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

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.

### MSC:

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

### Keywords:

strong invariance principle; random vector; stable law