scholarly journals An Upper Bound of Large Deviations for Capacities

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Xiaomin Cao
Keyword(s):  
Large Deviations ◽  
Poisson Process ◽  
Upper Bound ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Rate Function ◽  
Risk Theory ◽  
Model Uncertainties ◽  
New Process ◽  
Measures Of Risk

Up to now, most of the academic researches about the large deviation and risk theory are under the framework of the classical linear expectations. But motivated by problems of model uncertainties in statistics, measures of risk, and superhedging in finance, sublinear expectations are extensively studied. In this paper, we obtain a type of large deviation principle under the sublinear expectation. This result is a new expression of the Gärtner-Ellis theorem under the sublinear expectations which is in the classical theory of large deviations. In addition, we introduce a new process under the sublinear expectations, that is, theG-Poisson process. We give an application of our result and obtain the rate function of the compoundG-Poisson process in the upper bound of large deviations for capacities. The application of our result opens a new field for the research of risk theory in the future.

A LARGE DEVIATION FOR OCCUPATION TIME OF SUPER α-STABLE PROCESS

2008 ◽  
Vol 11 (01) ◽  
pp. 53-71 ◽  
Author(s):  
QIU-YUE LI ◽  
YAN-XIA REN
Keyword(s):  
Brownian Motion ◽  
Large Deviations ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Stable Process ◽  
Rate Function ◽  
Occupation Time ◽  
Occupation Times ◽  
Tail Probabilities ◽  
Super Brownian Motion

We derive a large deviation principle for occupation time of super α-stable process in ℝd with d > 2α. The decay of tail probabilities is shown to be exponential and the rate function is characterized. Our result can be considered as a counterpart of Lee's work on large deviations for occupation times of super-Brownian motion in ℝd for dimension d > 4 (see Ref. 10).

scholarly journals Sample Path Large Deviations of Poisson Shot Noise with Heavy-Tailed Semiexponential Distributions

2011 ◽  
Vol 48 (03) ◽  
pp. 688-698 ◽  
Author(s):  
Ken R. Duffy ◽  
Giovanni Luca Torrisi
Keyword(s):  
Large Deviations ◽  
Shot Noise ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Sample Path ◽  
Rate Function ◽  
Sample Paths ◽  
Heavy Tailed ◽  
Poisson Shot Noise ◽  
Sample Path Large Deviations

It is shown that the sample paths of Poisson shot noise with heavy-tailed semiexponential distributions satisfy a large deviation principle with a rate function that is insensitive to the shot shape. This demonstrates that, on the scale of large deviations, paths to rare events do not depend on the shot shape.

scholarly journals Matrix Liberation Process II: Relation to Orbital Free Entropy

2020 ◽  
pp. 1-49
Author(s):  
Yoshimichi Ueda
Keyword(s):  
Mutual Information ◽  
Upper Bound ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Rate Function ◽  
New Approach ◽  
Free Entropy ◽  
The Matrix ◽  
Orbital Free ◽  
Definition Of

Abstract We investigate the concept of orbital free entropy from the viewpoint of the matrix liberation process. We will show that many basic questions around the definition of orbital free entropy are reduced to the question of full large deviation principle for the matrix liberation process. We will also obtain a large deviation upper bound for a certain family of random matrices that is essential to define the orbital free entropy. The resulting rate function is made up into a new approach to free mutual information.

Small double limit with reflecting Wentzel boundary condition

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ibrahima Sane ◽  
Alassane Diedhiou
Keyword(s):  
Boundary Condition ◽  
Differential Equations ◽  
Large Deviations ◽  
Stochastic Differential Equations ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Rate Function ◽  
Deviation Principle ◽  
Two Parameters ◽  
Double Limit

Abstract We provide a large deviation principle on the stochastic differential equations with reflecting Wentzel boundary condition if δ ε {\frac{\delta}{\varepsilon}} tends to 0 when the two parameters δ (homogenization parameter) and ε (the large deviations parameter) tend to zero. Here, we suppose that the homogenization parameter converges sufficiently quickly more than the large deviations parameter. Furthermore, we will make explicit the associated rate function.

scholarly journals Sample Path Large Deviations of Poisson Shot Noise with Heavy-Tailed Semiexponential Distributions

2011 ◽  
Vol 48 (3) ◽  
pp. 688-698
Author(s):  
Ken R. Duffy ◽  
Giovanni Luca Torrisi
Keyword(s):  
Large Deviations ◽  
Shot Noise ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Sample Path ◽  
Rate Function ◽  
Sample Paths ◽  
Heavy Tailed ◽  
Poisson Shot Noise ◽  
Sample Path Large Deviations

