Trimmed Lévy processes and their extremal components

We analyze a stochastic process of the form ^(r)X_t=X_t−∑_i=1 ^rΔ_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.