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Convergence of the all-time supremum of a L'evy process in the heavy-traffic regime | | Kamil Marcin Kosinski
; Onno Boxma
; Bert Zwart
; | | Date: |
1 Jul 2010 | | Abstract: | In this paper we derive a technique of obtaining limit theorems for suprema
of L’evy processes from their random walk counterparts. That is, we show that
if ${Y^{(k)}_n:nge 0}$ is a sequence of independent identically distributed
random variables and ${X^{(k)}_t:tge 0}$ is a sequence of L’evy processes
such that $X_1^{(k)}de Y_1^{(k)}$, then, with $S^{(k)}_n=sum_{i=1}^n
Y^{(k)}_i$ and under some mild assumptions, $ Delta(k)max_{nge 0}
S_n^{(k)}stackrel{d}{ o} mathscr Riff Delta(k)sup_{tge 0}
X^{(k)}_tstackrel{d}{ o} mathscr R$, as $k oi$, for some random variable
$mathscr R$ and normalizing sequence $Delta(k)$. We utilize this result to
present a number of limiting theorems for suprema of L’evy processes in
heavy-traffic regime. | | Source: | arXiv, 1007.0155 | | Services: | Forum | Review | PDF | Favorites | |
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