## Abstract

We analyze a stochastic process of the form ^{(r)}X_{t}=X_{t}−∑_{i=1} ^{r}Δ_{t} ^{(i)}, where (X_{t})_{t≥0} is a driftless, infinite activity, subordinator on R_{+} with its jumps on [0,t] ordered as Δ_{t} ^{(1)}≥Δ_{t} ^{(2)}⋯. The r largest of these are “trimmed” from X_{t} to give ^{(r)}X_{t}. When r→∞, both ^{(r)}X_{t}↓0 and Δ_{t} ^{(r)}↓0 a.s. for each t>0, and it is interesting to study the weak limiting behavior of (^{(r)}X_{t},Δ_{t} ^{(r)}) in this case. We term this “large-trimming” behavior, and study the joint convergence of (^{(r)}X_{t},Δ_{t} ^{(r)}) as r→∞ under linear normalization, assuming extreme value-related conditions on the Lévy measure of X_{t} which guarantee that Δ_{t} ^{(r)} has a limit distribution with linear normalization. Allowing ^{(r)}X_{t} to have random centering and norming in a natural way, we first show that (^{(r)}X_{t},Δ_{t} ^{(r)}) has a bivariate normal limiting distribution as r→∞; then replacing the random normalizations with deterministic normings produces normal, and in some cases, non-normal, limits whose parameters we can specify.

Original language | English |
---|---|

Pages (from-to) | 2228-2249 |

Number of pages | 22 |

Journal | Stochastic Processes and their Applications |

Volume | 130 |

Issue number | 4 |

DOIs | |

Publication status | Published - Apr 2020 |