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, , for some random variable 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.
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The first author was supported by NWO Grant 613.000.701.
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Kosiński, K.M., Boxma, O.J. & Zwart, B. Convergence of the all-time supremum of a Lévy process in the heavy-traffic regime. Queueing Syst 67, 295–304 (2011). https://doi.org/10.1007/s11134-011-9215-4
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DOI: https://doi.org/10.1007/s11134-011-9215-4