Weak Convergence of Subordinators to Extremal Processes

TL;DR

This paper demonstrates that certain subordinators, when transformed appropriately, converge to extremal processes, with convergence established both for finite-dimensional distributions and in the Skorohod space of càdlàg functions.

Contribution

It establishes the weak convergence of transformed subordinators to extremal processes, extending understanding of their asymptotic behavior.

Findings

Convergence of $(-t ext{log} X_{ts})_{s>0}$ to an extremal process.

Weak convergence of truncated processes in Skorohod space.

Provides conditions under which subordinators approximate extremal processes.

Abstract

For certain subordinators $(X_t)_{t\ge 0}$ it is shown that the process $(-t\log X_{ts})_{s>0}$ tends to an extremal process $(\hat\eta_s)_{s>0}$ in the sense of convergence of the finite dimensional distributions. Additionally it is also shown that $(z\wedge(-t\log X_{ts}))_{s\ge 0}$ converges weakly to $(z\wedge\hat\eta_s)_{s\ge0}$ in $\mathcal{D}[0,\infty)$, the space of c\`{a}dl\`{a}g functions equipped with Skorohod's $J_1$ metric.