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

In summary, the conversation discusses proving that the vector X approaches the unit sphere as N goes to infinity. This can be shown through mean-squared convergence, where ||x||^2 becomes concentrated around the boundary of the sphere with a mean of 1 and a variance of 0 as N increases.
  • #1
alpines4
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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.
 
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  • #2


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
 

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

1. What is the unit sphere and why is it important in vector convergence?

The unit sphere is a mathematical concept that refers to a sphere with a radius of 1, centered at the origin. It is important in vector convergence because it serves as a reference point for measuring the distance of a vector from the origin.

2. How is the mean-squared error used to measure vector convergence?

The mean-squared error (MSE) is a measure of how far a set of data points are from the line of best fit. In the context of vector convergence, it is used to measure the average distance of a set of vectors from the unit sphere. A lower MSE indicates a higher degree of convergence.

3. What does it mean for a vector to be concentrated around the unit sphere?

A vector is said to be concentrated around the unit sphere when its values are close to the unit sphere, meaning that the vector is converging towards the unit sphere. This indicates that the vector is becoming more and more similar to the unit sphere, and the distance between them is decreasing.

4. How can the convergence of vectors around the unit sphere be visualized?

The convergence of vectors around the unit sphere can be visualized using a scatter plot, where the unit sphere is represented as a circle and the vectors are plotted as points around it. As the vectors converge towards the unit sphere, the points will become more tightly clustered around the circle.

5. What factors can affect the convergence of vectors around the unit sphere?

The convergence of vectors around the unit sphere can be affected by various factors such as the initial values of the vectors, the number of iterations used in the convergence process, and the convergence criteria set by the researcher. Other external factors, such as noise or outliers in the data, can also impact the convergence of vectors.

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