Abstract
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.
Highlights
A situation which often arises in probability theory is the necessity to generalize a known unconditional theorem to a setting which is not unconditional any more
We aim to show that the well-known set-theoretic techniques of Boolean valued analysis are perfectly suited to the type of applications of random set theory
By considering the Boolean valued model associated with the underlying probability space, we show that a random set corresponds to a Borel set in this model
Summary
A situation which often arises in probability theory is the necessity to generalize a known unconditional theorem to a setting which is not unconditional any more. The so-called transfer principle of Boolean valued models provides a tool for expanding the content of already available theorems to non-obvious analogues in random set theory. We study the Boolean valued representation of Markov kernels that stem from regular conditional distributions of real-valued random variables. By means of the previous Boolean valued representations and the Boolean valued transfer principle, we prove versions for sequences of Markov kernels of basics results in large deviations theory. We obtain versions of Varadhan’s and Bryc’s theorems and a conditional version of Cramér’s theorem for the sequence of sample means of a conditionally independent identically distributed sequence of real-valued random variables We emphasize that these results are just instances of application of the transfer principle, whose number can be increased.
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