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.
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Gallardo, L., Trimèche, K. Renewal Theorems for Singular Differential Operators. Journal of Theoretical Probability 15, 161–205 (2002). https://doi.org/10.1023/A:1013895502747
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DOI: https://doi.org/10.1023/A:1013895502747