  We study, using exact numerical simulations, the statistics of the longest
excursion l_{\max}(t) up to time t for the fractional Brownian motion with
Hurst exponent 0<H<1. We show that in the large t limit, < l_{\max}(t) >
\propto Q_\infty t where Q_\infty \equiv Q_\infty(H) depends continuously on H,
and in a non trivial way. These results are compared with exact analytical
results obtained recently for a renewal process with an associated persistence
exponent \theta = 1-H. This comparison shows that Q_\infty(H) carries the clear
signature of non-Markovian effects for H\neq 1/2. The pre-asymptotic behavior
of < l_{\max}(t)> is also discussed.
