## Brownian penalisations related to excursion lengths. VII.(English)Zbl 1181.60046

Limiting laws, as $$t \to \infty$$, for Brownian motion penalised by the longest length of excursions up to $$t$$, or up to the last zero before $$t$$, or again, up to the first zero after $$t$$, are shown to exist, and are characterized.
For part VI, cf. ESAIM, Probab. Stat. 13, 152–180 (2009; Zbl 1189.60069).

### MSC:

 60F17 Functional limit theorems; invariance principles 60F99 Limit theorems in probability theory 60G17 Sample path properties 60G40 Stopping times; optimal stopping problems; gambling theory 60G44 Martingales with continuous parameter 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60H20 Stochastic integral equations 60J25 Continuous-time Markov processes on general state spaces 60J55 Local time and additive functionals 60J60 Diffusion processes 60J65 Brownian motion

Zbl 1189.60069
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