Convergence in mean square (or L^2) sense

In summary, the conversation discusses the concept of convergence in "mean square" or L^2 sense. It includes an example from a textbook, where the function fn(x) is given and it is shown that the series telescopes. The discussion also mentions using a change of variable y=Nx and how it leads to the last line of the integral approaching infinity as N approaches infinity. The conversation ends with a request for clarification on the use of the change of variable.
  • #1
kingwinner
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Convergence in "mean square" (or L^2) sense

Homework Statement


This is an example from a textbook (with solutions) in which I am feeling confused.
Let fn(x) = [n/(1+n2x2)] - (n-1)/[1+(n-1)2x2] in the interval 0<x<L. This series telescopes so that
N
∑ fn(x) = N/(1+N2x2)
n=1

L
∫ [∑ fn(x)]2 dx =
0
L
∫ N2/(1+N2x2)2 dx =
0
NL
∫ N/(1+y2)2 dy (let y=Nx)
0
This last line -> +∞ as N->∞
Since it does not converge to 0, the series does NOT converge in the mean-square (or L2) sense to f(x)=0.

2. Homework Equations /concepts
Convergence in mean square/L2 sense

The Attempt at a Solution


N/A

(i) Now I don't understand why we have to use the change of variable y=Nx. What is the point of doing this?


(ii) Also, WHY as N->∞,
NL
∫ N/(1+y2)2 dy -> +∞ ?
0

Can someone please explain?
Thank you!
 
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  • #2


Why is it a good idea to bring N into the limits of integration? After the change of variable, N appears both in the upper limit of integration AND in the integrand...
 

1. What is convergence in mean square sense?

Convergence in mean square sense, also known as convergence in L^2 sense, is a type of convergence in probability theory. It measures how close a sequence of random variables is to its limit in terms of mean square error. Essentially, it measures the average distance between the sequence and its limit as the number of terms in the sequence increases.

2. How is convergence in mean square sense different from other types of convergence?

Convergence in mean square sense is different from other types of convergence, such as almost sure convergence or convergence in probability, because it takes into account the magnitude of the errors between the sequence and its limit, rather than just the probability of these errors occurring.

3. What are the necessary conditions for convergence in mean square sense?

In order for a sequence of random variables to converge in mean square sense, it must first converge in probability. Additionally, the sequence must have a finite second moment, meaning that the expected value of the square of the random variable is finite.

4. What is the significance of convergence in mean square sense?

Convergence in mean square sense is an important concept in probability theory as it allows us to make statements about the average behavior of a sequence of random variables, rather than just the behavior of individual terms. It is also often used in the analysis of statistical models and their performance.

5. How is convergence in mean square sense used in statistical analysis?

In statistical analysis, convergence in mean square sense is used to evaluate the performance of estimators and models. If a sequence of estimators or models converges in mean square sense, it means that the average error between the estimator/model and the true value is decreasing as the sample size increases. This is desirable as it indicates that the estimator/model is becoming more accurate.

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