Example of a non-Gaussian stochastic process?

[Jano L.]

Consider stochastic process $X(t)$ with properties

$$
\langle X(t) \rangle = 0,
$$

$$
\langle X(t) X(t-\tau) \rangle = C_0e^{-|\tau|/\tau_c}.
$$

For example, the position of a Brownian particle in harmonic potential can be described by $X$. In that case, the probability distribution will be Gaussian, i.e.

$$
\frac{dp}{dX} (X) = Ae^{-\frac{|X^2|}{2\sigma^2}}
$$

with some $A, \sigma$.

Do you know some example of a stochastic process $X$ with the above two properties (average value and correlation function) which however does not have Gaussian probability distribution?

 

[chiro]

Hey Jano L.

If the solution to this process implies a unique form for the Moment Generating Function then the answer to your question is no since a particular MGF implies a unique form of a distribution.

 

[Jano L.]

Well, I do not know the MGF for that process. I think I cannot determine it just from those two properties.

I just know those two averages. I think there are different processes than Gaussian with the above properties, so I was wondering whether there is some good example...

 

[chiro]

If you can show some kind of uniqueness for MGF, characteristic function, or any other attribute that unique describes a distribution (or family of distributions) then you can show it is unique.

Can you relate p or its derivative to one of the above attributes?

 

[Jano L.]

No, I do not think so. I think the properties I know (see OP) do not define the MGF or probability distribution completely. I was just wondering about some concrete example of such process, which would have those two properties from OP but be non-Gaussian.