具有斜积结构的各向同性自相似马尔可夫过程锥的出口时间的精确渐近性

IF 0.4 4区 数学 Q4 STATISTICS & PROBABILITY Probability and Mathematical Statistics-Poland Pub Date : 2016-10-02 DOI:10.37190/0208-4147.41.1.3
Z. Palmowski, Longmin Wang
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引用次数: 0

摘要

本文从具有斜积结构的各向同性$\alpha$ -自相似马尔可夫过程$X_t$的锥$C$中,确定了出口时间$\tau_C$分布的渐近尾部,即$X_t$是其径向过程与独立时变角分量$\Theta_t$的乘积。在一些附加的正则性假设下,角过程$\Theta_t$在离开锥体$C$时终止,其过渡密度可以用具有相应特征值的适当生成器的正交特征函数的完备集来表示。利用这一事实和与径向过程的Lamperti表示有关的被杀lsamvy过程的指数泛函的一些渐近性质,我们证明了对于$h$和$\kappa_1$的显式识别,$$\mathbb{P}_x(\tau_C>t)\sim h(x)t^{-\kappa_1}$$为$t\rightarrow\infty$。该结果扩展了DeBlassie(1988)、Bañuelos和Smits(1997)关于布朗运动的工作。
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On th exact asymptotics of exit time from a cone of an isotropic alpha-self-similar Markov process with a skew-product structure
In this paper we identify the asymptotic tail of the distribution of the exit time $\tau_C$ from a cone $C$ of an isotropic $\alpha$-self-similar Markov process $X_t$ with a skew-product structure, that is $X_t$ is a product of its radial process and independent time changed angular component $\Theta_t$. Under some additional regularity assumptions, the angular process $\Theta_t$ killed on exiting from the cone $C$ has the transition density that could be expressed in terms of a complete set of orthogonal eigenfunctions with corresponding eigenvalues of an appropriate generator. Using this fact and some asymptotic properties of the exponential functional of a killed L\'evy process related with Lamperti representation of the radial process, we prove that $$\mathbb{P}_x(\tau_C>t)\sim h(x)t^{-\kappa_1}$$ as $t\rightarrow\infty$ for $h$ and $\kappa_1$ identified explicitly. The result extends the work of DeBlassie (1988) and Ba\~nuelos and Smits (1997) concerning the Brownian motion.
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来源期刊
CiteScore
0.70
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0.00%
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0
审稿时长
>12 weeks
期刊介绍: PROBABILITY AND MATHEMATICAL STATISTICS is published by the Kazimierz Urbanik Center for Probability and Mathematical Statistics, and is sponsored jointly by the Faculty of Mathematics and Computer Science of University of Wrocław and the Faculty of Pure and Applied Mathematics of Wrocław University of Science and Technology. The purpose of the journal is to publish original contributions to the theory of probability and mathematical statistics.
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