# Exact asymptotics of large deviations of stationary Ornstein-Uhlenbeck processes for L p-functionals, p > 0

Exact asymptotics of large deviations of stationary Ornstein-Uhlenbeck processes for L... We prove a general result on the exact asymptotics of the probability % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFH0dXde9LqFHe9Lq% pepeea0xd9q8as0-LqLs-Jirpepeea0-as0Fb9pgea0lrP0xe9Fve9% Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaGaa8huam% aacmaabaWaa8qCaeaadaabdaqaaiabeE7aOnaaBaaaleaacqaHZoWz% aeqaaOGaaiikaiaadshacaGGPaaacaGLhWUaayjcSdWaaWbaaSqabe% aacaWGWbaaaOGaamizaiaadshacqGH+aGpcaWG1bWaaWbaaSqabeaa% caWGWbaaaaqaaiaaicdaaeaacaaIXaaaniabgUIiYdaakiaawUhaca% GL9baaaaa!4F53! $$P\left\{ {\int\limits_0^1 {\left| {\eta _\gamma (t)} \right|^p dt > u^p } } \right\}$$ as u → ∞, where p > 0, for a stationary Ornstein-Uhlenbeck process η γ(t), i.e., a Gaussian Markov process with zero mean and with the covariance function Eηγ(t)ηγ(s), t, s ∈ ℝ, γ > 0. We use the Laplace method for Gaussian measures in Banach spaces. Evaluation of constants is reduced to solving an extreme value problem for the rate function and studying the spectrum of a second-order differential operator of the Sturm-Liouville type. For p = 1 and p = 2, explicit formulas for the asymptotics are given. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Problems of Information Transmission Springer Journals

# Exact asymptotics of large deviations of stationary Ornstein-Uhlenbeck processes for L p-functionals, p > 0

, Volume 42 (1) – Apr 20, 2006
18 pages

/lp/springer_journal/exact-asymptotics-of-large-deviations-of-stationary-ornstein-uhlenbeck-Z0ilfDOg01
Publisher
Nauka/Interperiodica
Subject
Engineering; Communications Engineering, Networks; Electrical Engineering; Information Storage and Retrieval; Systems Theory, Control
ISSN
0032-9460
eISSN
1608-3253
D.O.I.
10.1134/S0032946006010054
Publisher site
See Article on Publisher Site

### Abstract

We prove a general result on the exact asymptotics of the probability % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFH0dXde9LqFHe9Lq% pepeea0xd9q8as0-LqLs-Jirpepeea0-as0Fb9pgea0lrP0xe9Fve9% Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaGaa8huam% aacmaabaWaa8qCaeaadaabdaqaaiabeE7aOnaaBaaaleaacqaHZoWz% aeqaaOGaaiikaiaadshacaGGPaaacaGLhWUaayjcSdWaaWbaaSqabe% aacaWGWbaaaOGaamizaiaadshacqGH+aGpcaWG1bWaaWbaaSqabeaa% caWGWbaaaaqaaiaaicdaaeaacaaIXaaaniabgUIiYdaakiaawUhaca% GL9baaaaa!4F53! $$P\left\{ {\int\limits_0^1 {\left| {\eta _\gamma (t)} \right|^p dt > u^p } } \right\}$$ as u → ∞, where p > 0, for a stationary Ornstein-Uhlenbeck process η γ(t), i.e., a Gaussian Markov process with zero mean and with the covariance function Eηγ(t)ηγ(s), t, s ∈ ℝ, γ > 0. We use the Laplace method for Gaussian measures in Banach spaces. Evaluation of constants is reduced to solving an extreme value problem for the rate function and studying the spectrum of a second-order differential operator of the Sturm-Liouville type. For p = 1 and p = 2, explicit formulas for the asymptotics are given.

### Journal

Problems of Information TransmissionSpringer Journals

Published: Apr 20, 2006

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