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 $P(X_n \leq x) = D(x)$.


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