Markov Chains and Stochastic Stability

“…The proof we now sketch follows a rather standard line of argument. We refer the reader to [16] [20] [17] for the missing details. The proof has three elements.…”

Section: Part IImentioning

confidence: 99%

Propagation of fluctuations in biochemical systems, II: nonlinear chains

Anderson

Mattingly

2007

IET Syst. Biol.

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We consider biochemical reaction chains and investigate how random external fluctuations, as characterized by variance and coefficient of variation, propagate down the chains. We perform such a study under the assumption that the number of molecules is high enough so that the behavior of the concentrations of the system is well approximated by differential equations. We conclude that the variances and coefficients of variation of the fluxes will decrease as one moves down the chain and, through an example, show that there is no corresponding result for the variances of the chemical species. We also prove that the fluctuations of the fluxes as characterized by their time averages decrease down reaction chains. The results presented give insight into how biochemical reaction systems are buffered against external perturbations solely by their underlying graphical structure and point out the benefits of studying the out-of-equilibrium dynamics of systems.

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“…The definition of null recurrence, positive recurrence, and transience as defined above are not generally a mutually exclusive partition of possibilities, however, this is true in the context of AR processes [for a more general treatment the reader is referred to Meyn and Tweedie (1993)]. An alternative definition of recurrence uses the notion of potential.…”

Section: Definition 1 (Recurrence)mentioning

confidence: 99%

“…Note that when b = 1 we recover an autoregressive sequence. Recurrence properties of Markov chains are discussed more generally in Borovkov (1998) and Meyn and Tweedie (1993).…”

Section: Introductionmentioning

confidence: 99%

Recurrence properties of autoregressive processes with super-heavy-tailed innovations

Zeevi¹

2004

J. Appl. Probab.

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This paper studies recurrence properties of autoregressive (AR) processes with "super-heavy tailed" innovations. Specifically, we study the case where the innovations are distributed, roughly speaking, as log-Pareto random variables (i.e., the tail decay is essentially a logarithm raised to some power). We show that these processes exhibit interesting and somewhat surprising behavior. In particular, we show that AR(1) processes, with the usual root assumption that is necessary for stability, can exhibit null-recurrent as well as transient dynamics when the innovations follow a log-Cauchy type distribution. In this regime, the recurrence classification of the process depends, somewhat surprisingly, on the value of the constant pre-multiplier of this distribution. More generally, for log-Pareto innovations, we provide a positive recurrence/null recurrence/transience classification of the corresponding AR processes.Short Title: AR processes with super-heavy tails

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Markov Chains and Stochastic Stability

Cited by 2,917 publications

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A Paradigm for Derivatives of Positive Systems

A Paradigm for Derivatives of Positive Systems

Propagation of fluctuations in biochemical systems, II: nonlinear chains

Recurrence properties of autoregressive processes with super-heavy-tailed innovations

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