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<p>In this paper, we investigate the scaling limit of heavy-tailed nearly unstable cumulative INAR(<InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\infty\)</EquationSource> </InlineEquation>) processes. These processes exhibit a power-law tail of the form <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(n^{-(1+\alpha )}\)</EquationSource> </InlineEquation> for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\alpha \in (\frac{1}{2}, 1)\)</EquationSource> </InlineEquation>, and the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\ell ^1\)</EquationSource> </InlineEquation> norm of the kernel vector converges to 1. We demonstrate that the discrete-time scaling limit retains a long-memory property and can be viewed as an integrated fractional Cox-Ingersoll-Ross process. Moreover, we present a method for simulating the fractional Cox-Ingersoll-Ross process. The simulation and Goodness-of-Fit Test code are available at <a href="https://github.com/gagawjbytw/INAR-rough-Heston">https://github.com/gagawjbytw/INAR-rough-Heston</a>.</p>

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Scaling Limit of Heavy-Tailed Nearly Unstable Cumulative INAR(\(\infty\)) Processes and Rough Fractional Diffusions

  • 期刊论文

  • 发表日期：

    2026年2月16日

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  • Chunhao Cai,
  • Ping He,
  • Qinghua Wang,
  • Yingli Wang

摘要

In this paper, we investigate the scaling limit of heavy-tailed nearly unstable cumulative INAR( \(\infty\) ) processes. These processes exhibit a power-law tail of the form \(n^{-(1+\alpha )}\) for \(\alpha \in (\frac{1}{2}, 1)\) , and the \(\ell ^1\) norm of the kernel vector converges to 1. We demonstrate that the discrete-time scaling limit retains a long-memory property and can be viewed as an integrated fractional Cox-Ingersoll-Ross process. Moreover, we present a method for simulating the fractional Cox-Ingersoll-Ross process. The simulation and Goodness-of-Fit Test code are available at https://github.com/gagawjbytw/INAR-rough-Heston.