Renewal structure of the Brownian taut string

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 (1989)). Using this renewal structure, we derive an expression for the constant in the L.L.N. given in Lifshits and Setterqvist (2015) . In addition, we provide an invariance principle for the (renormalized) energy spent by the Brownian taut string.