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Stable windings. (English) Zbl 0867.60045

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.

MSC:

60J99 Markov processes
60G50 Sums of independent random variables; random walks
60G18 Self-similar stochastic processes
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