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