# Confusion about random variables and convergence in probabilty and distribution

I'm studying statistical analysis and there's something fundamental I'm missing about random variables and how they are used in defining convergence in probability or distribution:

In my syllabus (which is in dutch, so the terms i use might be slightly off), when talking about samples, it says that

If we want to study the properties of a random variable $X$ in a target population, we take a random aselect sample of $n$ subjects from a collection of $n$ random variables $X_1, ... X_n$ that are all mutually (pairwise?) independent and which all have the same distribution, namely that of $X$ in the target population.

A bit further, discussing convergence, it says

An infinite row of random variables $X_1, X_2, ...$ on a probability space converges in probability to $X$ if the folowing is true for each $\epsilon > 0$: $\lim_{n\to\infty}{P(|X_n - X|)} \geq 0$

What I don't understand is what $X_i$ actually means in these two contexts. I read it as follows: In the first part, it is presented as one choice from the population: $X_i$ is the length of the $i$th of $n$ people, for example. In the second part, it seems as if now $X_i$ represents the distribution of all $X_k$ ($k \leq n$), which of course tends towards the actual distribution $X$.

Do these $X$ mean different things? Am I completely missing something? Can anyone help me make sense of this?

The second paragraph simply says that if $\lim_{n\to\infty} P(|X_n - X| \geq \epsilon) \to 0$ then we say $X_n \to X$ in probability. That's just a definition of what "convergence in probability means".
The first paragraph talks about specific sequences $X_i$ - namely of those which are independent and identically distributed. You may view those as samples drawn (with replacement) from a fixed population with cumulative distribution function (CDF) $D$.
You are correct that such a sequence of samples in general won't converge in probability. In fact, they never do unless the distribution of the $X_i$ is degenerate, i.e. there's an $x$ with $P(X_i = x) = 1$.
They do, however converge in distribution, which is a weaker form of convergence, and means that the cumulative distribution function (CDF) of $X_n$ converges pointwise to $D$, i.e. that $\lim_{n\to\infty} P(X_n \leq x) = D(x)$ for all $x \in \mathbb{R}$. They trivially do so - since all the $X_i$ have the same CDF, you don't even need a limit - you already have that $P(X_n \leq x) = D(x)$.