Asymptotic variance of the self-intersections of stable random walks using Darboux-Wiener theory

Abstract

We present a Darboux-Wiener type lemma as a powerful alternative to the classical Tauberian theorem when monotonicity is not known a priori. We apply it to obtain the exact asymptotics of the variance of the self-intersections of a one-dimensional stable random walk. Finally we prove a functional central limit theorem for stable random walk in random scenery conjectured in [1].

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Correspondence to G. Deligiannidis.

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Original Russian Text Copyright © 2011 Deligiannidis G. and Utev S. A.

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 52, No. 4, pp. 809–822, July–August, 2011.

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Deligiannidis, G., Utev, S.A. Asymptotic variance of the self-intersections of stable random walks using Darboux-Wiener theory. Sib Math J 52, 639 (2011). https://doi.org/10.1134/S0037446611040082

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Keywords

  • random walk
  • self-intersection
  • Darboux-Wiener theory