Coherent distributions: Hilbert space approach and duality
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916259610755072 |
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| author | Kravchenko, Egor |
| author_facet | Kravchenko, Egor |
| contents | Let $X$ be a Bernoulli random variable with the success probability $p$. We are interested in tight bounds on $\mathbb{E}[f(X_1,X_2)]$, where $X_i=\mathbb{E}[X| \mathcal{F}_i]$ and $\mathcal{F}_i$ are some sigma-algebras. This problem is closely related to understanding extreme points of the set of coherent distributions. A distribution on $[0,1]^2$ is called $\textit{coherent}$ if it can be obtained as the joint distribution of $(X_1, X_2)$ for some choice of $\mathcal{F}_i$. By treating random variables as vectors in a Hilbert space, we establish an upper bound for quadratic $f$, characterize $f$ for which this bound is tight, and show that such $f$ result in exposed coherent distributions with arbitrarily large support. As a corollary, we get a tight bound on $\mathrm{cov}\,(X_1,X_2)$ for $p\in [1/3,\,2/3]$. To obtain a tight bound on $\mathrm{cov}\,(X_1,X_2)$ for all $p$, we develop an approach based on linear programming duality. Its generality is illustrated by tight bounds on $\mathbb{E}[|X_1-X_2|^α]$ for any $α>0$ and $p=1/2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_04375 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Coherent distributions: Hilbert space approach and duality Kravchenko, Egor Probability Theoretical Economics Let $X$ be a Bernoulli random variable with the success probability $p$. We are interested in tight bounds on $\mathbb{E}[f(X_1,X_2)]$, where $X_i=\mathbb{E}[X| \mathcal{F}_i]$ and $\mathcal{F}_i$ are some sigma-algebras. This problem is closely related to understanding extreme points of the set of coherent distributions. A distribution on $[0,1]^2$ is called $\textit{coherent}$ if it can be obtained as the joint distribution of $(X_1, X_2)$ for some choice of $\mathcal{F}_i$. By treating random variables as vectors in a Hilbert space, we establish an upper bound for quadratic $f$, characterize $f$ for which this bound is tight, and show that such $f$ result in exposed coherent distributions with arbitrarily large support. As a corollary, we get a tight bound on $\mathrm{cov}\,(X_1,X_2)$ for $p\in [1/3,\,2/3]$. To obtain a tight bound on $\mathrm{cov}\,(X_1,X_2)$ for all $p$, we develop an approach based on linear programming duality. Its generality is illustrated by tight bounds on $\mathbb{E}[|X_1-X_2|^α]$ for any $α>0$ and $p=1/2$. |
| title | Coherent distributions: Hilbert space approach and duality |
| topic | Probability Theoretical Economics |
| url | https://arxiv.org/abs/2405.04375 |