Large deviations for the maximum local time of stable Lévy processes
Let X(t) be a strictly stable Lévy process of index \(\alpha\), \(1<\alpha \leq 2\), and skewness index h, \(| h| \leq 1\), and \(L^ x_ t\) its associated local time. Then \(L^*_ t=\sup_{x}L^ x_ t\) is known as the maximum local time. The author shows that \[ (1)\quad \lim_{\lambda \to +\infty}\lambda^{-\alpha} \log P(L_ 1>\lambda)=- C, \] where C is a positive constant depending upon h and \(\alpha\). This generalizes a result of \textit{E. Perkins} [Stochastic processes, Semin. Evanston/Ill. 1984, Prog. Probab. Stat. 9, 151-164 (1986; Zbl 0589.60065)] who showed that in the case of standard Brownian motion \(C=\).
scientific article; zbMATH DE number 4201316