Stochastic processes moments

Under current treatment of statistical method a set of the states of the Markovian stochastic process describing the ensemble of macromolecules with labeled units can be not only discrete but also continuous. So, for instance, when the description of the products of living anionic copolymerization is performed within the framework of a terminal model the role of the label characterizing the state of a monomeric unit is played by the moment when this unit forms in the course of a macroradical growth [25]. 

In the framework of this ultimate model [33] there are m2 constants of the rate of the chain propagation kap describing the addition of monomer to the radical Ra whose reactivity is controlled solely by the type a of its terminal unit. Elementary reactions of chain termination due to chemical interaction of radicals Ra and R is characterized by m2 kinetic parameters k f . The stochastic process describing macromolecules, formed at any moment in time t, is a Markov chain with transition matrix whose elements are expressed through the concentrations Ra and Ma of radicals and monomers at this particular moment in the following way [1,34] ... 

In this section, we begin the description of Brownian motion in terms of stochastic process. Here, we establish the link between stochastic processes and diffusion equations by giving expressions for the drift velocity and diffusivity of a stochastic process whose probability distribution obeys a desired diffusion equation. The drift velocity vector and diffusivity tensor are defined here as statistical properties of a stochastic process, which are proportional to the first and second moments of random changes in coordinates over a short time period, respectively. In Section VILA, we describe Brownian motion as a random walk of the soft generalized coordinates, and in Section VII.B as a constrained random walk of the Cartesian bead positions. 

We will present the equation of motion for a classical spin (the magnetic moment of a ferromagnetic single-domain particle) in the context of the theory of stochastic processes. The basic Langevin equation is the stochastic Landau-Lifshitz(-Gilbert) equation [5,45]. More details on this subject and various techniques to solve this equation can be found in the reviews by Coffey et al. [46] and Garcia-Palacios [8]. 

A stochastic process is called stationary when the moments are not affected by a shift in time, i.e., when... 

The generalization of the concept of a characteristic function to stochastic processes is the characteristic functional (In a different connection this idea was used in section II.3.) Let Y(t) be a given random process. Introduce an arbitrary auxiliary test function k(t). Then the characteristic or moment generating functional is defined as the following functional of k(t ... 

The extraction of a homogeneous process from a stationary Markov process is a familiar procedure in the theory of linear response. As an example take a sample of a paramagnetic material placed in a constant external magnetic field B. The magnetization Y in the direction of the field is a stationary stochastic process with a macroscopic average value and small fluctuations around it. For the moment we assume that it is a Markov process. The function Px (y) is given by the canonical distribution... 

One expects that the Langevin equation (1.1) is equivalent to the Fokker-Planck equation (VIII.4.6). This cannot be literally true, however, because the Fokker-Planck equation fully determines the stochastic process V(t), whereas the Langevin equation does not go beyond the first two moments. The reason is that the postulates (i), (ii), (iii) in section 1 say nothing about... 

For example, under kinetic modeling of "living" anionic copolymerization in the framework of the terminal model, a macromolecule is associated with the realization of a certain stochastic process. Its states (a,r) are monomeric units, each being characterized along with chemical type a and also by some label r. This random quantity equals the moment when this monomeric unit entered in a polymer chain as a result of the addition of o-type monomer to the terminal active center. It has been... 

The limitations associated with (7) are essentially a consequence of the stochastic nature of atmospheric transport and diffusion. Because the wind velocities are random functions of space and time, the airborne pollutant concentrations are random variables in space and time. Thus, the determination of the Cj, in the sense of being a specified quantity at any time, is not possible, but we can at best derive the probability density functions satisfied by the c. The complete specification of the probability density function for a stochastic process as complex as atmospheric diffusion is almost never possible. Instead, we must adopt a less desirable but more feasible approach, the determination of certain statisical moments of Ci, notably its mean, . (The mean concentration can be... 

In order to make the solution to the fractional Langevin equation a multifractal, we assume that the parameter a is a random variable. To construct the traditional measures of multifractal stochastic processes, we calculate the qt moment of the solution (148) by averaging over both the random force and the random parameter to obtain... 

The identities (7.63) and (7.64) are very useful because exponential functions of random variables of the forms that appear on the left sides of these identities are frequently encountered in practical applications. For example, we have seen (cf. Eq. (1.5)) that the average (e ), regarded as a function of a, is a generating function for the moments of the random variable z (see also Section 7.5.4 for a physical example). In this respect it is useful to consider extensions of (7.63) and (7.64) to non-Gaussian random variables and stochastic processes. Indeed, the identity (compare Problem 7.8)... 

Just as a random variable is characterized by the moments of its distribution, a stochastic process is characterized by its time correlation functions of various orders. In general, there are an infinite number of such functions, however we have seen that for the important class of Gaussian processes the first moments and the two-time coiTelation functions, simply referred to as time correlation functions, fully characterize the process. Another way to characterize a stationary stochastic process is by its spectral properties. This is the subject of this section. 

