### Abstract

Let L(x, T), x∈R^{d}, T∈R_{+}^{N}, be the local time of the N-parameter Wiener process W taking values in R^{d}. Even in the distribution valued cased d≧2 N, 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 of L(x, T) as |x|→0 and/or T∞ and of related occupation integrals {Mathematical expression} as T→∞. We obtain the rates of explosion in laws of the first order, i.e. normalized convergence laws for L(x, T) resp. X_{T}(f), and of the second order, i.e. normalized convergence laws for L(x, T)-E(L(x, T)) resp. X_{T}(f)-E(X_{T}(f)).

Original language | English |
---|---|

Pages (from-to) | 47-75 |

Number of pages | 29 |

Journal | Probability Theory and Related Fields |

Volume | 98 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 1994 |

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### Keywords

- Mathematics Subject Classification (1990): 60G60, 60J55, 60G15, 60H05

### ASJC Scopus subject areas

- Statistics and Probability
- Analysis
- Mathematics(all)

### Cite this

**The asymptotic behaviour of local times and occupation integrals of the N-parameter Wiener process in R ^{d}.** / Imkeller, Peter; Weisz, F.

Research output: Contribution to journal › Article

^{d}',

*Probability Theory and Related Fields*, vol. 98, no. 1, pp. 47-75. https://doi.org/10.1007/BF01311348

}

TY - JOUR

T1 - The asymptotic behaviour of local times and occupation integrals of the N-parameter Wiener process in Rd

AU - Imkeller, Peter

AU - Weisz, F.

PY - 1994/3

Y1 - 1994/3

N2 - Let L(x, T), x∈Rd, T∈R+N, be the local time of the N-parameter Wiener process W taking values in Rd. Even in the distribution valued cased d≧2 N, 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 of L(x, T) as |x|→0 and/or T∞ and of related occupation integrals {Mathematical expression} as T→∞. We obtain the rates of explosion in laws of the first order, i.e. normalized convergence laws for L(x, T) resp. XT(f), and of the second order, i.e. normalized convergence laws for L(x, T)-E(L(x, T)) resp. XT(f)-E(XT(f)).

AB - Let L(x, T), x∈Rd, T∈R+N, be the local time of the N-parameter Wiener process W taking values in Rd. Even in the distribution valued cased d≧2 N, 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 of L(x, T) as |x|→0 and/or T∞ and of related occupation integrals {Mathematical expression} as T→∞. We obtain the rates of explosion in laws of the first order, i.e. normalized convergence laws for L(x, T) resp. XT(f), and of the second order, i.e. normalized convergence laws for L(x, T)-E(L(x, T)) resp. XT(f)-E(XT(f)).

KW - Mathematics Subject Classification (1990): 60G60, 60J55, 60G15, 60H05

UR - http://www.scopus.com/inward/record.url?scp=21344487131&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=21344487131&partnerID=8YFLogxK

U2 - 10.1007/BF01311348

DO - 10.1007/BF01311348

M3 - Article

AN - SCOPUS:21344487131

VL - 98

SP - 47

EP - 75

JO - Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete

JF - Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete

SN - 0178-8051

IS - 1

ER -