My Math Forum Convergence of Random Sequence

 January 19th, 2018, 09:19 AM #1 Senior Member   Joined: Oct 2015 From: Antarctica Posts: 128 Thanks: 0 Convergence of Random Sequence 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 $X_n$?
 January 19th, 2018, 09:32 AM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,094 Thanks: 1088 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.
 January 19th, 2018, 07:03 PM #3 Senior Member   Joined: Oct 2015 From: Antarctica Posts: 128 Thanks: 0 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.

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