Laws of the iterated logarithm for the Brownian snake. (English) Zbl 0956.60012

Azéma, J. (ed.) et al., Séminaire de Probabilités XXXIV. Berlin: Springer. Lect. Notes Math. 1729, 302-312 (2000).
Summary: We consider the path-valued process \((W_s,\zeta_s)\) called the Brownian snake, with lifetime process \((\zeta_s)\) a reflected Brownian motion. We first give an estimate of the probability that this process exits a “big” ball. Then we show the following laws of the iterated logarithm for the Euclidean norm of the “terminal point” of the Brownian snake: \[ \limsup_{s\uparrow+\infty} {|W_s(\zeta_s)|\over s^{1/4}(\log\log s)^{3/4}}= c,\quad \limsup_{s\downarrow 0} {|W_s(\zeta_s)|\over s^{1/4}(\log\log(1/s))^{3/4}}= c, \] where \(c= 2.3^{-3/4}\).
For the entire collection see [Zbl 0940.00007].


60F15 Strong limit theorems
60F10 Large deviations
60G15 Gaussian processes
60G17 Sample path properties
60J25 Continuous-time Markov processes on general state spaces
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