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Doob's Theorem


A theorem proved by Doob (1942) which states that any random process which is both normal and Markov has the following forms for its correlation function C_y(tau), spectral density G_y(f), and probability densities p_1(y) and p_2(y_1|y_2,tau):

C_y(tau)=sigma_y^2e^(-tau/tau_r)
(1)
G_y(f)=(4tau_r^(-1)sigma_y^2)/((2pif)^2+tau_r^(-2))
(2)
p_1(y)=1/(sqrt(2pisigma_y^2))e^(-(y-y^_)^2/2sigma_y^2)
(3)
p_2(y_1|y_2,tau)=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)},
(4)

where y^_ is the mean, sigma_y the standard deviation, and tau_r the relaxation time.


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References

Doob, J. L. "The Brownian Movement and Stochastic Equations." Ann. Math. 43, 351-369, 1942. Reprinted in Selected Papers on Noise and Stochastic Processes (Ed. N. Wax). New York: Dover, pp. 319-337, 1954.Finch, S. "Ornstein-Uhlenbeck Process." May 15, 2004. http://algo.inria.fr/csolve/ou.pdf.

Referenced on Wolfram|Alpha

Doob's Theorem

Cite this as:

Weisstein, Eric W. "Doob's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DoobsTheorem.html

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