Existence of moments and an asymptotic result based on a mixture of exponential distributions

Let {Ft: 0 <st < t0 </ [infinity]} be a family of distribution associated with positive random variables {Yt}. The limiting distribution of Ytt, as t --> 0+, and the finiteness and form of E(Ytk), for any real k < 1, are obtained. The technique used to obtain these results is based on a mixture of exponential distributions, with Ft being the mixing distribution. Several illustrative examples are provided with an emphasis on distributions related to symmetric random walks and compound Poisson and extreme stable distributions.