Bertoin, Jean; Werner, Wendelin Stable windings. (English) Zbl 0867.60045 Ann. Probab. 24, No. 3, 1269-1279 (1996). It is considered a complex plane valued càdlàg process \(Z=(Z_t, t\geq 0)\) with stationary independent increments, such that for all \(z\in\mathbb{C}\), \(t\geq 0\) and some \(\alpha\in(0,2)\), \[ E[\exp\{i(z,Z_t)\}\mid Z_0= 0]=\exp\{- t|z|^\alpha\}, \] and the càdlàg process \(\theta= (\theta_t, t\geq 0)\) of the winding number around \(0\) of the process \(Z\) started from \(1\), is characterized by the properties that \(\theta\) has no jumps of absolute length greater than \(\pi\) and \(\exp\{i\theta_t\}= Z_t|Z_t|^{-1}\) for all \(t\geq 0\). It is proved that the sequence of stochastic processes \((n^{-1/2}\theta_{\exp(nt)}, t\geq 0)\) converges weakly in the Skorokhod topology to \((\beta_{c(\alpha)t}, t\geq 0)\), where \((\beta_t, t\geq 0)\) is a standard Brownian motion and \[ c(\alpha)={\alpha2^{-1-\alpha/2}\over\pi} \int_{\mathbb{C}}|z|^{-2-\alpha}|\arg(1+ z)|^2dz. \] An analogous asymptotic property is derived for the winding number around \(0\) of the stable random walk. Reviewer: B.Grigelionis (Vilnius) Cited in 1 ReviewCited in 12 Documents MSC: 60J99 Markov processes 60G50 Sums of independent random variables; random walks 60G18 Self-similar stochastic processes Keywords:stationary independent increments; winding number; Skorokhod topology; asymptotic property; stable random walk PDFBibTeX XMLCite \textit{J. Bertoin} and \textit{W. Werner}, Ann. Probab. 24, No. 3, 1269--1279 (1996; Zbl 0867.60045) Full Text: DOI