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<p>Consider the invariance principle for a random walk with a random environment (denoted by <i>μ</i>) in time on ℝ in a weak quenched sense. We show that a sequence of random probability measures on ℝ generated by <i>μ</i> and a bounded Lipschitz functional <i>f</i> will converge in distribution to another random probability measure, which can be represented by <i>f</i> and two independent Brownian motions. The upper bound of the convergence rate has been obtained. We also explain that in general, this convergence can not be strengthened to the almost surely sense.</p>

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Weak Quenched Invariance Principle for Random Walk with Random Environment in Time

  • 期刊论文

  • 发表日期：

    2025年2月25日

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  • You Lü,
  • Wenming Hong

摘要

Consider the invariance principle for a random walk with a random environment (denoted by μ) in time on ℝ in a weak quenched sense. We show that a sequence of random probability measures on ℝ generated by μ and a bounded Lipschitz functional f will converge in distribution to another random probability measure, which can be represented by f and two independent Brownian motions. The upper bound of the convergence rate has been obtained. We also explain that in general, this convergence can not be strengthened to the almost surely sense.