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

1. Mar 10, 2012

### alpines4

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. Mar 10, 2012

### sunjin09

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