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与稳定过程上确界有关的密度的渐近性态

The asymptotic behavior of densities related to the supremum of a stable process

R. A. Doney, M. S. Savov

备注：
Published in at http://dx.doi.org/10.1214/09-AOP479 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
发表信息：
Annals of Probability 2010, Vol. 38, No. 1, 316-326
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摘要：

如果$X$是一个指标$α\in(0,2)$的稳定过程，其Lévy测度在$(0,\infty)$上有密度$cx^{-α-1}$，且$S_1=\sup_{0<t\leq1}X_t$，则已知$P(S_1>x)\backsim Aα^{-1}x^{-α}$为$x\to\infty$，$P(S_1\leq x)\backsim Bα^{-1}ρ^{-1}x^{αρ}$为$x\downarrow0$。[Here $ρ=P(X_1>0)$和$A$和$B$是已知常数。] $m$说，众所周知，$S_1$具有连续的密度。本说明的要点是说明$m(x)\backsim Ax^{-(α+1)}$为$x\to\infty$，$m(x)\backsim Bx^{αρ-1}$为$x\downarrow0$。对于相关密度也得到了类似的结果。

英文摘要：

If $X$ is a stable process of index $α\in(0,2)$ whose Lévy measure has density $cx^{-α-1}$ on $(0,\infty)$, and $S_1=\sup_{0<t\leq1}X_t$, it is known that $P(S_1>x)\backsim Aα^{-1}x^{-α}$ as $x\to\infty$ and $P(S_1\leq x)\backsim Bα^{-1}ρ^{-1}x^{αρ}$ as $x\downarrow0$. [Here $ρ=P(X_1>0)$ 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 $m(x)\backsim Ax^{-(α+1)}$ as $x\to\infty$ and $m(x)\backsim Bx^{αρ-1}$ as $x\downarrow0$. Similar results are obtained for related densities.

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