scholarly journals Logarithmic Asymptotics for Multidimensional Extremes Under Nonlinear Scalings

2015 ◽  
Vol 52 (1) ◽  
pp. 68-81 ◽  
Author(s):  
K. M. Kosiński ◽  
M. Mandjes
Keyword(s):  
Large Deviation Principle ◽  
Large Deviation ◽  
Random Vectors ◽  
Deviation Principle ◽  
Logarithmic Asymptotics ◽  
Rate Functions ◽  
Regularly Varying ◽  
Positive Index

Let W = {Wn: n ∈ N} be a sequence of random vectors in Rd, d ≥ 1. In this paper we consider the logarithmic asymptotics of the extremes of W, that is, for any vector q > 0 in Rd, we find that logP(there exists n ∈ N: Wnuq) as u → ∞. We follow the approach of the restricted large deviation principle introduced in Duffy (2003). That is, we assume that, for every q ≥ 0, and some scalings {an}, {vn}, (1 / vn)logP(Wn / an ≥ uq) has a, continuous in q, limit JW(q). We allow the scalings {an} and {vn} to be regularly varying with a positive index. This approach is general enough to incorporate sequences W, such that the probability law of Wn / an satisfies the large deviation principle with continuous, not necessarily convex, rate functions. The equations for these asymptotics are in agreement with the literature.

scholarly journals Logarithmic Asymptotics for Multidimensional Extremes Under Nonlinear Scalings

2015 ◽  
Vol 52 (01) ◽  
pp. 68-81 ◽  
Author(s):  
K. M. Kosiński ◽  
M. Mandjes
Keyword(s):  
Large Deviation Principle ◽  
Large Deviation ◽  
Random Vectors ◽  
Deviation Principle ◽  
Logarithmic Asymptotics ◽  
Rate Functions ◽  
Regularly Varying ◽  
Positive Index

Let W = { W n : n ∈ N} be a sequence of random vectors in R d , d ≥ 1. In this paper we consider the logarithmic asymptotics of the extremes of W , that is, for any vector q > 0 in R d , we find that logP(there exists n ∈ N: W n u q ) as u → ∞. We follow the approach of the restricted large deviation principle introduced in Duffy (2003). That is, we assume that, for every q ≥ 0, and some scalings {a n }, {v n }, (1 / v n )logP( W n / a n ≥ u q ) has a, continuous in q , limit J W ( q ). We allow the scalings {a n } and {v n } to be regularly varying with a positive index. This approach is general enough to incorporate sequences W , such that the probability law of W n / a n satisfies the large deviation principle with continuous, not necessarily convex, rate functions. The equations for these asymptotics are in agreement with the literature.

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.

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

2008 ◽  
Vol 45 (1) ◽  
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.

Large deviation principle for the field in prequantum classical statistical field theory

2017 ◽  
Vol 20 (04) ◽  
pp. 1750025
Author(s):  
Andrei Khrennikov ◽  
Achref Majid
Keyword(s):  
Field Theory ◽  
Random Field ◽  
Field Model ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Background Field ◽  
Deviation Principle ◽  
Statistical Field ◽  
Statistical Field Theory

In this paper, we prove a large deviation principle for the background field in prequantum statistical field model. We show a number of examples by choosing a specific random field in our model.

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 Large Deviation Principle for the Greedy Exploration Algorithm over Erdös-Rényi Graphs

2022 ◽  
Vol 19 (1) ◽  
pp. 439
Author(s):  
Paola Bermolen ◽  
Valeria Goicoechea ◽  
Matthieu Jonckheere ◽  
Ernesto Mordecki
Keyword(s):  
Large Deviation Principle ◽  
Large Deviation ◽  
Deviation Principle
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