We consider subordinators X-alpha = (X-alpha(t))(t >= 0) in the domain of attraction
at 0 of a stable subordinator (S-alpha(t))(t >= 0) (where alpha is an element of (0,
1)); thus, with the property that (Pi) over bar (alpha), the tail function of the
canonical measure of X-alpha, is regularly varying of index -alpha is an element of
(-1, 0) as x down arrow 0. We also analyse the boundary case, alpha = 0, when (Pi)
over bar (alpha) is slowly varying at 0. When alpha is an element of (0, 1), we show
that (t (Pi) over bar (alpha)(X-alpha(t)))(-1) converges in distribution, as t down
arrow 0, to the random variable (S-alpha(1))(alpha). This latter random variable,
as a function of alpha, converges in distribution as alpha down arrow 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 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. (C) 2018 Elsevier B.V.
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