On joint exchangeability and conservative processes with stochastic rates

In a previous article Saunders (1975) investigated the form of transition probabilities for a generalization of conservative processes in which the usual transition rate parameters are replaced by time-dependent stochastic variables. The results of that investigation are given in terms of properties of exchangeable random variables and require that the process be in a particular initial state at time zero. This article removes the restriction on the initial state by using some properties of two sequences of jointly exchangeable variables. General results analogous to those obtained previously are shown to hold for general initial states.

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Conservative processes with stochastic rates

Journal of Applied Probability ◽

10.1017/s0021900200048257 ◽

1975 ◽

Vol 12 (03) ◽

pp. 447-456 ◽

Cited By ~ 3

Author(s):

Roy Saunders

Keyword(s):

Transition Rate ◽

Transition Probabilities ◽

Random Variables ◽

Time Dependent ◽

Independent Random Variables ◽

Exchangeable Random Variables

In this article we consider a generalisation of conservative processes in which the usual transition rate parameters are replaced by time-dependent stochastic variables. The main result of the article shows that these generalised processes which we call conservative processes with stochastic rates have transition probabilities which can be characterised in terms of exchangeable random variables in a manner similar to the characterisation of conservative processes in terms of independent random variables given by Bartlett (1949). We use this characterisation to obtain general expressions for the transition probabilities and to examine some limiting aspects of the processes. The carrier-borne epidemic is treated as a particular case of these generalised processes.

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Conservative processes with stochastic rates

Journal of Applied Probability ◽

10.2307/3212859 ◽

1975 ◽

Vol 12 (3) ◽

pp. 447-456 ◽

Cited By ~ 5

Author(s):

Roy Saunders

Keyword(s):

Transition Rate ◽

Transition Probabilities ◽

Random Variables ◽

Time Dependent ◽

Independent Random Variables ◽

Exchangeable Random Variables

In this article we consider a generalisation of conservative processes in which the usual transition rate parameters are replaced by time-dependent stochastic variables. The main result of the article shows that these generalised processes which we call conservative processes with stochastic rates have transition probabilities which can be characterised in terms of exchangeable random variables in a manner similar to the characterisation of conservative processes in terms of independent random variables given by Bartlett (1949). We use this characterisation to obtain general expressions for the transition probabilities and to examine some limiting aspects of the processes. The carrier-borne epidemic is treated as a particular case of these generalised processes.

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Local limit theorems for non-critical Galton–Watson processes by or without immigration

Journal of Applied Probability ◽

10.1017/s0021900200022750 ◽

1982 ◽

Vol 19 (02) ◽

pp. 262-271

Author(s):

R. Höpfner

Keyword(s):

Limit Theorems ◽

Critical Case ◽

Transition Probabilities ◽

Local Limit ◽

Limiting Distributions ◽

Initial State ◽

Final State ◽

Initial States ◽

Local Limit Theorems

From normal limiting distributions of suitably normed sequences of Galton–Watson processes or Galton-Watson processes with immigration, with initial states tending to ∞, we can derive local limit theorems for the transition probabilities Qn (i, j) and Pn (i, j) in the non-critical case, when initial state i and final state j tend to ∞ with n.

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A new strategy to analyze possible association structures between dynamic nocturnal hormone activities and sleep alterations in humans

AJP Regulatory Integrative and Comparative Physiology ◽

10.1152/ajpregu.90530.2008 ◽

2009 ◽

Vol 296 (4) ◽

pp. R1216-R1227 ◽

Cited By ~ 4

Author(s):

Stefanie Kalus ◽

Thomas Kneib ◽

Axel Steiger ◽

Florian Holsboer ◽

Alexander Yassouridis

Keyword(s):