If x(Z) is real then = x . Equation (7.69) resolves x(Z) into its spectral components, and associates with it a set of coefficients x such that x p is the strength or intensity of the spectral component of frequency However, since each realization of x(Z) in the interval 0,..., T yields a different set x , the variables x are themselves random, and characterized by some (joint) probability function P( x ). This distribution in turn is characterized by its moments, and these can be related to properties of the stochastic process x(Z). For example, the averages x satisfy... 

For our purpose the important moments are ( x p), sometimes referred to as the average strengths of the Fourier components The power spectrum I (co) of the stochastic process is defined as the T -> oo limit of the average intensity at frequency m ... 

We have already noted the difference between the Langevin description of stochastic processes in terms of the stochastic variables, and the master or Fokker-Planck equations that focus on their probabilities. Still, these descriptions are equivalent to each other when applied to the same process and variables. It should be possible to extract information on the dynamics of stochastic variables from the time evolution of their probabihty distribution, for example, the Fokker-Planck equation. Here we show that this is indeed so by addressing the passage time distribution associated with a given stochastic process. In particular we will see (problem 14.3) that the first moment of this distribution, the mean first passage time, is very useful for calculating rates. 

Ao and Rammer [166] obtained the same result (and more) on the basis of a fully quantum mechanical treatment. Frauenfelder and Wolynes [78] derived it from simple physical arguments. Equation (9.98) predicts a quasiadiabatic result, = h k/ v 1 and the Golden Rule result, Pk = k/ v, in the opposite limit, which is qualitatively similar to the Landau-Zener behavior of the transition probability but the implications are different. Equation (9.98) is the result of multiple nonadiabatic crossings of the delta sink although it does not depend on details of the stochastic process Xj- t). This can be understood from the following consideration. For each moment of time, the fast coordinate has a Gaussian distribution, p Xf, t) = (xy — Xj, transition region, the fast coordinate crosses it very frequently and thus forms an effective sink for the slow coordinate. 

The other class of fluctuation phenomena, well-known since the work of Gibbs (done in 1902, see Gibbs (1948)) and Einstein (1910), is equilibrium fluctuation. The theory of equilibrium (thermostatic) fluctuations considers the equilibrium state as a stationary stochastic process (see, for example, Tisza Quay (1963) and Tisza (1966)). By thermostatic fluctuation theory the statistical character (e.g. the distribution functions and moments derived from it) can be computed. 

From a mathematical point of view the approximation by a Gaussian process is appropriate, in the sense that for every stochastic process having first and second moment there exists a Gaussian process with the same first and second moment. 

The basic idea of the MC approach lies in the discrete representation of the joint PDF by an ensemble of stochastic particles. Each particle carries an array of properties denoting position, velocity and scalar composition. During a fractional time stepping procedure [6] the particles are submitted to certain deterministic and stochastic processes changing each particle s set of properties in accordance with the different terms in the PDF evolution equation. Afterwards the statistical moments may be derived in the simplest case by averaging from the ensemble of particles. 

The wind velocity is of random nature at a specific time and location, i.e., it is a stochastic process in time and space. The buffeting lift, L(x,t), and moment, Q(x,t), are linear functions of the horizontal and vertical components of the wind turbulence, and they are assumed to be stationary random processes. Hence, the buffeting loads are represented by the cross-spectra of the buffeting lift and moment, which, in turn, are expressed in terms of the cross - spectra of the horizontal and vertical components of the wind velocity [6, 7, 16, 18]. 

There are basically three categories of ENA visual examination, sequence-independent methods that treat the collection of voltage or current values without regard to their position in the sequence of readings (moments, mean, variance, standard deviation, skewness, and kurtosis), and those that take the sequence into account (autocorrelation, power spectra, fractal analysis, stochastic process analysis) [22],... 

According to Khinchin-Cramer s theorem, there exists a Gaussian process of which all first and second moments are equal to those of arbitrary stochastic processes . ). 

In the analysis of stochastic processes, an important role is played by the so-called spectral moments (SM), introduced by Vanmarcke (1972), important for the definition of some characteristics of stochastic processes and in reliability analysis (Michaelov et al. (1999)). These quantities are defined as the moments of the one-sided PSD with respect to the frequency origin. Let Sy (co) be the PSD of the earthquake acceleration with 5y(ffl) = 5y(—co). Let Gy(co) be the one-sided PSD defined... 

Let us consider now a zero-mean stationary stochastic process, F(t). This process is characterized by the feature that its statistical moments do not change over time and generally arise from any stable system which has achieved a steady state. Moreover, the probabilistic structure of a stationary process is invariant under a shift of the time origin. This is a consequence of the fact that a sample of the process will almost certainly not decay to zero at infinity. Then the stationary processes possess infinite energy. Since a stationary process possesses infinite energy, its th sample, F (t), cannot be represented by the Fourier transform. In fact in this case, the Dirichlet condition is not satisfied. It follows that the spectral representation of a sample of a stationary... 

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