By Anatolii Puhalskii
ISBN-10: 1420035800
ISBN-13: 9781420035803
ISBN-10: 1584881984
ISBN-13: 9781584881988
This Monograph bargains in-depth insurance of the new advancements within the learn of the massive deviation precept (LDP). the writer supplies a scientific remedy of a unique method of proving the LDP for sequences of chance measures - using rules and techniques of susceptible convergence concept to massive deviation settings. He comprehensively investigates the connection among huge deviations and vulnerable convergence and among chance idea and idempotent chance idea.
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Additional resources for Large Deviations and Idempotent Probability
Sample text
Typically, the probability law of interest is known only to lie in some known class of probability measures. For example, suppose we have a box containing 30 red balls and 60 other balls, some of which are white and the rest are black. We are going to draw a ball from this box. The payoffs for getting a red, black, and white ball are $10, $20, and $30, respectively. What is the expected payoff? Of course, there is not enough information to answer this question in the classical way since we do not know the probability distribution of the red, white, and black balls.
In the first step, “germs” are distributed according to a Poisson process in Rd , then in a second step, these germs cause sets of points (grains) modeled as random closed sets of Rd . The union of these grains is a random closed set in Rd , called the Boolean model. For practical aspects of the Boolean model as well as statistical inference involved, we refer the reader to [73, 78]. Here, we elaborate on its mathematical structure. We need to consider Poisson model in the space F\{∅} of nonempty closed sets of Rd .
A selection of S is a function X : Ω → U such that X(ω) ∈ S(ω), ∀ω ∈ Ω. The existence of a selection is guaranteed by the axiom of choice. In our case, there is more mathematical structure involved, namely, a probability space (Ω, A, P ) and U together with some σ-field B on it. We seek selections that are A − B-measurable as well as “almost sure selections” in the sense that the selection X of S is measurable and X ∈ S except on a P -null set of Ω. For existence theorems and further details, we refer the reader to [62, 124].
Large Deviations and Idempotent Probability by Anatolii Puhalskii
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