Asymptotics of sums of random variables from the domain of attraction of a stable law
This article extends the author's previous works, Dokl. Akad. Nauk Ukr. SSR, Ser. A 1984, No.7, 9-11 (1984; Zbl 0553.60035) and Teor. Veroyatn. Primen. 30, No.1, 132-136 (1985; Zbl 0574.60046), where the strong invariance principle (s.i.p.) for the partial sums \(S_ n=\sum_{1\leq i\leq n}\xi_ i\) of i.i.d. r.v. from the domain of normal attraction of the stable law \(G_{\alpha,\beta}\), \(| \beta | \leq 1\), \(0<\alpha <2\) is discussed. Here the error term in s.i.p. is decreased from \(o(n^{1/\alpha})\) to \(o(n^{1/\alpha -\rho})\), \(\rho >0.\) Let f(u) be the characteristic function (ch.f.) of \(\xi_ 1\) and \(g_{\alpha,\beta}(u)\) be the ch.f. of the stable law, \(E\xi_ 1=0\) for \(\alpha >1\). The main result is the following Theorem. If for some \(\ell >\alpha\), \(a_ 1,a_ 2>0\), \(| f(u)- g_{\alpha,\beta}(u)| \leq a_ 2| u|^{\ell}\) as \(| u| <a_ 1\), then we can redefine \(\{\xi_ i\}\) on a new probability space together with a stable process \(Y_{\alpha,\beta}(t)\), \(t>0\), such that \[ P\{| S_{[t]}-Y_{\alpha,\beta}(t)| =o(t^{1/\alpha - \rho})\}=1,\text{ for every } \rho \in (0,\quad 1/\alpha (A+1)). \] The conditions of this theorem are compared in details with assumptions of \textit{W. Stout} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 49, 69-70 (1974; Zbl 0283.60026)] where other methods are used to prove s.i.p.; see also \textit{E. Fisher} [ibid. 67, 461-471 (1984; Zbl 0535.60025)] and \textit{J. Mijnheer} [Information theory, statistical decision functions, random processes, Trans. 9th Prague Conf., Prague 1982, Vol. B, 83-89 (1983; Zbl 0539.60030)]. The article presents also some results about the rate of growth of \(S_ n\) as \(n\to \infty\) derived strictly from the properties of \(Y_{\alpha,\beta}(t)\).
scientific article; zbMATH DE number 4011557