## The asymptotic behavior of densities related to the supremum of a stable process.(English)Zbl 1185.60052

Summary: If $$X$$ is a stable process of index $$\alpha\in(0, 2)$$ whose Lévy measure has density $$cx^{-\alpha-1}$$ on $$(0,\infty)$$, and $$S_1=\sup_{0<t\leq 1}X_t$$, it is known that $$P(S_1>x)\sim A\alpha-1x^{-\alpha}$$ as $$x\to\infty$$ and $$P(S_1\leq x)\sim B\alpha^{-1}\rho^{-1}x^{\alpha\rho}$$ as $$x\downarrow 0$$. [Here $$\rho=P(X_1>0)$$ and $$A$$ and $$B$$ are known constants.] It is also known that $$S_1$$ has a continuous density, $$m$$ say. The main point of this note is to show that $$m(x)\sim Ax^{-(\alpha+1)}$$ as $$x\to\infty$$ and $$m(x)\sim Bx^{\alpha\rho-1}$$ as $$x\downarrow 0$$. Similar results are obtained for related densities.

### MSC:

 60G52 Stable stochastic processes 60F15 Strong limit theorems 60G70 Extreme value theory; extremal stochastic processes 60E99 Distribution theory
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### References:

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