Cumulant Generating Functions

The coefficients of the multivariable Taylor series expansion of G J) about the point where the Schwinger probes vanish are elements of the ROMs. Thus G J) is known as the generating functional for ROMs. Mathematically, the RDMs of the functional G J) are known as the moments. The moment-generating functional G(y) may be used to define another functional W J), known as the cumulant-generating functional, by the relation... 

Successive terms in this sequence can be obtained from a type of cumulant generating function. If, in the expansion of... 

The term (h — Xm), being multiplied by —N T, can now be interpreted as the entropy of mixing per particle in the small phase. It arises from the deviation of the (generalized) mean size m = m in the small phase from the mean size m of the parent, and it is given by the Legendre transform of the generalized cumulant generating function ... 

The basic problem of the FCS is to calculate a probability Pto(N) for N particles to pass a system during an observation time to- Equivalently, one can find a cumulant generating function (CGF) S(x),... 

A very useful tool for finding analytically the distribution of Nft) is to obtain and solve partial differential equations for the associated cumulant generating functions. The moment generating function, denoted by Ai (0, t), is defined for a multivariate integer-valued variable N (t) as... 

There are also a number of advantages to using cumulant generating functions instead of probability or moment generating functions. For instance, in the univariate case ... 

The boundary condition for this partial differential equation is obtained from (9.36). Multiplying both sides of this relationship by A x and using the definition of the cumulant generating function, the partial differential equation of the... 

To illustrate how to proceed using the cumulant generating functions, the well-known two-compartment model and the enzymatic reaction will be presented as examples of linear and nonlinear systems, respectively. In these examples, there are two interacting populations (m = 2) and the cumulant generating function is... 

Renshaw, E., Saddlepoint approximations for stochastic processes with truncated cumulant generating functions, Journal of Mathematical Applications in Medicine and Biology, Vol. 15, 1998, pp. 1—12. 

The cumulative generating function that is actually a formal series can be obtained from the moment generating function by division through 1 — exp(x). It is easily seen that the partial sums are obtained when the divisor is expanded in a geometric series and resolved term by term, namely... 

An uncomplicated solution technique that can be applied to equation (16) is the method of cumulants. In 1972 Koppel showed that the logarithm of the normalized autocorrelation function was identical to the cumulant generating function for the distribution of decay constantsl. The coefficients of the cumulant expansion can be related to the moments of the F(r) distribution. The Koppel equation can be expressed by... 

The cumulants can then be obtained from the cumulants generating function (logarithm of the moments generating function) with the central moments to be standardized. The results are Ki= =l,K2= =l,Ki= = 2,... 

The cumulants method is based on the formalism of the statistical cumulant generating function ... 

However, in general, the evaluation of the above integrals becomes cumbersome, especially in the case of higher-order moments. Then, in the case of a Poisson process, it is much easier to handle the cumulants, which can be obtained directly from the log-characteristic function, called also a cumulant-generating function. In the case of a filtered renewal process, the... 

Figure 3.8. The transformation of a rectangular into a normal distribution. The rectangle at the lower left shows the probability density (idealized observed frequency of events) for a random generator versus x in the range 0 < jc < 1. The curve at the upper left is the cumulative probability CP versus deviation z function introduced in Section 1.2.1. At right, a normal distribution probability density PD is shown. The dotted line marked with an open square indicates the transformation for a random number smaller or equal to 0.5, the dot-dashed line starting from the filled square is for a random number larger than 0.5.

In the last equation Hi(x) is the th Hermite polynomial. The reader may readily recognize that the functions look familiar. Indeed, these functions are identical to the wave functions for the different excitation levels of the quantum harmonic oscillator. Using the expansion (2.56), it is possible to express AA as a series, as has been done before for the cumulant expansion. To do so, one takes advantage of the linearization theorem for Hermite polynomials [42] and the fact that exp(-t2 + 2tx) is the generating function for these polynomials. In practice, however, it is easier to carry out the integration in (2.12) numerically, using the representation of Po(AU) given by expressions (2.56) and (2.57). 

The reconstruction functionals, derived in the previous section through the particle-hole duality, may also be produced through the theory of cumulants [21,22,24,26,39,55-57]. We begin by constructing a functional whose derivatives with respect to probe variables generate the reduced density matrices in second quantization. Because we require that additional derivatives increase the number of second quantization operators, we are led to the following exponential form ... 

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