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Central limit theorems for divergent higher-order Hermite integrals of Brownian motion

Author

Listed:
  • Chen, Yinmeng
  • Xia, Xiaoyu
  • Yan, Litan

Abstract

Let B={Bt,t≥0} be a standard Brownian motion. This paper establishes the asymptotic normality of renormalized Hermite integrals Υɛ(m)≔ɛm−1∫ɛ1H2m(Bs,s)s2mds,m∈{2,3,…}, where H2m(x,y)=ymh2m(x/y) with y>0, and hm denotes the classical Hermite polynomial of order m. We prove that as ɛ→0+, Υɛ(m)⟶N0,σm2,σm2=(2m)!(2m−1)(m−1), in distribution. This result quantifies the Gaussian limit behavior of divergent Hermite integrals through renormalization.

Suggested Citation

  • Chen, Yinmeng & Xia, Xiaoyu & Yan, Litan, 2026. "Central limit theorems for divergent higher-order Hermite integrals of Brownian motion," Statistics & Probability Letters, Elsevier, vol. 231(C).
    • Handle: RePEc:eee:stapro:v:231:y:2026:i:c:s0167715225002810
    DOI: 10.1016/j.spl.2025.110636
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