Gaussian processes

The Shot Noise Process.—In this and the next section we shall discuss two specific random processes—the shot noise process53 and the gaussian process. These processes play a central role in many physical applications of the theory of random processes as well as being of considerable theoretical interest in themselves. 

The Gaussian Process.—A gaussian process was defined in the last section to be a process all of whose finite-order distributions are multi-dimensional gaussian distributions. This means that the multi-dimensional characteristic function of Px.fK must be of the form... 

We can conveniently summarize our results at this point by stating that if X(t) is a gaussian process with mean mx and autocorrelation... 

In this connection, it should be carefully noted that, even if X(t) is not a gaussian process, the mean and the autocorrelation function of the output of a linear, time-invariant filter are related to the mean and autocorrelation function of the input process according to Eqs. (3-293) and (3-294).64 This is an important fact of which use will be made in the next section. 

A. N. Malakhov, Cumulant Analysis of Random Non-Gaussian Processes and Its Transformations, Sovetskoe Radio, Moscow, 1978, in Russian. 

While the form of this term is the same as the viscous-dissipation term in the conditional acceleration, the modeling approach is very different. Indeed, while the velocity field in a homogeneous turbulent flow is well described by a multi-variate Gaussian process, the scalar fields are very often bounded and, hence, non-Gaussian. Moreover, joint scalar... 

Obrezanova, O., Csanyi, G., Gola, J.M.R., Segall, M.D. Gaussian processes a method for automatic QSAR modeling of ADME properties. J. Chem. Inf. Model. 2007, 47, 1847-57. 

In the second example of a Gaussian modulation, the frequency Q takes continuous values and is a Gaussian process. If further it is assumed to be Markovian, the Doob theorem8 tells us that its correlation function has a simple exponential decay,... 

In a condensed system, the local field on a magnetic spin can be considered as a stochastic process. If a constant magnetic field H0 is present in the z-direction, the local field H( ) can be decomposed into the parallel and the perpendicular components. If the constant field is strong enough, this decomposition is meaningful the parallel component Hz(t) causes adiabatic shifts of the resonance frequency, whereas the perpendicular component H t) produces nonadiabatic effects.6 If only the adiabatic part is considered, the problem is just that treated in Section II, and if the local field Hz(t) is assumed to be a Gaussian process, then the Gaussian model of Section III can be adopted. 

A process is called a Gaussian process if all its Pn are (multivariate) Gaussian distributions. In that case all cumulants beyond m = 2 are zero and... 

Thus a Gaussian process is fully specified by its average and its second moment [Pg.63]

Exercise. The definition of a Gaussian process would be moot if it were incompatible with (iii). Show, however, that when some P is Gaussian, so are all the lower ones. Also that the conditional probabilities are Gaussian. 

Each Pn is a multivariate Gaussian distribution, so that we are dealing with a Gaussian process. This enables one to use the equations of 1.6. It is then readily found that = 0, and xt x2,... 

J.L. Doob, Annals of Math. 43, 351 (1942) reprinted in wax. Other theorems about Gaussian processes are given in J.L. Doob, Annals Mathem. Statist. 15, 229 (1944). 

Exercise. Prove that for any Gaussian process (with zero mean and unit variance) the conditional average at t2, given the value at tl9 is... 

Exercise. When Y(t) is a Gaussian process and R(u y) is a Gauss distribution, then Pu(y, t) is also Gaussian, provided that the initial Pu(y, 0) is a delta function or Gaussian. 

Exercise. From the previous Exercise it follows that even for non-Markovian Gaussian processes the conditional probability P(x,t x0, t0) obeys an equation of type (6.16). However, this is not a master equation, as is betrayed by the fact that the coefficients depend on t0. Compare V.l. Indicate this dependence in the Fokker-Planck equation used by S.A. Adelman, J. Chem. Phys. 64, 124 (1976). 

The equivalence of the Langevin equation (1.1) to the Fokker-Planck equation (VIII.4.6) for the velocity distribution of our Brownian particle now follows simply by inspection. The solution of (VIII.4.6) was also a Gaussian process, see (VIII.4.10), and its moments (VIII.4.7) and (VIII.4.8) are the same as the present (1.5) and (1.6). Hence the autocorrelation function (1.8) also applies to both, so that both solutions are the same process. Q.E.D. 

Assumption (a) implies that V(t) is a stationary Gaussian process. The Langevin equation when solved subject to assumption (b) yields the velocity autocorrelation function... 

Doob s theorem states that a Gaussian process is Markovian if and only if its time correlation function is exponential. It thus follows that V is a Gaussian-Markov Process. From this it follows that the probability distribution, P(V, t), in velocity space satisfies the Fokker-Planck equation,... 

With identical arguments as previously used to show that E(t) is normal, it can be demonstrated that X(t) is a gaussian process with expectation and variance equal to... 

The stochastic process af t) is related to the random force Q (see details in Section 3.4). Both and cr (t) are assumed to be independent Gaussian processes. 

---