Distributions of suprema of Lévy processes via the heavy traffic invariance principle. (English) Zbl 1048.60038
Summary: We study the relationship between the distribution of the supremum functional \(M_X=\sup_{0\leq t<\infty} (X(t)-\beta t)\) for a process \(X\) with stationary, but not necessarily independent increments, and the limiting distribution of an appropriately normalized stationary waiting time for G/G/1 queues in heavy traffic. As a by-product we obtain explicit expressions for the distribution of \(M_X\) in several special cases of Lévy processes.
MSC:
60G51 | Processes with independent increments; Lévy processes |
60G10 | Stationary stochastic processes |
60E07 | Infinitely divisible distributions; stable distributions |
60G18 | Self-similar stochastic processes |
60K25 | Queueing theory (aspects of probability theory) |