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

2006 ◽  
Vol 42 (1) ◽  
pp. 46-63 ◽  
Author(s):  
V. R. Fatalov
Keyword(s):  
Large Deviations ◽  
Exact Asymptotics

scholarly journals Asymptotics for the First Passage Times of Lévy Processes and Random Walks

2013 ◽  
Vol 50 (1) ◽  
pp. 64-84 ◽  
Author(s):  
Denis Denisov ◽  
Vsevolod Shneer
Keyword(s):  
Large Deviations ◽  
Random Walks ◽  
Levy Processes ◽  
Levy Process ◽  
First Passage Times ◽  
Exact Asymptotics ◽  
First Passage ◽  
Heavy Tailed ◽  
First Time ◽  
Passage Times

We study the exact asymptotics for the distribution of the first time, τx, a Lévy process Xt crosses a fixed negative level -x. We prove that ℙ{τx >t} ~V(x) ℙ{Xt≥0}/t as t→∞ for a certain function V(x). Using known results for the large deviations of random walks, we obtain asymptotics for ℙ{τx>t} explicitly in both light- and heavy-tailed cases.

scholarly journals Asymptotics for the First Passage Times of Lévy Processes and Random Walks

2013 ◽  
Vol 50 (01) ◽  
pp. 64-84 ◽  
Author(s):  
Denis Denisov ◽  
Vsevolod Shneer
Keyword(s):  
Large Deviations ◽  
Random Walks ◽  
Levy Processes ◽  
Levy Process ◽  
First Passage Times ◽  
Exact Asymptotics ◽  
First Passage ◽  
Heavy Tailed ◽  
First Time ◽  
Passage Times

We study the exact asymptotics for the distribution of the first time, τ x , a Lévy process X t crosses a fixed negative level -x. We prove that ℙ{τ x >t} ~V(x) ℙ{X t ≥0}/t as t→∞ for a certain function V(x). Using known results for the large deviations of random walks, we obtain asymptotics for ℙ{τ x >t} explicitly in both light- and heavy-tailed cases.

EXACT ASYMPTOTICS OF SAMPLE-MEAN-RELATED RARE-EVENT PROBABILITIES

2017 ◽  
Vol 32 (2) ◽  
pp. 207-228 ◽  
Author(s):  
Julia Kuhn ◽  
Michel Mandjes ◽  
Thomas Taimre
Keyword(s):  
Large Deviations ◽  
Importance Sampling ◽  
Rare Event ◽  
Random Variable ◽  
Exact Asymptotics ◽  
Sample Mean ◽  
Packing Problems ◽  
Polynomial Term ◽  
Image Position ◽  
Two Populations

Relying only on the classical Bahadur–Rao approximation for large deviations (LDs) of univariate sample means, we derive strong LD approximations for probabilities involving two sets of sample means. The main result concerns the exact asymptotics (asn→∞) of$$ {\open P}\left({\max_{i\in\{1,\ldots,d_x\}}\bar X_{i,n} \les \min_{i\in\{1,\ldots,d_y\}}\bar Y_{i,n}}\right),$$with the${\bar X}_{i,n}{\rm s}$(${\bar Y}_{i,n}{\rm s}$, respectively) denotingdx(dy) independent copies of sample means associated with the random variableX(Y). Assuming${\open E}X \gt {\open E}Y$, this is a rare event probability that vanishes essentially exponentially, but with an additional polynomial term. We also point out how the probability of interest can be estimated using importance sampling in a logarithmically efficient way. To demonstrate the usefulness of the result, we show how it can be applied to compare the order statistics of the sample means of the two populations. This has various applications, for instance in queuing or packing problems.

scholarly journals Run-and-Tumble Motion: The Role of Reversibility

2021 ◽  
Vol 183 (3) ◽  
Author(s):  
Bart van Ginkel ◽  
Bart van Gisbergen ◽  
Frank Redig
Keyword(s):  
Free Energy ◽  
Diffusion Coefficient ◽  
Large Deviations ◽  
Energy Function ◽  
Internal State ◽  
Rate Function ◽  
Free Energy Function ◽  
Large Deviations Principle ◽  
Preferred Direction ◽  
State Process

AbstractWe study a model of active particles that perform a simple random walk and on top of that have a preferred direction determined by an internal state which is modelled by a stationary Markov process. First we calculate the limiting diffusion coefficient. Then we show that the ‘active part’ of the diffusion coefficient is in some sense maximal for reversible state processes. Further, we obtain a large deviations principle for the active particle in terms of the large deviations rate function of the empirical process corresponding to the state process. Again we show that the rate function and free energy function are (pointwise) optimal for reversible state processes. Finally, we show that in the case with two states, the Fourier–Laplace transform of the distribution, the moment generating function and the free energy function can be computed explicitly. Along the way we provide several examples.

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