# Higher-Order Time Correlation Functions of White Noise?

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• Opus_723
In summary, the 4th-order correlation of Gaussian white noise can be calculated using the standard result from probability theory for zero-mean jointly-Gaussian random variables. The Kronecker delta is not necessary in this case, and the resulting equation can be derived using characteristic functions. However, the Kronecker delta may be necessary in certain cases, such as when dealing with uncorrelated forces acting on different particles.
Opus_723
Suppose I have Gaussian white noise, with the usual dirac-delta autocorrelation function,

<F1(t1)F2(t2)> = s2*d(t1-t2)*D12

Where s is the standard deviation of the Gaussian, little d is the delta function, and big D is the kronecker delta. For concreteness and to keep track of units, say F represents a force.

What is the 4th-order correlation of this same white noise?

<F(t1)F(t2)F(t3)F(t4)> = ?

My first guess, would be to simply let s2 become 3s4 to capture the higher moment in the Gaussian distribution, and add a couple of dirac-deltas and kroneckers to make sure it's only nonzero when all t1,t2,t3,t4 are the same:

<F(t1)F(t2)F(t3)F(t4)> = 3s4*d(t1-t2)d(t2-t3)d(t3-t4)*D12*D23*D34

But this is, of course, wrong. We would need three dirac deltas to ensure that the result is nonzero only when all times are the same, but this introduces too many units of 1/time. It no longer makes any sense as a 4th-order correlation of forces. Somehow this must be accomplished with a different structure.

So what is the right way to do this? Do we need to specify additional properties of the white noise in order to calculate this?

First, the Kronecker delta is not needed for your autocorrelation function. The Dirac delta is all that you need.

Anyway, since your white noise is Gaussian, I would use a standard result from probability theory that if you have four zero-mean jointly-Gaussian random variables ##x_1, x_2, x_3, x_4##, then
$$\left\langle x_1 x_2 x_3 x_4 \right\rangle = \left\langle x_1 x_2\right\rangle \left\langle x_3 x_4\right\rangle + \left\langle x_1 x_3\right\rangle \left\langle x_2 x_4\right\rangle + \left\langle x_1 x_4\right\rangle \left\langle x_2 x_3\right\rangle$$

This can be derived using characteristic functions - it isn't a difficult derivation but it is messy.

Jason

Opus_723
jasonRF said:
First, the Kronecker delta is not needed for your autocorrelation function. The Dirac delta is all that you need.

Anyway, since your white noise is Gaussian, I would use a standard result from probability theory that if you have four zero-mean jointly-Gaussian random variables ##x_1, x_2, x_3, x_4##, then
$$\left\langle x_1 x_2 x_3 x_4 \right\rangle = \left\langle x_1 x_2\right\rangle \left\langle x_3 x_4\right\rangle + \left\langle x_1 x_3\right\rangle \left\langle x_2 x_4\right\rangle + \left\langle x_1 x_4\right\rangle \left\langle x_2 x_3\right\rangle$$

This can be derived using characteristic functions - it isn't a difficult derivation but it is messy.

Jason

Thank you. This is exactly what I needed, and in retrospect makes a lot of sense. I will note that the Kronecker delta *is* necessary in my particular problem as I have uncorrelated forces acting on different particles, but I probably didn't make that very clear.

Thanks again!

## 1. What are higher-order time correlation functions?

Higher-order time correlation functions are mathematical tools used in the analysis of random signals, such as white noise. They measure the statistical dependence between different points in time within a signal.

## 2. How are higher-order time correlation functions different from autocorrelation functions?

Higher-order time correlation functions take into account the statistical dependence between more than two points in time, while autocorrelation functions only consider the dependence between two points.

## 3. What is white noise?

White noise is a type of random signal that has a flat power spectrum, meaning it contains equal energy at all frequencies. It is often used as a model for random disturbances in scientific experiments and engineering systems.

## 4. How are higher-order time correlation functions calculated?

Higher-order time correlation functions are typically calculated using mathematical formulas or algorithms, depending on the specific function being used. They may involve integrals, convolutions, or other mathematical operations.

## 5. What is the significance of higher-order time correlation functions in scientific research?

Higher-order time correlation functions are important in many fields of science, such as physics, chemistry, and engineering. They can provide insights into the underlying dynamics and behavior of complex systems, and are especially useful in the analysis of noisy data.

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