## The Annals of Probability

### Integrated Brownian motions and exact $L_2$-small balls

#### Abstract

We will introduce a class of m-times integrated Brownian motions. The exact asymptotic expansions for the $L_2$-small ball probabilities will be discussed for members of this class, of which the usual m-times integrated Brownian motion is an example. Another example will be what we call an Euler-integrated Brownian motion. We will also find very sharp estimates for the asymptotics of the eigenvalues of the covariance operator of integrated Brownian motions and will, therefore, obtain exact, not just logarithmic, asymptotics.

#### Article information

Source
Ann. Probab. Volume 31, Number 3 (2003), 1320-1337.

Dates
First available in Project Euclid: 12 June 2003

http://projecteuclid.org/euclid.aop/1055425782

Digital Object Identifier
doi:10.1214/aop/1055425782

Mathematical Reviews number (MathSciNet)
MR1989435

Zentralblatt MATH identifier
1047.60030

Subjects
Primary: 60G15: Gaussian processes

#### Citation

Gao, F.; Hannig, J.; Torcaso, F. Integrated Brownian motions and exact $L_2$-small balls. Ann. Probab. 31 (2003), no. 3, 1320--1337. doi:10.1214/aop/1055425782. http://projecteuclid.org/euclid.aop/1055425782.

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• MOSCOW, IDAHO 83844-1103 E-MAIL: fuchang@uidaho.edu J. HANNIG DEPARTMENT OF STATISTICS COLORADO STATE UNIVERSITY FT. COLLINS, COLORADO 80523-1877 E-MAIL: hannig@stat.colostate.edu F. TORCASO DEPARTMENT OF MATHEMATICAL SCIENCES JOHNS HOPKINS UNIVERSITY
• BALTIMORE, MARy LAND 21218-2682 E-MAIL: torcaso@mts.jhu.edu