
Advanced Statistics Advanced Probability and Statistics Math Forum 
 LinkBack  Thread Tools  Display Modes 
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: 2,009 Thanks: 1042 
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. 

Tags 
convergence, random, sequence 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Sequence convergence (Leibniz and absolute convergence)  Chemist@  Calculus  7  November 1st, 2014 02:16 PM 
Almost sure convergence of i.i.d. random variables  ale88ssia  Advanced Statistics  0  May 3rd, 2014 02:20 AM 
Convergence of Random Variable  ananda  Algebra  1  November 25th, 2013 12:01 AM 
help with random variable convergence  sneaky  Algebra  0  November 19th, 2010 06:39 PM 
Almost sure convergence of sequences of random variables?  boot20  Advanced Statistics  3  July 9th, 2010 01:20 PM 