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

Loading next page...
 
/lp/springer_journal/exact-asymptotics-of-large-deviations-of-stationary-ornstein-uhlenbeck-Z0ilfDOg01
Publisher
Springer Journals
Copyright
Copyright © 2006 by Pleiades Publishing, Inc.
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

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off