  In this paper, we deal with the asymptotic distribution of the maximum
increment of a random walk with a regularly varying jump size distribution.
This problem is motivated by a long-standing problem on change point detection
for epidemic alternatives. It turns out that the limit distribution of the
maximum increment of the random walk is one of the classical extreme value
distributions, the Fr\'{e}chet distribution. We prove the results in the
general framework of point processes and for jump sizes taking values in a
separable Banach space.
