Abstract
We are concerned with the general problem of proving the existence of joint distributions of two discrete random variables M and N subject to infinitely many constraints of the form . In particular, the variable M has a countably infinite range and the other variable N is uniformly distributed with finite range. The constraints placed on the joint distributions will require, for most elements j in the range of N, for infinitely many values of i in the range of M, where the corresponding values of i depend on j. To prove the existence of such joint distributions, we apply a theorem proved by Strassen on the existence of joint distributions with prespecified marginal distributions.
We consider some combinatorial structures that can be decomposed into components. Given , consider an assembly, multiset, or selection among elements of , and consider a uniformly distributed random variable on . For each , denote by the number of components of of size i so that . In each of these combinatorial structures, there exists infinitely many processes , indexed by a real parameter x, consisting of non-negative independent variables such that the distribution of the vector equals the distribution of the vector conditional on the event . Let denote a random variable whose components are given by . We introduce the notion of pivot mass which is then combined with Strassen’s work to provide couplings of and with desired properties. For each of these combinatorial structures, we prove that there exists a real number for which we can couple and with when . We are providing a partial answer to the question “how much dependence is there in the process ?”
Acknowledgments
Thanks to Prof. Michael Cranston and Prof. Nathan Kaplan for their support in this project. Thanks to Prof. Richard Arratia and the referee for the helpful suggestions.
Citation
Joseph Squillace. "On the dependence of the component counting process of a uniform random variable." Electron. Commun. Probab. 26 1 - 12, 2021. https://doi.org/10.1214/21-ECP399
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