then $$\Bbb{P}(X_n = 0) = 1 \rightarrow \forall i \in \{1\dots n\}\quad \Bbb{P}(X_i = 0) = 1 \rightarrow Var(X_i) = 0$$ Which is a contradiction again.

It can' be $+ \infty$ either, because that can only occur if $X_n \ne 0$ and $X_{n-1} = 0$, but as we've already discussed,

$$\lim_{n \to \infty}\frac{X^{(n)}}{X^{(n-1)}} = \lim_{n \to \infty}\frac{X^{(n-1)}}{X^{(n-2)}} (= + \infty)$$

Which also means that $X_{n-1} \ne 0$ and $X_{n-2} = 0$, and we get a contradiction from $X_{n-1} = 0$ and $X_{n-1} \ne 0$.

The only option in this case is for the limit not to exist, which isn't technically a contradiction, but it further proves our point.

Result:

Both cases lead to contradiction, which means that the given limit does not exist.