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Doob's Theorem

A theorem proved by Doob (1942) which states that any random process which is both Gaussian and Markov has the following forms for its correlation function, spectral density, and probability densities:

$\displaystyle C_y(\tau)$ $\textstyle =$ $\displaystyle {\sigma_y}^2e^{-\tau/\tau_r}$ (1)
$\displaystyle G_y(f)$ $\textstyle =$ $\displaystyle {4{\tau_r}^{-1} {\sigma_y}^2\over (2\pi f)^2+{\tau_r}^{-2}}$ (2)
$\displaystyle p_1(y)$ $\textstyle =$ $\displaystyle {1\over \sqrt{2\pi{\sigma_y}^2}} e^{-(y-\bar y)^2/2{\sigma_y}^2}$ (3)
$\displaystyle p_2(y_1\vert y_2,\tau)$ $\textstyle =$ $\displaystyle {1\over \sqrt{2\pi(1-e^{-2\tau/\tau_r}){\sigma_y}^2}}\mathop{\rm ...
...-\tau/\tau_r}(y_1-\bar y)]^2\over 2(1-e^{-2\tau/\tau_r}){\sigma_y}^2}}\right\},$ (4)

where $\bar y$ is the Mean, $\sigma_y$ the Standard Deviation, and $\tau_r$ the relaxation time.


Doob, J. L. ``Topics in the Theory of Markov Chains.'' Trans. Amer. Math. Soc. 52, 37-64, 1942.

© 1996-9 Eric W. Weisstein