Small Time Convergence of Subordinators with Regularly or Slowly Varying Canonical Measure

We consider subordinators $X_\alpha=(X_\alpha(t))_{t\ge 0}$ in the domain of attraction at 0 of a stable subordinator $(S_\alpha(t))_{t\ge 0}$ (where $\alpha\in(0,1)$); thus, with the property that $\overline{\Pi}_\alpha$, the tail function of the canonical measure of $X_\alpha$, is regularly varying of index $-\alpha\in (-1,0)$ as $x\downarrow 0$. We also analyse the boundary case, $\alpha=0$, when $\overline{\Pi}_\alpha$ is slowly varying at 0. When $\alpha\in(0,1)$, we show that $(t \overline{\Pi}_\alpha (X_\alpha(t)))^{-1}$ converges in distribution, as $t\downarrow 0$, to the random variable $(S_\alpha(1))^\alpha$. This latter random variable, as a function of $\alpha$, converges in distribution as $\alpha\downarrow 0$ to the inverse of an exponential random variable. We prove these convergences, also generalised to functional versions (convergence in $\mathbb{D}[0,1]$), and to trimmed versions, whereby a fixed number of its largest jumps up to a specified time are subtracted from a process. The $\alpha=0$ case produces convergence to an extremal process constructed from ordered jumps of a Cauchy subordinator. Our results generalise random walk and stable process results of Darling, Cressie, Kasahara, Kotani and Watanabe.