  In a recent paper, M. Lifshits and E. Setterqvist introduced the taut string
of a Brownian motion $w$, defined as the function of minimal quadratic energy
on $[0,T]$ staying in a tube of fixed width $h>0$ around $w$. The authors
showed a Law of Large Number (L.L.N.) for the quadratic energy spent by the
string for a large time $T$.
  In this note, we exhibit a natural renewal structure for the Brownian taut
string, which is directly related to the time decomposition of the Brownian
motion in terms of its $h$-extrema (as first introduced by Neveu and Pitman).
Using this renewal structure, we derive an expression for the constant in the
L.L.N. given by M. Lifshits and E. Setterqvist. In addition, we provide a
Central Limit Theorem (C.L.T.) for the fluctuations of the energy spent by the
Brownian taut string.
