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Extra info for Applied Stochastic Processes in Science and Engineering
Xn ; t1 , . . , tn ) = f (xn , tn |xn−1 , . . , x1 ; tn−1 , . . , t1 )× . . × f (x2 , t2 |x1 , t1 ) × f (x1 , t1 ). Intuitively, the relationship between the conditional and joint probabilities is very straightforward. Focusing upon the two variable case, f (x1 , x2 ; t1 , t2 ) = f (x2 , t2 |x1 , t1 ) × f (x1 , t1 ), simply means the probability that the state is x2 at time t2 and x1 at t1 is equal to the probability that the state is x2 at time t2 given that it was x1 at t1 multiplied by the probability that the state actually was x1 at t1 .
If we multiply η(t + τ ) by y ⋆ (t) (where the gate) and take the average, then η(t + τ )y ⋆ (t) = ∞ −∞ ⋆ denotes the complex conju- η(t + τ )η ⋆ (t − τ ′ ) h⋆ (τ ′ )dτ ′ . After a change of variables, and a Fourier transform, Sηy (ω) = Sηη (ω)H ⋆ (ω). 17) In a similar way, we also have, Syy (ω) = H(ω)Sηy (ω). 18) 42 Applied stochastic processes Combining Eqs. 18, yields the fundamental relation, Syy (ω) = Sηη (ω)H(ω)H ⋆ (ω) = Sηη (ω) × |H(ω)|2 . 20) the impulse response H(ω) is simply H(ω) = 1/(iω + β), so that Syy (ω) = Sηη (ω) 1 Γ = .
2) The algorithm can be continued. This property makes Markov processes manageable, and in many applications (for example Einstein’s study of Brownian motion), this property is approximately satisfied by the process over the coarse-groaned time scale (see p. 59). Notice, too, the analogy with ordinary differential equations. Here, we have a process f (x1 , x2 , x3 , . . ; t1 , t2 , t3 , . ) with a propagator f (xi+1 , ti+1 |xi , ti ) carrying the system forward in time, beginning with the initial distribution f (x1 , t1 ).