It is shown that the sample paths of Poisson shot noise with heavy-tailed semiexponential distributions satisfy a large deviation principle with a rate function that is insensitive to the shot shape. This demonstrates that, on the scale of large deviations, paths to rare events do not depend on the shot shape.

scholarly journals LARGE DEVIATIONS FOR FLOWS OF INTERACTING BROWNIAN MOTIONS

2010 ◽  
Vol 10 (03) ◽  
pp. 315-339 ◽  
Author(s):  
A. A. DOROGOVTSEV ◽  
O. V. OSTAPENKO
Keyword(s):  
Brownian Motion ◽  
Large Deviations ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Stochastic Flows ◽  
Brownian Motions ◽  
Deviation Principle ◽  
And Brownian Motion

We establish the large deviation principle (LDP) for stochastic flows of interacting Brownian motions. In particular, we consider smoothly correlated flows, coalescing flows and Brownian motion stopped at a hitting moment.

scholarly journals The Large Deviations of Estimating Rate Functions

2005 ◽  
Vol 42 (01) ◽  
pp. 267-274 ◽  
Author(s):  
Ken Duffy ◽  
Anthony P. Metcalfe
Keyword(s):  
Large Deviation Principle ◽  
Large Deviation ◽  
Length Distribution ◽  
Rate Function ◽  
Partial Sums ◽  
Single Server ◽  
Mixing Condition ◽  
Deviation Principle ◽  
Queue Length Distribution ◽  
Empirical Estimates

Given a sequence of bounded random variables that satisfies a well-known mixing condition, it is shown that empirical estimates of the rate function for the partial sums process satisfy the large deviation principle in the space of convex functions equipped with the Attouch-Wets topology. As an application, a large deviation principle for estimating the exponent in the tail of the queue length distribution at a single-server queue with infinite waiting space is proved.

A large deviations heuristic made precise

2000 ◽  
Vol 128 (3) ◽  
pp. 561-569 ◽  
Author(s):  
NEIL O'CONNELL
Keyword(s):  
Large Deviations ◽  
Bin Packing ◽  
Symmetric Functions ◽  
Large Deviation ◽  
Random Variables ◽  
Packing Problem ◽  
Rate Function ◽  
Empirical Measures ◽  
Underlying Distribution ◽  
Uniformly Lipschitz

Sanov's Theorem states that the sequence of empirical measures associated with a sequence of i.d.d. random variables satisfies the large deviation principle (LDP) in the weak topology with rate function given by a relative entropy. We present a derivative which allows one to establish LDPs for symmetric functions of many i.d.d. random variables under the condition that (i) a law of large numbers holds whatever the underlying distribution and (ii) the functions are uniformly Lipschitz. The heuristic (of the title) is that the LDP follows from (i) provided the functions are ‘sufficiently smooth’. As an application, we obtain large deviations results for the stochastic bin-packing problem.

Occupation time processes of super-Brownian motion with cut-off branching

2004 ◽  
Vol 41 (4) ◽  
pp. 984-997 ◽  
Author(s):  
Zhao Dong ◽  
Shui Feng
Keyword(s):  
Brownian Motion ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Rate Function ◽  
Occupation Time ◽  
Time Behavior ◽  
Long Time ◽  
Super Brownian Motion ◽  
Dimension One ◽  
Occupation Time Process

In this article we investigate a class of superprocess with cut-off branching, studying the long-time behavior of the occupation time process. Persistence of the process holds in all dimensions. Central-limit-type theorems are obtained, and the scales are dimension dependent. The Gaussian limit holds only when d ≤ 4. In dimension one, a full large deviation principle is established and the rate function is identified explicitly. Our result shows that the super-Brownian motion with cut-off branching in dimension one has many features that are similar to super-Brownian motion in dimension three.

scholarly journals The Large Deviation Principle for the On-Off Weibull Sojourn Process

2008 ◽  
Vol 45 (01) ◽  
pp. 107-117 ◽  
Author(s):  
Ken R. Duffy ◽  
Artem Sapozhnikov
Keyword(s):  
Renewal Process ◽  
Time Scale ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Rate Function ◽  
Compact Set ◽  
Natural Processes ◽  
Deviation Principle ◽  
Rate Functions ◽  
Nonlinear Scale

This article proves that the on-off renewal process with Weibull sojourn times satisfies the large deviation principle on a nonlinear scale. Unusually, its rate function is not convex. Apart from on a compact set, the rate function is infinite, which enables us to construct natural processes that satisfy the large deviation principle with nontrivial rate functions on more than one time scale.

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