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On the Lower Tail Probabilities of Some Random Sequences in l p

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Abstract

We investigate the behaviour of the logarithmic small deviation probability of a sequence (σ n θ n ) in l p , 0<p≤∞, where (θ n ) are i.i.d. random variables and (σ n ) is a decreasing sequence of positive numbers. In particular, the example σ n ∼n −μ(1+log n)−ν is studied thoroughly. Contrary to the existing results in the literature, the rate function and the small deviation constant are expressed expli- citly in the present treatment. The restrictions on the distribution of θ 1 are kept to an absolute minimum. In particular, the usual variance assumption is removed. As an example, the results are applied to stable and Gamma-distributed random variables.

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  1. Institut für Stochastik, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 3, 07737, Jena, Germany

    Frank Aurzada

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  1. Frank Aurzada
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Correspondence to Frank Aurzada.

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Aurzada, F. On the Lower Tail Probabilities of Some Random Sequences in l p . J Theor Probab 20, 843–858 (2007). https://doi.org/10.1007/s10959-007-0095-9

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  • DOI: https://doi.org/10.1007/s10959-007-0095-9

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