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Mean squared convergnce of normal random variables to the boundary of the unit sphere

  1. Mar 10, 2012 #1
    1. The problem statement, all variables and given/known data
    Let Y_i be standard normal random variables, and let X be an N vector of random variables, X=(X_1, ..., X_N) where X_i = 1/{sqrt{N}} * Y_i. I want to show that as N goes to infinity, the vector X becomes "close" to the unit sphere.



    2. Relevant equations



    3. The attempt at a solution
    I want to show for N large, ||X||^2 is concentrated around the boundary of the sphere, and I am told that I can frame this in terms of convergence of mean-squared. I have no idea how to formulate this problem in terms of mean-squared convergence.
     
  2. jcsd
  3. Mar 10, 2012 #2
    Re: Mean squared convergnce of normal random variables to the boundary of the unit sp

    You want to prove that ||x||^2→1 in the mean square sense, which means as a random variable, the mean of ||x||^2 is 1 and variance is 0 as N→inf
     
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