Slow Wave Sleep ◽

Transition Probabilities ◽

Sleep Stage ◽

Selection Criterion ◽

Time Dependent ◽

Initial State ◽

Endocrine Activity ◽

Human Sleep ◽

Hormone Effects ◽

Probability Of Transition

The human sleep process shows dynamic alterations during the night. Methods are needed to examine whether and to what extent such alterations are affected by internal, possibly time-dependent, factors, such as endocrine activity. In an observational study, we examined simultaneously sleep EEG and nocturnal levels of renin, growth hormone (GH), and cortisol (between 2300 and 0700) in 47 healthy volunteers comprising 24 women (41.67 ± 2.93 yr of age) and 23 men (37.26 ± 2.85 yr of age). Hormone concentrations were measured every 20 min. Conventional sleep stage scoring at 30-s intervals was applied. Semiparametric multinomial logit models are used to study and quantify possible time-dependent hormone effects on sleep stage transition courses. Results show that increased cortisol levels decrease the probability of transition from rapid-eye-movement (REM) sleep to wakefulness (WAKE) and increase the probability of transition from REM to non-REM (NREM) sleep, irrespective of the time in the night. Via the model selection criterion Akaike's information criterion, it was found that all considered hormone effects on transition probabilities with the initial state WAKE change with time. Similarly, transition from slow-wave sleep (SWS) to light sleep (LS) is affected by a “hormone-time” interaction for cortisol and renin, but not GH. For example, there is a considerable increase in the probability of SWS-LS transition toward the end of the night, when cortisol concentrations are very high. In summary, alterations in human sleep possess dynamic forms and are partially influenced by the endocrine activity of certain hormones. Statistical methods, such as semiparametric multinomial and time-dependent logit regression, can offer ambitious ways to investigate and estimate the association intensities between the nonstationary sleep changes and the time-dependent endocrine activities.

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Markov Processes Involving q-Stirling Numbers

Combinatorics Probability Computing ◽

10.1017/s0963548397002964 ◽

1997 ◽

Vol 6 (2) ◽

pp. 165-178 ◽

Cited By ~ 11

Author(s):

D. CRIPPA ◽

K. SIMON ◽

P. TRUNZ

Keyword(s):

Markov Process ◽

Normal Distribution ◽

Markov Processes ◽

Random Graphs ◽

Transition Probabilities ◽

Analysis Of Algorithms ◽

Random Variables ◽

Stirling Numbers ◽

Time Dependent

In this paper we consider the Markov process defined byP1,1=1, Pn,[lscr ]=(1−λn,[lscr ]) ·Pn−1,[lscr ] +λn,[lscr ]−1 ·Pn−1,[lscr ]−1for transition probabilities λn,[lscr ]=q[lscr ] and λn,[lscr ]=qn−1. We give closed forms for the distributions and the moments of the underlying random variables. Thereby we observe that the distributions can be easily described in terms of q-Stirling numbers of the second kind. Their occurrence in a purely time dependent Markov process allows a natural approximation for these numbers through the normal distribution. We also show that these Markov processes describe some parameters related to the study of random graphs as well as to the analysis of algorithms.

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Local limit theorems for non-critical Galton–Watson processes by or without immigration

Journal of Applied Probability ◽

10.2307/3213479 ◽

1982 ◽

Vol 19 (2) ◽

pp. 262-271 ◽

Cited By ~ 2

Author(s):

R. Höpfner

Keyword(s):

Limit Theorems ◽

Critical Case ◽

Transition Probabilities ◽

Local Limit ◽

Limiting Distributions ◽

Initial State ◽

Final State ◽

Initial States ◽

Local Limit Theorems

From normal limiting distributions of suitably normed sequences of Galton–Watson processes or Galton-Watson processes with immigration, with initial states tending to ∞, we can derive local limit theorems for the transition probabilities Qn (i, j) and Pn (i, j) in the non-critical case, when initial state i and final state j tend to ∞ with n.

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Photoinduced Charge Transfer Dynamics in Carotenoid-Porphyrin-C60 Triad via the Linearized Semiclassical Nonequilibrium Fermi's Golden Rule

10.26434/chemrxiv.12519485.v1 ◽

2020 ◽

Author(s):

Zhengqing Tong ◽

Margaret S. Cheung ◽

Barry D. Dunietz ◽

Eitan Geva ◽

Xiang Sun

Keyword(s):

Charge Transfer ◽

Golden Rule ◽

Time Dependent ◽

Rate Coefficients ◽

Initial State ◽

Photoinduced Charge Transfer ◽

Transfer Rates ◽

Fermi's Golden Rule ◽

Photoinduced Charge ◽

Fermi’S Golden Rule

The nonequilibrium Fermi’s golden rule (NE-FGR) describes the time-dependent rate coefficient for electronic transitions, when the nuclear degrees of freedom start out in a nonequilibrium state. In this letter, the linearized semiclassical (LSC) approximation of the NE-FGR is used to calculate the photoinduced charge transfer rates in the carotenoid-porphyrin-C60 molecular triad dissolved in explicit tetrahydrofuran. The initial nonequilibrium state corresponds to impulsive photoexcitation from the equilibrated ground-state to the ππ* state, and the porphyrin-to-C60 and the carotenoid-to-C60 charge transfer rates are calculated. Our results show that accounting for the nonequilibrium nature of the initial state significantly enhances the transition rate of the porphyrin-to-C60 CT process. We also derive the instantaneous Marcus theory (IMT) from LSC NE-FGR, which casts the CT rate coefficients in terms of a Marcus-like expression, with explicitly time-dependent reorganization energy and reaction free energy. IMT is found to reproduce the CT rates in the system under consideration remarkably well.

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