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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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