Scholarlink

<p>We study how conditioning on the first <i>k</i> steps, which we think of as training, affects the long-term behavior of the Elephant Random Walk. When the elephant is conditioned to be at position <i>k</i> at time <i>k</i>, the first return time to the origin scales as <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(k^{(4-4p)/(3-4p)}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>k</mi> <mrow> <mo stretchy="false">(</mo> <mn>4</mn> <mo>-</mo> <mn>4</mn> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>-</mo> <mn>4</mn> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msup> </math></EquationSource> </InlineEquation> in the diffusive regime, and grows exponentially in the critical regime. We loosely interpret this as a measurement of the rate at which the elephant forgets its training.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

How Long does it Take to Train an Elephant Random Walk

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2026年6月19日

查看全文
  • Zheng Fang

摘要

We study how conditioning on the first k steps, which we think of as training, affects the long-term behavior of the Elephant Random Walk. When the elephant is conditioned to be at position k at time k, the first return time to the origin scales as \(k^{(4-4p)/(3-4p)}\) k ( 4 - 4 p ) / ( 3 - 4 p ) in the diffusive regime, and grows exponentially in the critical regime. We loosely interpret this as a measurement of the rate at which the elephant forgets its training.