John Travolski

I'm trying to understand exactly what the convergence of a sequence of random variables means. So if I have a sequence of random variables, $\lbrace X_n\rbrace$, which converges in probability to $X$, what exactly is $X_n$? I know that $X_n$ is a random variable, but how is it related to the random sequence? What exactly does the $n$ subscript denote when we say $X_n$?

romsek

You're overthinking things.

You know what a sequence is. You know what a random variable is.

A sequence of random variables is exactly what you'd expect it to be.

$n$ is just an identifier for one of the rvs in that sequence.

Suppose that $X$ is uniform over $[a,b]$

let $X_n$ be uniform over $\left[a+\dfrac 1 n,b-\dfrac 1 n\right]$

Then $X_n$ converges to $X$ in probability.

John Travolski

Okay, that makes perfect sense. They key that I was missing was this:

The $X_n$ refers to a random variable whose probability mass/density functions may or may not be dependent on $n$. Hence they're not identically distributed and the limit as n approaches infinity actually means something and isn't totally arbitrary.