The strong invariance principle for the sums of random vectors from the domain of attraction of stable law. (English. Ukrainian original) Zbl 0946.60019

Zinchenko, N. M.

Theory Probab. Math. Stat. 53, 57-63 (1996); translation from Teor. Jmovirn. Mat. Stat. 53, 51-57 (1995).

Let \(\{\xi_i, i\geq 1\}\) be \(d\)-dimensional i.i.d. r.v. (independent identically distributed random vectors) that belong to the domain of attraction of a stable law \(G_{\alpha}, \alpha \neq 1,\) of Lévy-Feldheim type. The author considers the strong invariance principle and proves the following approximation for the sums \(S_N = \sum_{i=1}^n \xi_i.\) If the absolute pseudomoment \(\nu (l)<\infty\) for \(l=1+[\alpha]\) and all the mixed pseudomoments \(\mu (s_1,\ldots, s_d)=0\) for \(s_1,\ldots, s_d =0,\ldots, l-1,\) then \(\{\xi_i, i\geq 1\}\) may be defined on the probability space where the sequence of i.i.d. stable r.v. \(\{\eta_i, i\geq 1\}\) are defined such that \(\left|S_N - \sum_{i=1}^n \eta_i\right|= o(n^{1/\alpha - \lambda}), \lambda=\lambda(\alpha,d),\) a.s.

Reviewer: Yu.S.Mishura (Kyïv)

MSC:

60F15	Strong limit theorems
60F17	Functional limit theorems; invariance principles

Keywords:

strong invariance principle; random vector; stable law