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- Title
Convergence of the all-time supremum of a Lévy process in the heavy-traffic regime.
- Authors
Kosiński, K. M.; Boxma, O. J.; Zwart, B.
- Abstract
In this paper we derive a technique for obtaining limit theorems for suprema of Lévy processes from their random walk counterparts. For each a>0, let $\{Y^{(a)}_{n}:n\ge1\}$ be a sequence of independent and identically distributed random variables and $\{X^{(a)}_{t}:t\ge0\}$ be a Lévy process such that $X_{1}^{(a)}\stackrel{d}{=}Y_{1}^{(a)}$, $\mathbb{E}X_{1}^{(a)}<0$ and $\mathbb{E}X_{1}^{(a)}\uparrow0$ as a↓0. Let $S^{(a)}_{n}=\sum _{k=1}^{n} Y^{(a)}_{k}$. Then, under some mild assumptions, [InlineEquation not available: see fulltext.], for some random variable [InlineEquation not available: see fulltext.] and some function Δ(⋅). We utilize this result to present a number of limit theorems for suprema of Lévy processes in the heavy-traffic regime.
- Subjects
LEVY processes; ACCELERATION of convergence in numerical analysis; COMPETING risks; STRUCTURAL equation modeling; QUINTIC equations; RANDOM walks
- Publication
Queueing Systems, 2011, Vol 67, Issue 4, p295
- ISSN
0257-0130
- Publication type
Article
- DOI
10.1007/s11134-011-9215-4