On the Lower Tail Probabilities of Some Random Sequences in lp

We investigate the behaviour of the logarithmic small deviation probability of a sequence (σnθn) in lp, 0<p≤∞, where (θn) are i.i.d. random variables and (σn) is a decreasing sequence of positive numbers. In particular, the example σn∼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 θ1 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.