## 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

### MSC:

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

### Citations:

Zbl 0338.60031; Zbl 0351.60007
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