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 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 Sample Path Large Deviations for Order Statistics

2011 ◽  
Vol 48 (01) ◽  
pp. 238-257 ◽  
Author(s):  
Ken R. Duffy ◽  
Claudio Macci ◽  
Giovanni Luca Torrisi
Keyword(s):  
Order Statistics ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Sample Path ◽  
Sample Paths ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Trimmed Means ◽  
Hill Plots ◽  
Sample Path Large Deviations

We consider the sample paths of the order statistics of independent and identically distributed random variables with common distribution function F. If F is strictly increasing but possibly having discontinuities, we prove that the sample paths of the order statistics satisfy the large deviation principle in the Skorokhod M 1 topology. Sanov's theorem is deduced in the Skorokhod M'1 topology as a corollary to this result. A number of illustrative examples are presented, including applications to the sample paths of trimmed means and Hill plots.

scholarly journals Large deviations for multidimensional state-dependent shot-noise processes

2015 ◽  
Vol 52 (04) ◽  
pp. 1097-1114 ◽  
Author(s):  
Amarjit Budhiraja ◽  
Pierre Nyquist
Keyword(s):  
Large Deviations ◽  
Shot Noise ◽  
Large Deviation ◽  
Physical Systems ◽  
Sample Paths ◽  
State Dependent ◽  
Light Tails ◽  
Engineering Sciences ◽  
Weak Convergence Approach ◽  
Poisson Shot Noise

Shot-noise processes are used in applied probability to model a variety of physical systems in, for example, teletraffic theory, insurance and risk theory, and in the engineering sciences. In this paper we prove a large deviation principle for the sample-paths of a general class of multidimensional state-dependent Poisson shot-noise processes. The result covers previously known large deviation results for one-dimensional state-independent shot-noise processes with light tails. We use the weak convergence approach to large deviations, which reduces the proof to establishing the appropriate convergence of certain controlled versions of the original processes together with relevant results on existence and uniqueness.

scholarly journals Sample Path Large Deviations for Order Statistics

2011 ◽  
Vol 48 (1) ◽  
pp. 238-257 ◽  
Author(s):  
Ken R. Duffy ◽  
Claudio Macci ◽  
Giovanni Luca Torrisi
Keyword(s):  
Order Statistics ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Sample Path ◽  
Sample Paths ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Trimmed Means ◽  
Hill Plots ◽  
Sample Path Large Deviations

We consider the sample paths of the order statistics of independent and identically distributed random variables with common distribution functionF. IfFis strictly increasing but possibly having discontinuities, we prove that the sample paths of the order statistics satisfy the large deviation principle in the SkorokhodM1topology. Sanov's theorem is deduced in the SkorokhodM'1topology as a corollary to this result. A number of illustrative examples are presented, including applications to the sample paths of trimmed means and Hill plots.

scholarly journals Large deviations for multidimensional state-dependent shot-noise processes

2015 ◽  
Vol 52 (4) ◽  
pp. 1097-1114 ◽  
Author(s):  
Amarjit Budhiraja ◽  
Pierre Nyquist
Keyword(s):  
Large Deviations ◽  
Shot Noise ◽  
Large Deviation ◽  
Physical Systems ◽  
Sample Paths ◽  
State Dependent ◽  
Light Tails ◽  
Engineering Sciences ◽  
Weak Convergence Approach ◽  
Poisson Shot Noise

Shot-noise processes are used in applied probability to model a variety of physical systems in, for example, teletraffic theory, insurance and risk theory, and in the engineering sciences. In this paper we prove a large deviation principle for the sample-paths of a general class of multidimensional state-dependent Poisson shot-noise processes. The result covers previously known large deviation results for one-dimensional state-independent shot-noise processes with light tails. We use the weak convergence approach to large deviations, which reduces the proof to establishing the appropriate convergence of certain controlled versions of the original processes together with relevant results on existence and uniqueness.

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 in Credit Risk

2011 ◽  
Vol 2011 ◽  
pp. 1-28 ◽  
Author(s):  
V. J. G. Leijdekker ◽  
M. R. H. Mandjes ◽  
P. J. C. Spreij
Keyword(s):  
Credit Risk ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Sample Path ◽  
Rare Event ◽  
Asymptotic Results ◽  
Loss Process ◽  
Logarithmic Decay ◽  
Sample Path Large Deviations ◽  
Deviation Theory

The event of large losses plays an important role in credit risk. As these large losses are typically rare, and portfolios usually consist of a large number of positions, large deviation theory is the natural tool to analyze the tail asymptotics of the probabilities involved. We first derive a sample-path large deviation principle (LDP) for the portfolio's loss process, which enables the computation of the logarithmic decay rate of the probabilities of interest. In addition, we derive exact asymptotic results for a number of specific rare-event probabilities, such as the probability of the loss process exceeding some given function.

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.

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