Small Parameter Behavior of Families of Distributions

For each t > 0 let Yt be a positive random variable with distribution Ft and Laplace Stieltjes transform (LST) ψt. In [1] it was shown that as t → 0, Y −t t converges in distribution to some Y with distribution F as t → 0 if and only if ψt(u) → 1 − F (u) for all continuity points u of F . The applicability of this result was demonstrated for numerous examples, all in which the limiting distribution F was found to be Pareto or exponential. In [3] these results were used to show that when (Yt)t≥0 is a subordinator (a nondecreasing Lévy process) and the limit law of Y −t t is nondeterministic, then it must be of the Pareto type. Such results necessarily raises the following questions. Are there other natural interesting classes of families (Yt)t>0 for which Y −t t has a nontrivial limit law as t → 0 and what are the conditions that should be imposed? What “centralizing” transformations gt other than gt(x) = x −t can be used in order to identify a simple relationship between the limit laws of gt(Yt) and limits associated with the LSTs of Yt and what are some examples of possible limit laws that can be obtained via this procedure? In this paper we deal with these two questions. We first provide easily verifiable conditions under which various limiting laws of Y −t t as t→ 0 are obtained. We then derive necessary and/or sufficient conditions under which various centralizing transformations gt can be used together with the LST of Yt to give a simple streamlined procedure for obtaining nontrivial limiting laws of gt(Yt) as t→ 0.