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Laws of the iterated logarithm for the Brownian snake

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Part of the Lecture Notes in Mathematics book series (SEMPROBAB,volume 1729)

Abstract

We consider the path-valued process (W s , ζ s ) called the Brownian snake, with lifetime process (ζ 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:

$$\mathop {\lim \sup }\limits_{s \uparrow + \infty } \frac{{\left| {W_s (\zeta _s )} \right|}}{{s^{1/4} (\log \log s)^{3/4} }} = c,\mathop {\lim \sup }\limits_{s \downarrow 0} \frac{{\left| {W_s (\zeta _s )} \right|}}{{s^{1/4} (\log \log (1/s))^{3/4} }} = c$$

where c = 2.3−3/4.

AMS Classification numbers

  • 60F10
  • 60F15
  • 60G15
  • 60G17
  • 60J25
  • Keywords
  • Brownian snake
  • large deviations
  • law of the iterated logarithm
  • Borel-Cantelli lemma

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References

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© 2000 Springer-Verlag

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Serlet, L. (2000). Laws of the iterated logarithm for the Brownian snake. In: Azéma, J., Ledoux, M., Émery, M., Yor, M. (eds) Séminaire de Probabilités XXXIV. Lecture Notes in Mathematics, vol 1729. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0103809

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  • DOI: https://doi.org/10.1007/BFb0103809

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67314-9

  • Online ISBN: 978-3-540-46413-6

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