I Higher-Order Time Correlation Functions of White Noise?

[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?

 

[jasonRF]

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]

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!