scholarly journals Boolean Valued Representation of Random Sets and Markov Kernels with Application to Large Deviations

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1848
Author(s):  
Antonio Avilés López ◽  
José Miguel Zapata García
Keyword(s):  
Large Deviations ◽  
Random Sets ◽  
Transfer Principle ◽  
Random Set ◽  
Borel Sets ◽  
Large Deviations Theory ◽  
Boolean Valued Analysis ◽  
Markov Kernels ◽  
Borel Functions ◽  
Borel Probability

We establish a connection between random set theory and Boolean valued analysis by showing that random Borel sets, random Borel functions, and Markov kernels are respectively represented by Borel sets, Borel functions, and Borel probability measures in a Boolean valued model. This enables a Boolean valued transfer principle to obtain random set analogues of available theorems. As an application, we establish a Boolean valued transfer principle for large deviations theory, which allows for the systematic interpretation of results in large deviations theory as versions for Markov kernels. By means of this method, we prove versions of Varadhan and Bryc theorems, and a conditional version of Cramér theorem.

scholarly journals Large-Deviations Theory and Empirical Estimator Choice

2008 ◽  
Vol 27 (4-6) ◽  
pp. 513-525 ◽  
Author(s):  
Marian Grendár ◽  
George Judge
Keyword(s):  
Large Deviations ◽  
Large Deviations Theory ◽  
Empirical Estimator

Estimating the reduced moments of a random measure

1996 ◽  
Vol 28 (02) ◽  
pp. 335-336
Author(s):  
Kiên Kiêu ◽  
Marianne Mora
Keyword(s):  
Transformation Group ◽  
Smooth Boundary ◽  
Random Measure ◽  
Random Sets ◽  
Random Set ◽  
Counting Measure ◽  
Geometrical Properties ◽  
Measure Decomposition ◽  
Moment Measures ◽  
Curvature Measures

Random measures are commonly used to describe geometrical properties of random sets. Examples are given by the counting measure associated with a point process, and the curvature measures associated with a random set with a smooth boundary. We consider a random measure with an invariant distribution under the action of a standard transformation group (translatioris, rigid motions, translations along a given direction and so on). In the framework of the theory of invariant measure decomposition, the reduced moments of the random measure are obtained by decomposing the related moment measures.

MULTIVARIATE MODELS AND VARIABILITY INTERVALS: A LOCAL RANDOM SET APPROACH

2011 ◽  
Vol 19 (05) ◽  
pp. 799-823 ◽  
Author(s):  
THOMAS FETZ
Keyword(s):  
Confidence Level ◽  
Structural Mechanics ◽  
Random Sets ◽  
Confidence Limits ◽  
Random Set ◽  
Multivariate Functions ◽  
Multivariate Models ◽  
Multivariate Case ◽  
Fréchet Bounds

This article is devoted to the propagation of families of variability intervals through multivariate functions comprising the semantics of confidence limits. At fixed confidence level, local random sets are defined whose aggregation admits the calculation of upper probabilities of events. In the multivariate case, a number of ways of combination is highlighted to encompass independence and unknown interaction using random set independence and Fréchet bounds. For all cases we derive formulas for the corresponding upper probabilities and elaborate how they relate. An example from structural mechanics is used to exemplify the method.

scholarly journals Stationary countable dense random sets

2000 ◽  
Vol 32 (01) ◽  
pp. 86-100 ◽  
Author(s):  
Wilfrid S. Kendall
Keyword(s):  
Probability Theory ◽  
Polish Space ◽  
Measurable Subset ◽  
Locally Compact Group ◽  
Random Sets ◽  
Random Set ◽  
Locally Compact ◽  
Polish Spaces ◽  
Probability Zero ◽  
Group Of Symmetries

We study the probability theory of countable dense random subsets of (uncountably infinite) Polish spaces. It is shown that if such a set is stationary with respect to a transitive (locally compact) group of symmetries then any event which concerns the random set itself (rather than accidental details of its construction) must have probability zero or one. Indeed the result requires only quasi-stationarity (null-events stay null under the group action). In passing, it is noted that the property of being countable does not correspond to a measurable subset of the space of subsets of an uncountably infinite Polish space.

Law of small numbers as concentration inequalities for sums of independent random setsand random set valued mappings

2012 ◽  
Author(s):  
Vladimir I. Norkin ◽  
Roger J-B Wets
Keyword(s):  
Hilbert Space ◽  
Discrete Distributions ◽  
Random Sets ◽  
Concentration Inequalities ◽  
Random Set ◽  
Infinite Dimensional ◽  
Random Functions ◽  
The Monte Carlo Method ◽  
Exponential Bounds ◽  
Set Valued Mappings

In the paper we study concentration of sample averages (Minkowski's sums) of independent bounded random sets and set valued mappings around their expectations. Sets and mappings are considered in a Hilbert space. Concentration is formulated in the form of exponential bounds on probabilities of normalized large deviations. In a sense, concentration phenomenon reflects the law of small numbers, describing non-asymptotic behavior of the sample averages. We sequentially consider concentration inequalities for bounded random variables, functions, vectors, sets and mappings, deriving next inequalities from preceding cases. Thus we derive concentration inequalities with explicit constants for random sets and mappings from the sharpest available (Talagrand type) inequalities for random functions and vectors. The most explicit inequalities are obtained in case of discrete distributions. The obtained results contribute to substantiation of the Monte Carlo method in infinite dimensional spaces.

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