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Strong invariance principles for partial sums of independent random vectors. (English) Zbl 0637.60041
Let Q be a Borel probability measure on the Euclidean space \(R^ d\) with mean zero and covariance matrix \(\Sigma\) and let H be a continuous nonnegative function on [0,\(\infty)\) satisfying some regularity conditions. It is shown that there exist sequences of independent random vectors \(\{X_ n\}\), \(\{Y_ n\}\) such that \(X_ n\) has distribution Q, \(Y_ n\) has distribution \(N(0,\Sigma_ n)\), \(\| \Sigma_ n-\Sigma \| =o((H^{-1}(n))^ 2/n)\) and \((X_ 1+...+X_ n)-(Y_ 1+...+Y_ n)=o(H^{-1}(n))\) a.s. Here \(\| \cdot \|\) denotes the Euclidean matrix norm and \(H^{-1}\) the inverse function of H.
This result immediately implies the multidimensional version of Strassen’s invariance principle, P. Major’s strong invariance principle [Z. Wahrscheinlichkeitstheor. verw. Geb. 35, 213-220 (1976; Zbl 0338.60031)] under weakened assumptions as well as a new strong invariance principle which “interpolates” between these two invariance principles. It is shown that the assumptions are nearly necessary.
The main tools in the proof are the Strassen-Dudley theorem and a new estimate of the Prokhorov distance in the multidimensional central limit theorem which is a modification of a well-known theorem of V. V. Yurinskij [Teor. Verojatn. Primen. 20, 3-12 (1975; Zbl 0351.60007); English translation in Theor. Probab. Appl. 20, 1-10 (1975)].
Reviewer: T.Inglot

60F17 Functional limit theorems; invariance principles
60F15 Strong limit theorems
60G50 Sums of independent random variables; random walks
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