Consider a random walk S=(Sn: n≥0) that is `perturbed'
by a stationary sequence (ξn: n≥0) to produce the
process S=(Sn+ξn: n≥0). In this paper, we are
concerned with developing limit theorems and approximations for
the distribution of Mn=max{Sk+ξk:
0≤k≤n}
when the random walk has a drift close to 0. Such maxima are of
interest in several modeling contexts, including operations
management and insurance risk theory. The associated limits
combine features of both conventional diffusion approximations for
random walks and extreme-value limit theory.
Primary Subjects: 60F17, 60J60, 60G99
Secondary Subjects: 60G70, 90B30
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