用固定半径或生长随机球覆盖紧凑空间

IF 0.6 4区 数学 Q4 STATISTICS & PROBABILITY
D. Aldous
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引用次数: 2

摘要

. 简单的随机覆盖模型,在欧氏空间中得到了很好的研究,也可以在一般紧度量空间S上定义。在一个特定的模型中,“种子”作为泊松过程(在时间上)到达随机位置,在S上有一些分布θ,并产生半径以恒定速率增加的球。通过标准化速率,覆盖时间C只取决于θ。值χ (S) = min θ E θ C是紧空间S的一个数值特征,并给出了S的覆盖数的弱一般上界和下界。这暗示了一个未来的研究计划,改进这样的一般界限,并估计χ (S)为熟悉的紧空间的例子。我们处理一个例子,无限积空间[0,1]∞与积拓扑。在另一个主题上,通过类比几何模型、离散券集问题和有限马尔可夫链的覆盖时间,我们期望C分布的“弱集中”界在最小假设下成立。我们用一个关于集值递增马尔可夫过程的一般结果的一个简单结论来证明这一点。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Covering a compact space by fixed-radius or growing random balls
. Simple random coverage models, well studied in Euclidean space, can also be defined on a general compact metric space S . In one specific model, “seeds" arrive as a Poisson process (in time) at random positions with some distribution θ on S , and create balls whose radius increases at constant rate. By standardizing rates, the cover time C depends only on θ . The value χ ( S ) = min θ E θ C is a numerical characteristic of the compact space S , and we give weak general upper and lower bounds in terms of the covering numbers of S . This suggests a future research program of improving such general bounds, and estimating χ ( S ) for familiar examples of compact spaces. We treat one example, infinite product space [0 , 1] ∞ with the product topology. On a different theme, by analogy with the geometric models, and with the discrete coupon collector’s problem and with cover times for finite Markov chains, one expects a “weak concentration" bound for the distribution of C to hold under minimal assumptions. We prove this as a simple consequence of a general result for increasing set-valued Markov processes.
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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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