I do a numerical experiment and find something strange. Suppose I have a unit vector on a sphere $(m_1,m_2,m_3),(m_1^2+m_2^2+m_3^2=1)$, then I rotate the vector on the sphere according to the following rules:
1> I chose uniformly a vector from the unit sphere $(v_1,v_2,v_3)$;
2> I rotate the vector $(m_1,m_2,m_3)$ around the axis $(v_1,v_2,v_3)$ at a uniform speed and sample its value uniformly, for example, I sample its value every theta angle;
3> every time I sampled N samples(eg, $N=10$), I go back to step 1 and repeat.
After I get enough sample, I compute the auto-correlation function of $m_1,m_2,m_3$ separately. I find that they behave like this:
Initially, I choose $(m_1,m_2,m_3)=(-1,-1,1)/\sqrt{3}$.
The peculiar points are:
1> Two of these functions collapse;
2>The don't go to zero but oscillate at a fixed amplitude as time $x$ goes to $\infty$.
3>They decay to the fixed amplitude at the time $x= N$($N$ is the number of samples I take everytime I choose a vector $(v_1,v_2,v_3)$).
Can someone give a explanation about what happens here?