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:

$$C_y(\tau) = {\sigma_y}^2e^{-\tau/\tau_r}$$ (1)

$$G_y(f) = {4{\tau_r}^{-1} {\sigma_y}^2\over (2\pi f)^2+{\tau_r}^{-2}}$$ (2)

$$p_1(y) = {1\over \sqrt{2\pi{\sigma_y}^2}} e^{-(y-\bar y)^2/2{\sigma_y}^2}$$ (3)

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

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

References

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