Small time convergence of subordinators with regularly or slowly varying canonical measure

We consider subordinators X_α=(X_α(t))_t≥0 in the domain of attraction at 0 of a stable subordinator (S_α(t))_t≥0 (where α∈(0,1)); thus, with the property that Π¯_α, the tail function of the canonical measure of X_α, is regularly varying of index −α∈(−1,0) as x↓0. We also analyse the boundary case, α=0, when Π¯_α is slowly varying at 0. When α∈(0,1), we show that (tΠ¯_α(X_α(t)))⁻¹ converges in distribution, as t↓0, to the random variable (S_α(1))^α. This latter random variable, as a function of α, converges in distribution as α↓0 to the inverse of an exponential random variable. We prove these convergences, also generalised to functional versions (convergence in D[0,1]), and to trimmed versions, whereby a fixed number of its largest jumps up to a specified time are subtracted from the process. The α=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.