Self-Standardized Central Limit Theorems for Trimmed Subordinators

We prove under general conditions that a trimmed subordinator satisfies a
self-standardized central limit theorem [CLT]. Our basic tools are a classic
representation for subordinators and a distributional approximation result
of Zaitsev (1987). Among other results, we obtain as special cases of our
main result the recent self-standardized CLTs of Ipsen, Maller and Resnick
(2019) for trimmed subordinators and a trimmed subordinator analog of a CLT
of S. Csorgo, Horvath and Mason (1986) for intermediate trimmed
sums in the domain of attraction of a stable law. We then discuss how our
methods extend to proving similar theorems for spectrally positive L\'{e}vy
processes and then to general L\'{e}vy processes.