Fractional Brownian motion stationary and self-similar increments. It has stationary increments: For "H" > ½ the process exhibits long-range dependence, Sample-paths are almost nowhere differentiable. However, almost-all trajectories are locally Hölder continuous of any order strictly less than "H": for each such trajectory, for every "T" > 0 and for every "ε" > 0 there exists a (random) constant "c" such that for 0 < "s","t" < "T". With probability 1, the graph of "B"("t") has both Hausdorff dimension and box dimension of 2−"H". As for regular Brownian motion, one can define stochastic integrals with respect to fractional Brownian motion, usually called "fractional
Brownian excursion Groeneboom (1983) and Pitman (1983) give decompositions of Brownian motion formula_5 in terms of i.i.d Brownian excursions and the least concave majorant (or greatest convex minorant) of formula_5. For an introduction to Itô's general theory of Brownian excursions and the Itô Poisson process of excursions, see Revuz and Yor (1994), chapter XII. The Brownian excursion area arises in connection with the enumeration of connected graphs, many other problems in combinatorial theory; see e.g.,, and the limit distribution of the Betti numbers of certain varieties in cohomology theory. Takacs (1991a) shows that formula_19 has density where formula_21 are the zeros of