Level crossings of absolutely continuous stationary symmetric \(\alpha\)-stable processes
The mean rate is described at which a general absolutely continuous stationary symmetric \(\alpha\)-stable (S\(\alpha\)S) process \((0<\alpha<2)\) crosses a high level. More precisely, let this process be representable as \(X(t)= \int_S f(t,x)M(dx)\). Under smooth assumptions (notably \(f(0,x)\) absolutely continuous) it is proved that if \(C_\lambda(1)\) is the number of upcrossings of the level \(\lambda\) in the interval \([0,1]\), we have \[ \begin{multlined}\lim_{\lambda\to+\infty} \lambda^\alpha E(C_\lambda(1))= \alpha 2^{\alpha-1/2}\pi^{-(1+1/\alpha)/2}\Biggl(\Gamma\Biggl({\alpha+1\over 2}\Biggr)\Biggr)^{1+1/\alpha} C^{-1/\alpha}_\alpha\times\\ \int_s|\dot f(0,x)||f(0,x)|^{\alpha-1}\text{\textbf{1}}_{(f(0,x)\dot f(0,x)> 0)}m(dx)\end{multlined} \] with \(C_\alpha= \left(\int^{+\infty}_0 x^{-\alpha}\sin x dx\right)^{-1}\). As an example the result for the two-sided Ornstein-Uhlenbeck process is given. This result generalizes the previous one of the authors and \textit{T. Gadrich} [ibid. 3, No. 2, 553-575 (1993; Zbl 0779.60034)].
scientific article; zbMATH DE number 1051772