The local limit problem is investigated for sequences (p,) of probability densities with stable limit densities q~ having characteristic exponent c~e(0, 2].It is shown that certain continuity-properties (H61der-continuity) are necessary and -under appropriate additional conditions -sufficient for suplp,(x)-q~(x)l~O as n~oe. In this sense the speed of convergence is x also studied.
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