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Renewal Theorems for Singular Differential Operators

Author

Listed:
  • Léonard Gallardo

    (Université de Tours)

  • Khalifa Trimèche

    (Faculté des Sciences de Tunis)

Abstract

Let * be the convolution on M( $${\mathbb{R}}$$ +) associated with a second order singular differential operator L on ]0, +∞[. If μ is a probability measure on $${\mathbb{R}}$$ + with suitable moment conditions, we study how to normalize the measures μ* n ; n∈ $${\mathbb{N}}$$ } (resp. $$\left\{ {\varepsilon _x * \sum _{n\; = \;0}^\infty \mu ^{ * n} } \right\}$$ ) in order to get vague convergence if n→+∞ (resp. x→+∞). The results depend on the asymptotic drift of the operator L and on a precise study of the asymptotic behaviour of its eigenfunctions.

Suggested Citation

  • Léonard Gallardo & Khalifa Trimèche, 2002. "Renewal Theorems for Singular Differential Operators," Journal of Theoretical Probability, Springer, vol. 15(1), pages 161-205, January.
    • Handle: RePEc:spr:jotpro:v:15:y:2002:i:1:d:10.1023_a:1013895502747
    DOI: 10.1023/A:1013895502747
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