Bhattacharya, Rabi; Rao, B. V. Asymptotics of iteration of i.i.d. symmetric stable processes. (English) Zbl 0881.60005 Brunner, E. (ed.) et al., Research developments in probability and statistics. Festschrift in honor of Madan L. Puri on the occasion of his 65th birthday. Utrecht: VSP. 3-10 (1996). Summary: It is shown that the finite-dimensional distributions of the process \(\{X_n (x): =Z_n(X_{n-1}(x))\), \(x\in \mathbb{R}^1 \backslash \{0\}\}\), obtained by successive iterations by an i.i.d. sequence of symmetric stable processes \(Z_n\) \((n\geq 1)\) of exponent \(\alpha>1\) on \(\mathbb{R}^1 \backslash \{0\}\), converge to those of an exchangeable process on \(\mathbb{R}^1 \backslash \{0\}\). But \(X_n\) does not converge weakly in the Skorokhod topology. Convergence of one-dimensional distributions of \(X_n (\cdot)\) was earlier established by K. B. Athreya [in: Stoch. Processes relat. Top. Vol. 1, 239-248 (1975; Zbl 0334.60035)]. We also show that for the case of Brownian motion, i.e., \(\alpha=2\), the limit of the one-dimensional distributions is the double exponential \(\exp \{-2 |y|\} dy\).For the entire collection see [Zbl 0866.00070]. MSC: 60B10 Convergence of probability measures 60E07 Infinitely divisible distributions; stable distributions Keywords:finite-dimensional distributions; exchangeable process; Skorokhod topology; Brownian motion Citations:Zbl 0334.60035 PDFBibTeX XMLCite \textit{R. Bhattacharya} and \textit{B. V. Rao}, in: Research developments in probability and statistics. Festschrift in honor of Madan L. Puri on the occasion of his 65th birthday. Utrecht: VSP. 3--10 (1996; Zbl 0881.60005)