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Multidimensional version of the results of Komlós, Major and Tusnády for vectors with finite exponential moments. (English) Zbl 0897.60033
Consider centered independent random vectors (r.v.’s) $$\xi_k$$ in $$\mathbb{R}^d$$ having the covariance matrix cov $$\xi_k=I$$ (identity operator), $$k=1,\dots,n$$. Assume that, for some $$\tau\geq 1$$ and $$k=1,\dots,n$$, the distributions $${\mathcal L}(\xi_k)$$ belong to the class $${\mathcal A}_d(\tau)$$ of probability measures introduced by the author [Theory Probab. Appl. 31, No. 2, 203-220 (1997); translation from Teor. Veroyatn. Primen. 31, No. 2, 246-265 (1986; Zbl 0604.60021)]. Then one can construct, on a probability space, independent r.v.’s $$X_1,\dots,X_n$$ and corresponding independent r.v.’s $$Y_1,\dots,Y_n$$ such that $${\mathcal L}(X_k)={\mathcal L}(Y_k)$$, $$k=1,\dots,n$$, and for $$\alpha > 0$$, ${\mathbf E} \exp\left(\frac{c_1(\alpha)\Delta (X,Y)}{\tau d^3 L(d)}\right) \leq \exp(c_2(\alpha) d^{9/4+\alpha} L(n/\tau^2)).$ Here positive $$c_1(\alpha)$$ and $$c_2(\alpha)$$ depend only on $$\alpha$$, $$L(b)=\max \{1,\log b\}$$ for $$b>0$$ and $$\Delta(X,Y)=\max_{1\leq k \leq n} | \sum_{i=1}^{k} X_i-\sum_{i=1}^{k} Y_i|, | x|=\max_{1\leq j\leq d}| x_j|$$ for $$x=(x_1,\dots,x_d)\in\mathbb{R}^d$$. As a corollary the conditions guaranteeing the rate of strong approximation $\sum_{j=1}^n X_j-\sum_{j=1}^n Y_j=O(\log n)\quad \text{a.s.}$ are provided in the case of i.i.d. r.v.’s $$\xi_j$$, $$j\in\mathbb{N}$$. The contributions of various authors to this research field are also discussed starting from the classical results for $$d=1$$.

##### MSC:
 60F15 Strong limit theorems 60F17 Functional limit theorems; invariance principles
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##### References:
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