Generalized Cumulants Representing a Transient Random Process in a Linear System

1977 ◽  
Vol 99 (2) ◽  
pp. 103-108 ◽  
Author(s):  
Arthur Mayer
Keyword(s):  
Linear System ◽  
Generating Function ◽  
State Vector ◽  
Tracking System ◽  
Normal Modes ◽  
Moment Generating Function ◽  
Initial Distribution ◽  
The State ◽  
Symmetric Tensors ◽  
The Moment

The state vector of a linear system responding to gaussian noise satisfies a Langevin equation, and the moment-generating function of the probability distribution of the state vector satisfies a partial differential equation. The logarithm of the moment-generating function is expanded in a power series, whose coefficients are organized into a sequence of symmetric tensors. These are the generalized cumulants of the time-dependent distribution of the state vector. They separately satisfy an infinite sequence of uncoupled ordinary differential tensor equations. The normal modes of each of the generalized cumulants are given by an easy formula. This specifies transient response and proves that all cumulants of a stable system are stable. Also, all cumulants of an unstable system are unstable. As an example, a particular non-gaussian initial distribution is assumed for the state vector of a second-order tracking system, and the transient fourth cumulant is calculated.

scholarly journals Open Markov Chains: Cumulant Dynamics, Fluctuations and Correlations

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 256
Author(s):  
Raúl Salgado-García
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Generating Function ◽  
Moment Generating Function ◽  
The State ◽  
Interacting Particles ◽  
The Moment ◽  
Dependent Correlation ◽  
Number Of Particles

In this work we propose a model for open Markov chains that can be interpreted as a system of non-interacting particles evolving according to the rules of a Markov chain. The number of particles in the system is not constant, because we allow the particles to arrive or leave the state space according to prescribed protocols. We describe this system by looking at the population of particles on every state by establishing the rules of time-evolution of the distribution of particles. We show that it is possible to describe the distribution of particles over the state space through the corresponding moment generating function. This description is given through the dynamics ruling the behavior of such a moment generating function and we prove that the system is able to attain the stationarity under some conditions. We also show that it is possible to describe the dynamics of the two first cumulants of the distribution of particles, which in some way is a simpler technique to obtain useful information of the open Markov chain for practical purposes. Finally we also study the behavior of the time-dependent correlation functions of the number of particles present in the system. We give some simple examples of open chains that either, can be fully described through the moment generating function or partially described through the exact solution of the cumulant dynamics.

Problèmes de premier passage résolubles par la méthode de séparation des variables

1995 ◽  
Vol 32 (02) ◽  
pp. 337-348
Author(s):  
Mario Lefebvre
Keyword(s):  
Brownian Motion ◽  
Stochastic Processes ◽  
Generating Function ◽  
Moment Generating Function ◽  
First Passage Times ◽  
Standard Brownian Motion ◽  
First Passage ◽  
The Moment ◽  
Passage Times

In this paper, bidimensional stochastic processes defined by ax(t) = y(t)dt and dy(t) = m(y)dt + [2v(y)]1/2 dW(t), where W(t) is a standard Brownian motion, are considered. In the first section, results are obtained that allow us to characterize the moment-generating function of first-passage times for processes of this type. In Sections 2 and 5, functions are computed, first by fixing the values of the infinitesimal parameters m(y) and v(y) then by the boundary of the stopping region.

scholarly journals Mellin’s Transform and Application to Some Time Series Models

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
S. S. Appadoo ◽  
A. Thavaneswaran ◽  
S. Mandal
Keyword(s):  
Time Series ◽  
Generating Function ◽  
Mellin Transform ◽  
Fuzzy Numbers ◽  
Moment Generating Function ◽  
Time Series Models ◽  
Form Expression ◽  
The Moment ◽  
Close Form ◽  
Skewness And Kurtosis

This paper uses the Mellin transform to establish the means, variances, skewness, and kurtosis of fuzzy numbers and applied them to the random coefficient autoregressive (RCA) time series models. We also give a close form expression to the moment generating function related to fuzzy numbers. It is shown that the results of the proposed time series models are consistent with those of the conventional time series models and that the developed concepts are straightforward and easily implemented.

scholarly journals On a Dual Model with Barrier Strategy

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Yuzhen Wen ◽  
Chuancun Yin
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Generating Function ◽  
Risk Model ◽  
Moment Generating Function ◽  
Dual Model ◽  
Barrier Strategy ◽  
Dividend Payments ◽  
The Moment

We consider the dual of the generalized Erlang(n)risk model with a barrier dividend strategy. We derive integro-differential equations with boundary conditions satisfied by the expectation of the sum of discounted dividends until ruin and the moment-generating function of the discounted dividend payments until ruin, respectively. The results are illustrated by several examples.

scholarly journals Errata: Estimation of non-Gaussian noise parameters in the wavelet domain using the moment-generating function

2012 ◽  
Vol 21 (3) ◽  
pp. 039802-1
Author(s):  
Jan Švihlík ◽  
Karel Fliegel ◽  
Jaromír Kukal ◽  
Eva Jerhotová ◽  
Petr Páta ◽  
...  
Keyword(s):  
Generating Function ◽  
Gaussian Noise ◽  
Moment Generating Function ◽  
Wavelet Domain ◽  
Noise Parameters ◽  
The Moment ◽  
Non Gaussian

The Moment-Generating Function and Negative Integer Moments

1981 ◽  
Vol 35 (3) ◽  
pp. 148-150 ◽  
Author(s):  
Noel Cressie ◽  
Anne S. Davis ◽  
J. Leroy Folks ◽  
J. Leroy Folks
Keyword(s):  
Generating Function ◽  
Moment Generating Function ◽  
Negative Integer ◽  
The Moment
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