Characteristic functions, Bochner's Thm.

In summary, the conversation revolves around the introduction of measure sets and characteristic functions in an online course on functional integration. The professor talked about the properties of characteristic functions and Bochner's Theorem, which states that if a function satisfies the properties, it is a Fourier transform of a measure. An example was given where the function c(t) = 1 satisfied the properties and was a Fourier transform of some measure H(x). The professor mentioned that H(x) is discrete and related to the Helmholtz function. The conversation then shifts to the question of what H(x) is and the speaker's attempt to find an answer through searching the Helmholtz function. However, they consider the possibility that H(x) is actually the heavy
  • #1
KrugalSausage
2
0
Hi,

I am taking an online course on functional integration and the professor is introducing measure sets and characteristic functions.

He introduced properties of characteristic functions and then gave us Bochner's Thm. which basically says that if a function satisfies the properties he listed, then it is a Fourier transform of a measure.

As an example he showed that the function c(t) = 1 is a function that satisfies the properties listed, so 1 is a Fourier transform of some measure H(x)

He then said that it is discrete (as opposed to a gaussian measure which is absolutely continuous), and that it is related to the Helmholtz function. Then he moved on to another topic.

My question is, what is H(x)? What function can you take the Fourier transform of and get the value 1? I think the hint was that it is discrete, but I can't think of what it might be.

I tried searching Helmholtz function but the search results give me the differential equation resulting from the Fourier transform of another differential equation that also has time dependence. I only mention this because I know that somewhere there is a link that I am not seeing as these transformations have been mentioned in the context of probability measures.

Thanks a lot!
-k
 
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  • #2
Thinking about it more,

It might be possible that me misspoke, and H(x) is just the heavy side step function, and not some Helmholtz function.

So that dH(x) is the delta function, so that the Fourier transform of the delta function would be 1.

Does this sound okay?
 

1. What are characteristic functions?

Characteristic functions are mathematical functions that are used to describe the probability distribution of a random variable. They are also known as moment generating functions and have a wide range of applications in statistics, probability, and signal processing.

2. How are characteristic functions related to Bochner's Theorem?

Bochner's Theorem is a fundamental result in probability theory that states that a function is a characteristic function if and only if it satisfies certain conditions, such as being continuous, positive definite, and having a specific limit at infinity. This theorem provides a powerful tool for analyzing and understanding characteristic functions.

3. What is the significance of Bochner's Theorem?

Bochner's Theorem is significant because it establishes a link between the properties of characteristic functions and the underlying probability distributions. This allows for the use of characteristic functions in a wide range of statistical and probabilistic applications, such as hypothesis testing, estimation, and simulation.

4. Can Bochner's Theorem be applied to any type of random variable?

Yes, Bochner's Theorem can be applied to any type of random variable, including discrete, continuous, and mixed distributions. This is because the conditions for a function to be a characteristic function are general and do not depend on the specific form of the underlying distribution.

5. How are characteristic functions used in practice?

Characteristic functions have a variety of practical applications, such as in finance for modeling stock prices and in engineering for analyzing signals. They are also used in probability and statistics for deriving the properties of random variables and for solving complex probability problems.

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