On the small-time behavior of subordinators

Abstract

We prove several results on the behavior near t=0 of $Y_t^{-t}$ for certain $(0,\infty)$-valued stochastic processes $(Y_t)_{t>0}$. In particular, we show for L\'{e}vy subordinators that the Pareto law on $[1,\infty)$ is the only possible weak limit and provide necessary and sufficient conditions for the convergence. More generally, we also consider the weak convergence of $tL(Y_t)$ as $t\to0$ for a decreasing function $L$ that is slowly varying at zero. Various examples demonstrating the applicability of the results are presented.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ363 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Parametric Estimation of Lévy Processes

“…for some b ≥ 0 and ν supported by R + . Recently, [9] characterized the class of drift-free (b = 0) subordinators X for which…”

“…where P γ (γ > 0) denotes the Pareto distribution corresponding to the density x → γx −γ−1 1 [1,∞) (x). For example, [9] proved that the above weak convergence holds if L (X 1 ) admits a Lebesgue density p 1 (x) such that…”

Weak convergence of subordinators to extremal processes

“…Also note that this condition cannot hold for a compound Poisson process, so that when it does hold then necessarily ν(0, ǫ] = ∞, which in turn implies that X t > 0 almost surely for each t > 0 and thus −t log X t is well defined for all t > 0. Several examples of subordinators fulfilling these conditions are given in [2]. A prominent member is the gamma process, where…”

“…Several examples of subordinators fulfilling these conditions are given in [2]. A prominent member is the gamma process, where…”

“…It was shown in [1] that if (X t ) t>0 is a family of positive random variables and if X is a non-constant random variable with distribution function F , then X −t t converges weakly to X as t → 0 if and only if ψ t (u 1/t ) → 1 − F (u) as t → 0 at all continuity points u of F , where ψ t is the Laplace transform of X t . In [2] it was found that for the convolution family ψ t (u) = ϕ(u) t , where ϕ is the Laplace transform of an infinitely divisible random variable, i.e. if the process X t is a subordinator, the limit distribution, if not concentrated on a single point, is always a Pareto distribution.…”