Convex rearrangements of stable processes
The paper is concerned with a monotonic rearrangement of the homogeneous stable process with independent increments. We say that \(Tf\) is the monotonic rearrangement of a measurable real function on \([0,1]\) if the function \(Tf\) is nondecreasing continuous from the right and \(\lambda f^{-1}=\lambda (Tf)^{-1}\), where \(\lambda (\cdot)\) is the Lebesgue measure and \(\lambda g^{-1} (A)= \lambda\{x: g(x)\in A\}\). The operator \(V\), which is defined for any absolutely continuous on \([0,1]\) function \(f\) as follows \(Vf(t)=f(0)+ \int^t_0 T(f')(s)ds\), is called the operator of convex rearrangement. (The function \(Vf (t)\) is convex on the interval \([0,1].)\) For any process \(\xi(t)\), \(t\in [0,1]\), let \(\xi_n(t)\) be its polygonal approximation, i.e., \[ \xi_n (t)=\xi (k/n)+ (tn-k) \biggl(\xi \bigl((k+1)/n \bigr)-\xi (k/n)\biggr), \quad t\in\bigl[ k/n, (k+1) /n\bigr]. \] The asymptotic behavior of the process \(V\xi_n\) is studied for stable processes with independent increments and, in particular, for Brownian motion. One of the results looks as follows: if \(\xi(t)\) is a stable process with exponent \(\alpha\in (1,2)\), then \(n^{1/ \alpha-1}V \xi_n(t) \to \int^t_0 F^{-1} (s)ds\) a.s., where \(F(\cdot)\) is the distribution function of \(\xi(1)\).
scientific article; zbMATH DE number 1289182