A probabilistic proof of Cooper and Frieze's First Visit Time Lemma

IF 0.6 4区 数学 Q4 STATISTICS & PROBABILITY
F. Manzo, Matteo Quattropani, E. Scoppola
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引用次数: 9

Abstract

We present an alternative proof of the so-called First Visit Time Lemma (FVTL), originally presented by Cooper and Frieze in its first formulation in Cooper and Frieze (2005), and then used and refined in a list of papers by Cooper, Frieze and coauthors. We work in the original setting, considering a growing sequence of irreducible Markov chains on n states. We assume that the chain is rapidly mixing and with a stationary measure with no entry being either too small nor too large. Under these assumptions, the FVTL shows the exponential decay of the distribution of the hitting time of a given state x—for the chain started at stationarity—up to a small multiplicative correction. While the proof by Cooper and Frieze is based on tools from complex analysis, and it requires an additional assumption on a generating function, we present a completely probabilistic proof, relying on the theory of quasi-stationary distributions and on strong-stationary times arguments. In addition, under the same set of assumptions, we provide some quantitative control on the Doob’s transform of the chain on the complement of the state x.
Cooper和Frieze首次访问时间引理的概率证明
我们提出了所谓的首次访问时间引理(FVTL)的另一种证明,FVTL最初是由Cooper和Frieze(2005)在Cooper和Frieze的第一个公式中提出的,然后在Cooper、Frieze和合作者的一系列论文中使用和改进。我们在原始环境中工作,考虑n个状态上的不可约马尔可夫链的增长序列。我们假设链正在快速混合,并且具有平稳的度量,没有入口太小或太大。在这些假设下,FVTL显示了给定状态x的命中时间分布的指数衰减-对于从平稳开始的链-直到一个小的乘修正。虽然Cooper和Frieze的证明是基于复杂分析的工具,并且它需要对生成函数进行额外的假设,但我们提出了一个完全概率的证明,依赖于准平稳分布理论和强平稳时间参数。此外,在相同的假设集合下,我们对状态x的补上的链的Doob变换提供了一些定量控制。
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来源期刊
CiteScore
1.10
自引率
0.00%
发文量
48
期刊介绍: ALEA publishes research articles in probability theory, stochastic processes, mathematical statistics, and their applications. It publishes also review articles of subjects which developed considerably in recent years. All articles submitted go through a rigorous refereeing process by peers and are published immediately after accepted. ALEA is an electronic journal of the Latin-american probability and statistical community which provides open access to all of its content and uses only free programs. Authors are allowed to deposit their published article into their institutional repository, freely and with no embargo, as long as they acknowledge the source of the paper. ALEA is affiliated with the Institute of Mathematical Statistics.
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