$L^p$-Boundedness of the Overshoot in Multidimensional Renewal Theory
| Autor: | Terry R. McConnell, Philip S. Griffin |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 1995 |
| Předmět: |
Statistics and Probability
Discrete mathematics Limit of a function 60G50 $L^p$-boundedness Euclidean distance Multidimensional model exit condition multidimensional renewal theory 60K05 Mathematics::K-Theory and Homology 60J15 Ball (mathematics) Renewal theory Statistics Probability and Uncertainty Random variable Overshoot Mathematics |
| Zdroj: | Ann. Probab. 23, no. 4 (1995), 2022-2056 |
| Popis: | Let $T_r$ be the first time a sum $S_n$ of nondegenerate i.i.d. random variables leaves a ball of radius $r$ in some given norm on $\mathbb{R}^d$. In the case of the Euclidean norm we completely characterize $L^p$-boundedness of the overshoot $\|S_{T_r}\| - r$ in terms of the underlying distribution. For more general norms we provide a similar characterization under a smoothness condition on the norm which is shown to be very nearly sharp. One of the key steps in doing this is a characterization of the possible limit laws of $S_{T_r}/\|S_{T_r}\|$ under the weaker condition $\|S_{T_r}\|/r \rightarrow_p 1$. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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