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Almost sure functional limit theorems. II: The case of independent random variables. (English) Zbl 0993.60020
In the first part of this paper [ibid. 34, No. 1-3, 273-304 (1998; Zbl 0921.60033)] general almost sure functional limit theorems (ASFLT) were proved for self-similar processes. These results are used to obtain almost sure versions of some functional limit theorems in \(D([0,1])\). Theorem 1 states that for the step line process constructed from the partial sums of independent random variables satisfying the Lindeberg condition the ASFLT holds with Wiener limit measure. In Theorem 3 such conditions are imposed under which the normalized partial sums of independent identically distributed random variables converge in distribution to a stable law. It is proved that these conditions imply the ASFLT with the distribution of a stable process as the limit measure.

MSC:
60F05 Central limit and other weak theorems
28D05 Measure-preserving transformations
60F17 Functional limit theorems; invariance principles
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