How does asymptotic approximation follow in this scenario?

In summary, the author claims that as M approaches infinity, the double sum of the squared magnitude of random variables hi(l) approaches the value of M, where the expected value of the squared magnitude of hi(l) is equal to 1. This is due to the law of large numbers and the independence of hi and hj for i ≠ j.
  • #1
EngWiPy
1,368
61
Hello,

I am reading a paper, and the author claimed that in asymptotic sense as M goes to infinite:

[tex]\sum_{i=1}^M\sum_{l=0}^L|h_i(l)|^2=M[/tex]

where:

[tex]\sum_{l=0}^L\mathbb{E}\left\{|h_i(l)|^2\right\}=1[/tex].

How is that asymptotic follows?

Thanks in advance
 
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  • #2
There are undefined symbols. What kind of things are hi(l)? What is the meaning of E?
 
  • #3
mathman said:
There are undefined symbols. What kind of things are hi(l)? What is the meaning of E?

E is the expectation, and h are random variables.

I got it, it is just by using the law of large numbers.

Thanks
 
  • #4
In the equation cited, what is asymptotic? Ratio -> 1 (true) or difference -> 0 (false)?
 
  • #5
mathman said:
In the equation cited, what is asymptotic? Ratio -> 1 (true) or difference -> 0 (false)?

as M goes to infinite.
 
  • #6
S_David said:
as M goes to infinite.

I know you mean as M -> ∞, but my question is what is supposed to happening as M -> ∞, expression divided by M -> 1 (true) or expression minus M -> 0 (false)?
 
  • #7
mathman said:
I know you mean as M -> ∞, but my question is what is supposed to happening as M -> ∞, expression divided by M -> 1 (true) or expression minus M -> 0 (false)?

I am sorry, I did not understand you quiet well. Can you say it in different way, please?
 
  • #8
I have the feeling that he is dividing by M.
 
  • #9
chiro said:
I have the feeling that he is dividing by M.

If he is dividing by the M the result would be 1 not M.
 
  • #10
S_David said:
I am sorry, I did not understand you quiet well. Can you say it in different way, please?
Let ∑(M) be the double sum you are talking about. There are two ways of expressing the limit as M -> ∞, {∑(M)}/M -> 1 (true) or ∑(M) - M -> 0 (false).
 
  • #11
mathman said:
Let ∑(M) be the double sum you are talking about. There are two ways of expressing the limit as M -> ∞, {∑(M)}/M -> 1 (true) or ∑(M) - M -> 0 (false).

[tex]\lim_{M\xrightarrow{}\infty}\frac{1}{M}\sum_{i=1}^M\sum_{l=0}^L|h(l)|^2=1[/tex]
 
  • #12
S_David said:
[tex]\lim_{M\xrightarrow{}\infty}\frac{1}{M}\sum_{i=1}^M\sum_{l=0}^L|h(l)|^2=1[/tex]
True. The author is making use of one of the fundamental theorems of probability theory - the law of large numbers.
 
  • #13
mathman said:
True. The author is making use of one of the fundamental theorems of probability theory - the law of large numbers.

Yes, right. Thanks
 
  • #14
S_David said:
Yes, right. Thanks

There is one caveat: hi independent of hj for i ≠ j.
 

Related to How does asymptotic approximation follow in this scenario?

1. What is asymptotic approximation?

Asymptotic approximation is a mathematical technique used to approximate a function or a set of data points using a simpler function or set of data points. It is commonly used when the original function or data set is too complex to work with, or when a simpler representation is needed for practical purposes.

2. How is asymptotic approximation different from other approximation methods?

Asymptotic approximation is different from other approximation methods because it focuses on the behavior of the function or data set as it approaches a particular point or value. It does not try to fit the function or data set to a specific curve or line, but rather looks at its overall behavior. This makes it useful for approximating complex functions or data sets that may not fit a specific curve or line.

3. What are some common applications of asymptotic approximation?

Asymptotic approximation is commonly used in fields such as engineering, physics, and computer science. It is used to approximate complex functions or data sets in various applications, including signal processing, control systems, and data analysis. It is also used in numerical methods for solving differential equations and in machine learning algorithms.

4. Can asymptotic approximation be used for all types of functions or data sets?

No, asymptotic approximation is not suitable for all types of functions or data sets. It works best for functions or data sets that exhibit a specific behavior as they approach a particular point or value. If the function or data set does not exhibit this behavior, other approximation methods may be more appropriate.

5. What are some limitations of asymptotic approximation?

Asymptotic approximation is not always accurate and may introduce errors in the approximation. It also requires a good understanding of the behavior of the function or data set being approximated and may not be suitable for beginners. Additionally, it may not work well for functions or data sets with multiple variables or complicated behavior near the point being approximated.

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