Summary
We obtain a strong approximation theorem for partial sums of i.i.d.d-dimensional r.v.'s with possibly infinite second moments. Using this result, we can extend Philipp's strong invariance principle for partial sums of i.i.d.B-valued r.v.'s satisfying the central limit theorem toB-valued r.v.'s which are only in the domain of attraction of a Gaussian law. This new strong invariance principle implies a compact as well as a functional law of the iterated logarithm which improve some recent results of Kuelbs (1985).
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Einmahl, U. Strong approximations for partial sums of i.i.d.B-valued r.v.'s in the domain of attraction of a Gaussian law. Probab. Th. Rel. Fields 77, 65–85 (1988). https://doi.org/10.1007/BF01848131
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DOI: https://doi.org/10.1007/BF01848131