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July, 1991 $L_2$ Rates of Convergence for Attractive Reversible Nearest Particle Systems: The Critical Case
Thomas M. Liggett
Ann. Probab. 19(3): 935-959 (July, 1991). DOI: 10.1214/aop/1176990330


Reversible nearest particle systems are certain one-dimensional interacting particle systems whose transition rates are determined by a probability density $\beta(n)$ with finite mean on the positive integers. The reversible measure for such a system is the distribution $\nu$ of the stationary renewal process corresponding to this density. In an earlier paper, we proved under certain regularity conditions that the system converges exponentially rapidly in $L_2(\nu)$ if and only if the system is supercritical. This in turn is equivalent to $\beta(n)$ having exponential tails. In this paper, we consider the critical case, and give moment conditions on $\beta(n)$ which are separately necessary and sufficient for the convergence of the process in $L_2(\nu)$ at a specified algebraic rate. In order to do so, we develop conditions on the generator of a general Markov process which correspond to algebraic $L_2$ convergence of the process. The use of these conditions is also illustrated in the context of birth and death chains on the positive integers.


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Thomas M. Liggett. "$L_2$ Rates of Convergence for Attractive Reversible Nearest Particle Systems: The Critical Case." Ann. Probab. 19 (3) 935 - 959, July, 1991.


Published: July, 1991
First available in Project Euclid: 19 April 2007

zbMATH: 0737.60092
MathSciNet: MR1112402
Digital Object Identifier: 10.1214/aop/1176990330

Primary: 60K35

Keywords: algebraic rates of convergence for semigroups , interacting particle systems , nearest particle systems , renewal theory

Rights: Copyright © 1991 Institute of Mathematical Statistics


Vol.19 • No. 3 • July, 1991
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