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January 19th, 2018, 10:19 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, 10:32 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,299 Thanks: 1218 
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, 08:03 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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convergence, random, sequence 
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