The convergence rate in the invariance principle for differently distributed variables with exponential moments. (Russian) Zbl 0541.60024

Sakhanenko, A. I.

Predel’nye Teoremy dlya Summ Sluchajnykh Velichin, Tr. Inst. Mat. 3, 4-49 (1984).

Let \((\eta_ j)\) be a sequence of normally distributed r.v.’s with \(E\eta_ j=0\) and \(E\eta^ 2_ j<\infty\). Let \(F_ j(x)\) be distribution functions such that \(\int xF_ j(dx)=0\), \(\int x^ 2F_ j(dx)<\infty\) and for some \(\lambda>0 \lambda \int | x|^ 3\exp(\lambda | x|)F_ j(dx)\leq \int x^ 2F_ j(dx)\quad \forall j.\) The sequence of independent r.v.’s \((\xi_ j)\) with distribution functions \((F_ j(x))\) and such that \(E \exp(e\lambda \Delta_ n)\leq 1+\lambda B_ n,\) where \(c>0\) is some absolute constant, \(B^ 2_ n=\sum_{j\leq n}D\xi_ j, \Delta_ n=\max_{m\leq n}| \sum_{j\leq m}\xi_ j-\sum_{j\leq m}\mu_ j|,\) is constructed. This result generalizes the earlier one of J. Komlos, P. Major, and G. Tusnady, Z. Wahrscheinlichkeitstheor. Verw. Geb. 34, 33-58 (1976; Zbl 0307.60045).

Reviewer: K.Kubilius

MSC:

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

Keywords:

convergence rate; invariance principle