## The Annals of Probability

### Lower tail probabilities for Gaussian processes

#### Abstract

Let $X=(X_t)_{t \in S}$ be a real-valued Gaussian random process indexed by S with mean zero. General upper and lower estimates are given for the lower tail probability $\mathbb{P}(\sup_{t \in S} (X_t-X_{t_0}) \leq x )$ as $x \to 0$, with $t_0\in S$ fixed. In particular, sharp rates are given for fractional Brownian sheet. Furthermore, connections between lower tail probabilities for Gaussian processes with stationary increments and level crossing probabilities for stationary Gaussian processes are studied. Our methods also provide useful information on a random pursuit problem for fractional Brownian particles.

#### Article information

Source
Ann. Probab., Volume 32, Number 1A (2004), 216-242.

Dates
First available in Project Euclid: 4 March 2004

https://projecteuclid.org/euclid.aop/1078415834

Digital Object Identifier
doi:10.1214/aop/1078415834

Mathematical Reviews number (MathSciNet)
MR2040781

Zentralblatt MATH identifier
1052.60028

Subjects
Primary: 60G15: Gaussian processes
Secondary: 60G17: Sample path properties 60G60: Random fields

#### Citation

Li, Wenbo V.; Shao, Qi-Man. Lower tail probabilities for Gaussian processes. Ann. Probab. 32 (2004), no. 1A, 216--242. doi:10.1214/aop/1078415834. https://projecteuclid.org/euclid.aop/1078415834

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