Summary
LetL(x, T),x∈R d,T∈R N+ , be the local time of theN-parameter Wiener processW taking values inR d. Even in the distribution valued casedd≧2N,L can be described in a series representation by means of multiple Wiener-Ito integrals. This setting proves to be a good starting point for the investigation of the asymptotic behaviour ofL(x, T) as |x|→0 and/orT∞ and of related occupation integrals\(X_T (f) = \int\limits_{[0,T]} f (W_S )\) asT→∞. We obtain the rates of explosion in laws of the first order, i.e. normalized convergence laws forL(x, T) resp.X T (f), and of the second order, i.e. normalized convergence laws forL(x, T)−E(L(x, T)) resp.X T (f)−E(X T (f)).
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This research was made during a stay at the LMU in München supported by DAAD
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Imkeller, P., Weisz, F. The asymptotic behaviour of local times and occupation integrals of theN-parameter Wiener process in Rd . Probab. Th. Rel. Fields 98, 47–75 (1994). https://doi.org/10.1007/BF01311348
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DOI: https://doi.org/10.1007/BF01311348
Mathematics Subject Classification (1990)
- 60G60
- 60J55
- 60G15
- 60H05