Random process, stationary

The mean values of the. (t) are zero and each is assumed to be stationary Gaussian white noise. The linearity of these equations guarantees that the random process described by the a. is also a stationary Gaussian-... 

Nagode, M. and Fajdiga, M. 1998 A General Multi-Modal Probability Density Function Suitable for the Rainflow Ranges of Stationary Random Processes. Int. Journal of Fatigue, 20(3), 211-223. 

To bring our terminology into accord -with that commonly used by mathematicians, the nomenclature stationary random process should be used here to distinguish our models from the more general models mentioned at the end of Section 3.1. This distinction is not meaningful in the context of this chapter, but should be borne in mind when consulting other treatments of the subject. 

At this point in the development of the theory of (stationary) random processes, we encounter what appears to be the only major... 

The mean velocity components are expressed as //, v, and ii>. We assume that the velocity components are stationary Gaussian random processes, so that, based on the preceding discussion, the autocovariances of u, V, and w can be written as (Papoulis, 1965, p. 397)... 

Exercise. Apply the result to the harmonic oscillator (1.3) with frequency co2(t) = cog l H- a (t), where t) is a stationary random process with zero mean and autocorrelation time tc. The answer is... 

A well known result states that the values of the discrete Fourier transform of a stationary random process are normally distributed complex variables when the length of the Fourier transform is large enough (compared to the decay rate of the noise correlation function) [Brillinger, 1981], This asymptotic normal behavior leads to a Rayleigh distributed magnitude and a uniformly distributed phase (see [McAulay and Malpass, 1980, Ephraim andMalah, 1984] and [Papoulis, 1991]). 

We observe that / is also called the statistically homogeneous (i.e. stationary) random process. Statistical homogeneity means that two geometric points of the space are statistically undistinguishable, or the statistical properties of the medium are invariant under the action of translation. Then we have a group C/x x lRn of isometries on L2(fi) = L2(Q,F, pt) defined by... 

In Eq. (22), the Langevin force F(t) may be considered as a Gaussian stationary random process of zero mean with correlation function given by Eq. (20). 

As stated above, the Langevin force F(t) can be viewed as corresponding to a stationary random process. Clearly, the same is true of the solution v(f) of the generalized Langevin equation (22), an equation which is valid once the limit ti —> —oo has been taken. Thus, Fourier analysis and the Wiener-Khintchine theorem can be used to obtain the velocity correlation function, which only depends on the observation time Cvv(t, t2) = Cvv(t —12). As in the classical case, the velocity does not age. 

In Eq. (141), it is assumed that the diffusing particle and the surrounding medium have been put in contact in an infinitely remote past, as pictured by the lower integration bound —oo in the retarded friction term. Both F(t) and v(t) can be viewed as stationary random processes. Their spectral densities are linked by... 

A random process is weakly stationary if its mean value and autocorrelation function are independent of r. Thus, for a weakly stationary random process, the mean value is a constant [fJiy r) = fiy] and the autocorrelation function depends only on the spatial lag 6 [e.g., Ryir, r + 6) = Ry d)]. A random process is strongly stationary if the infinite collection of higher order statistical moments and joint moments are space invariant. Most geophysical phenomena are not strongly stationary. However, the random process under study must be at least weakly stationary, otherwise the results of the space- or time-series analysis can be suspect. An extensive treatment of these statistical concepts is available 45, 46). A detailed re-... 

Let us now examine the theory of coagulation in a greater detail. In agreement with the theory of random processes, one of the two particles involved in Brownian motion may be viewed as stationary. It is thus possible for the one to bind the coordinate system origin to this stationary n-dimensional particle and say that the diffusion coefficient of the second particles is given by the mutual diffusion coefficient, Dmn = Dm + Dn (see... 

Thus we have found that the mean concentration of a tracer released in a flow where the velocity is a stationary, Gaussian random process has a distribution that is, itself, Gaussian. This is an important result. 

The well known probability density function can be considered as the limiting value of the probability that an amplitude of noise n(t) lies in an interval around a certain value, divided by the width of that interval. The shape, which is often Gaussian, and the width of the PDF, expressed in the standard deviation 0 j, are used for statistical calculations of detection limit etc. A random signal, or in general a family of functions of time (random process) of which the values vary randomly even if it is stationary, is not uniquely specified by a PDF, as is demonstrated in Figure 2. Both random signals have the same PDF, but they are obviously different. 

Under the assumption that the low frequency component is a stationary Gaussian random process, let us proceed with the specification of a detection algorithm. The detection limit indicates the performance of the detection algorithm. Detection of peaks of interest in the adjusted chromatogram involves removal of the low frequency component to the extent possible and perhaps the search in time for the desired peak. 

Expanding (t + At) in power series in At and assuming the random process to be stationary, we can transform the relation (11.132) into... 

On the basis of the Viner-Hinchin theorem the autocorrelation function of the stationary random process is represented by the Fourier integral ... 

It is well known that the stationary response x is a zero-mean Gaussian random process with the auto-correlation function ... 

The data that can go into computing factor returns will of course depend on what the factors are. It can include bond and index level data as well as currency exchange rates. Assume that we have the factor return series. To construct covariances, we could postulate that the underlying random processes are time stationary and compute covariances using equally weighted factor returns. We actually know that mar-... 

The Cauchy problem involving the Riesz-Feller derivative was analyzed in [166, 260]. In the next section we discuss the general Markov random processes with independent and stationary increments, the Levy processes, for which the characteristic function is known explicitly. 

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