Showing X Vector Concentrated Around Unit Sphere Convergence of Mean-Squared

[alpines4]

Homework Statement

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.

Homework Equations

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.

 

[sunjin09]

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