How Is Non-Gaussian Noise Calculated in Stochastic Processes?

[unica]

Dear friends,I have a question about non-gaussian noise as follow:

R(t) is a white noise and Y1,Y2,Y3 are three uncorrelated Gaussian Variables with average 0 and standard deviation 1, so that
$$Z_{1}=\int^h_{0}R(t)dt=h^{1/2}Y_{1}$$
$$Z_{2}=\int^h_{0}Z_{1}(t)dt=\int^h_{0}(\int^t_{0}R(s)ds)dt=h^{3/2}(Y_{1}/2+Y_{2}/(2\sqrt{3})$$
I know how to deduce above formulas, as Z1,Z2 are both Gaussian noises, so we can calculate the covariance matrix to get them.

But a question emerged, when I calculated the following formula.

$$Z_{3}=\int^h_{0}Z_{1}^2(t)dt=\int^h_{0}(\int^t_{0}R(s)ds\int^t_{0}R(y)dy)dt\approx h^{2}/3(Y_{1}^2+Y_{3}+1/2)$$

Because Z3 is not a gaussian noise, and it refer to calculate the higher moments, so how to deduce the last formula?

 

[anathema]

Dear OP,

Thank you for your question. The formula for Z3 can be deduced by using the properties of white noise and Gaussian variables. Let's break down the equation step by step:

1. Z3 is defined as the integral of Z1^2 from 0 to h. This means that Z3 is the sum of all the squared values of Z1 over the interval of 0 to h.

2. Z1 is defined as the integral of R(t) from 0 to h. This means that Z1 is the sum of all the values of R(t) over the interval of 0 to h.

3. Since R(t) is a white noise, it has a mean of 0 and a variance of h. Therefore, the sum of all the values of R(t) over the interval of 0 to h is approximately h^2/3.

4. Now, let's consider the integral of R(t) from 0 to t, which is part of the definition of Z1. This integral can be written as the sum of all the values of R(t) over the interval of 0 to t.

5. Since R(t) is a white noise, the values of R(t) over the interval of 0 to t are uncorrelated. This means that the value of R(t) at any specific time t is independent of the value of R(s) at any other time s.

6. Therefore, when we square the integral of R(t) from 0 to t, we are essentially multiplying the sum of all the values of R(t) over the interval of 0 to t by itself. This results in the sum of all the squared values of R(t) over the interval of 0 to t.

7. Since R(t) has a variance of h, the sum of all the squared values of R(t) over the interval of 0 to t is approximately t^2h/3.

8. Now, we can substitute this value into the original equation for Z3 and simplify to get h^2/3(Y1^2 + Y3 + 1/2).

I hope this helps to clarify the derivation of the formula for Z3. Please let me know if you have any further questions.

 

[Moetasim]

Hello,

Non-Gaussian noise refers to any type of random signal that does not follow a Gaussian or normal distribution. In your example, Z3 is a non-Gaussian noise because it is not normally distributed. This can occur when the noise is highly correlated or has non-linear characteristics.

To deduce the last formula, you can use the fact that Z1 and Z2 are Gaussian noises with zero mean and calculate the higher moments using the formula h^(2n)(m2n/(2n)!) where m2n is the (2n)th moment of the Gaussian noise. This will give you the approximation of h^2/3(Y1^2 + Y3 + 1/2) for Z3.

It is important to note that in real-world situations, non-Gaussian noise is common and can significantly impact the accuracy of data analysis and modeling. Therefore, it is important for scientists to understand and account for non-Gaussian noise in their research and experiments.