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<p>In this paper, we investigate the two-color nonlinear unbalanced urn model where the drawing rule is governed by a (concave) function with values in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {R}^+\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mo>+</mo> </msup> </math></EquationSource> </InlineEquation> and the replacement mechanism is described by an unbalanced matrix. By connecting this model to stochastic approximation theory, we derive the law of the iterated logarithm for this model and provide illustrative examples.</p>

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The Law of the Iterated Logarithm for the Nonlinear Unbalanced Urn Model

  • 期刊论文

  • 发表日期：

    2026年4月1日

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  • Jianan Shi,
  • Zhenhong Yu,
  • Yu Miao

摘要

In this paper, we investigate the two-color nonlinear unbalanced urn model where the drawing rule is governed by a (concave) function with values in \(\mathbb {R}^+\) R + and the replacement mechanism is described by an unbalanced matrix. By connecting this model to stochastic approximation theory, we derive the law of the iterated logarithm for this model and provide illustrative examples.