Doob's theorem (Doob 1942) states that any random process which is both normal and Markov has the following forms for its correlation
function 
,
spectral density 
,
and probability densities 
and 
:
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  | ![1/(sqrt(2pi(1-e^(-2tau/tau_r))sigma_y^2))exp{-([(y_2-y^_)-e^(-tau/tau_r)(y_1-y^_)]^2)/(2(1-e^(-2tau/tau_r))sigma_y^2)},](/images/equations/DoobsTheorem/Inline16.svg) |
(4)
|
where 
is the mean, 
the standard deviation,
and 
the relaxation time.
