L2 RATES OF CONVERGENCE FOR ATTRACTIVE REVERSIBLE NEAREST PARTICLE-SYSTEMS - THE CRITICAL CASE

• 成果类型: Article
• 署名作者: LIGGETT, TM
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176990330
• 发表日期: 1991
• 页码: 935-959
• 关键词: central limit-theorem; invariance-principle

摘要

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 upsilon 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 L2(upsilon) 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 L2(upsilon) 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 L2 convergence of the process. The use of these conditions is also illustrated in the context of birth and death chains on the positive integers.