On the Moment-Generating Functions of Extrema and Their Complements for Almost Semicontinuous Integer-Valued Poisson Processes on Markov Chains

2016 ◽  
Vol 67 (8) ◽  
pp. 1164-1182
Author(s):  
M. S. Herych ◽  
D. V. Husak
Keyword(s):  
Markov Chains ◽  
Generating Functions ◽  
Poisson Processes ◽  
Moment Generating Functions ◽  
The Moment

The Properties Related to the Moment Generating Function of the Fuzzy Variable

2014 ◽  
Vol 519-520 ◽  
pp. 863-866
Author(s):  
Sheng Ma
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Uncertainty Theory ◽  
Fuzzy Variable ◽  
Moment Generating Function ◽  
Distribution Functions ◽  
Moment Generating Functions ◽  
The Moment

In the paper, some properties related to the moment generating function of a fuzzy variable are discussed based on uncertainty theory. And we obtain the result that the convergence of moment generating functions to an moment generating function implies convergence of credibility distribution functions. Thats, the moment generating function characterizes a credibility distribution.

scholarly journals Quasimodularity of the kth residual cranks

2020 ◽  
pp. 1-13
Author(s):  
Thomas Morrill ◽  
Aleksander Simonič
Keyword(s):  
Generating Functions ◽  
Second Moments ◽  
Moment Generating Functions ◽  
The Moment

We study a family of residual crank generating functions defined on overpartitions, the so-called [Formula: see text]th residual cranks. Specifically, the moment generating functions associated to these cranks exhibit quasimodularity properties which are dependent on the choice of [Formula: see text]. We also show that the second moments of these cranks admit a combinatoric interpretation as weighted overpartition counts. This interpretation gives a refinement to an existing inequality between crank moments of differing modulus [Formula: see text].

The structured coalescent process with weak migration

2001 ◽  
Vol 38 (1) ◽  
pp. 1-17 ◽  
Author(s):  
Morihiro Notohara
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Approximate Solutions ◽  
Coalescence Time ◽  
Migration Rates ◽  
Structured Population ◽  
Moment Generating Functions ◽  
Structured Coalescent ◽  
The Moment ◽  
Segregating Sites

The aim of this paper is to study genealogical processes in a geographically structured population with weak migration. The coalescence time for sampled genes from different colonies diverges to infinity as the migration rates among colonies are close to zero. We investigate the moment generating functions of the coalescence time, the number of segregating sites and the number of allele types in sampled genes when there is low migration. Employing a perturbation method, we obtain a system of recurrence relations for the approximate solutions of these moment generating functions and solve them in some cases.

The structured coalescent process with weak migration

2001 ◽  
Vol 38 (01) ◽  
pp. 1-17 ◽  
Author(s):  
Morihiro Notohara
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Approximate Solutions ◽  
Coalescence Time ◽  
Migration Rates ◽  
Structured Population ◽  
Moment Generating Functions ◽  
Structured Coalescent ◽  
The Moment ◽  
Segregating Sites

The aim of this paper is to study genealogical processes in a geographically structured population with weak migration. The coalescence time for sampled genes from different colonies diverges to infinity as the migration rates among colonies are close to zero. We investigate the moment generating functions of the coalescence time, the number of segregating sites and the number of allele types in sampled genes when there is low migration. Employing a perturbation method, we obtain a system of recurrence relations for the approximate solutions of these moment generating functions and solve them in some cases.

Spectral analysis of M/G/1 and G/M/1 type Markov chains

1996 ◽  
Vol 28 (01) ◽  
pp. 114-165 ◽  
Author(s):  
H. R. Gail ◽  
S. L. Hantler ◽  
B. A. Taylor
Keyword(s):  
Markov Chains ◽  
Generating Functions ◽  
Complex Analysis ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
Equilibrium Analysis ◽  
Transform Method ◽  
Technical Problems ◽  
The Matrix ◽  
Transient Case

When analyzing the equilibrium behavior of M/G/1 type Markov chains by transform methods, restrictive hypotheses are often made to avoid technical problems that arise in applying results from complex analysis and linear algebra. It is shown that such restrictive assumptions are unnecessary, and an analysis of these chains using generating functions is given under only the natural hypotheses that first moments (or second moments in the null recurrent case) exist. The key to the analysis is the identification of an important subspace of the space of bounded solutions of the system of homogeneous vector-valued Wiener–Hopf equations associated with the chain. In particular, the linear equations in the boundary probabilities obtained from the transform method are shown to correspond to a spectral basis of the shift operator on this subspace. Necessary and sufficient conditions under which the chain is ergodic, null recurrent or transient are derived in terms of properties of the matrix-valued generating functions determined by transitions of the Markov chain. In the transient case, the Martin exit boundary is identified and shown to be associated with certain eigenvalues and vectors of one of these generating functions. An equilibrium analysis of the class of G/M/1 type Markov chains by similar methods is also presented.

On identifiability and order of continuous-time aggregated Markov chains, Markov-modulated Poisson processes, and phase-type distributions

1996 ◽  
Vol 33 (3) ◽  
pp. 640-653 ◽  
Author(s):  
Tobias Rydén
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Continuous Time ◽  
Poisson Processes ◽  
Hitting Times ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Minimum Number ◽  
Markov Modulated ◽  
Two Parameters

An aggregated Markov chain is a Markov chain for which some states cannot be distinguished from each other by the observer. In this paper we consider the identifiability problem for such processes in continuous time, i.e. the problem of determining whether two parameters induce identical laws for the observable process or not. We also study the order of a continuous-time aggregated Markov chain, which is the minimum number of states needed to represent it. In particular, we give a lower bound on the order. As a by-product, we obtain results of this kind also for Markov-modulated Poisson processes, i.e. doubly stochastic Poisson processes whose intensities are directed by continuous-time Markov chains, and phase-type distributions, which are hitting times in finite-state Markov chains.

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