
#1
Jan213, 02:27 PM

P: 1,027

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? 



#2
Jan213, 06:47 PM

P: 4,570

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. 



#3
Jan313, 05:58 PM

P: 1,027

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... 



#4
Jan313, 07:22 PM

P: 4,570

Example of a nonGaussian stochastic process?
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? 



#5
Jan413, 09:31 AM

P: 1,027

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 nonGaussian.



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