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Guess first, then run Monte Carlo simulations to make a table of P2r,2n for n = 10 and r = 0, 1, 2, . . , 10. Chapter 2 Random processes By the time the Ornstein and Uhlenbeck paper appeared, it had become clear that Brownian motion was a phenomenon that could not be handled in a rigorous fashion by classical probability theory. For one thing, although Brownian motion is certainly a random affair, the variables that describe it are not the ordinary random variables of classical probability theory, but rather they are random functions.

Tn , we then identify several examples of stochastic processes: 1. e. f (x1 , . . , xn ; t1 , . . , tn ) = f (x1 , t1 ) · f (x2 , t2 ) · · · f (xn , tn ); in other words, all the information about the process is contained in the 1st -order density. g. from f (x1 , x2 , x3 ; t1 , t2 , t3 ) = f (x1 , t1 ) · f (x2 , t2 ) · f (x3 , t3 ), it follows that, f (x1 , x2 ; t1 , t2 ) = f (x1 , t1 ) · f (x2 , t2 ). However, the converse is not true. 2. Markov Process: Defined by the fact that the conditional probability density enjoys the property, f (xn , tn |x1 , .

35) 48 Applied stochastic processes where Syy (ω) is the matrix of the spectra of Cij (t − t′ ) and H is the matrix Fourier transform. 3. e. Y (t) is stationary). It is often easier to measure the t transported charge Z(t) = 0 Y (t′ )dt′ . Show that the spectral density of Y , SY Y (ω), is related to the transported charge fluctuations by MacDonald’s theorem, ω SY Y (ω) = π ∞ sin ωt d Z 2 (t) dt. dt 0 Hint: First show (d/dt) Z 2 (t) = 2 t 0 Y (0)Y (t′ ) dt′ . 4. Let the stochastic process X(t) be defined by X(t) = A cos(ωt) + B sin(ωt), where ω is constant and A and B are random variables.