A note on maximal conditional entropy on Lebesgue spaces
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917646124974080 |
|---|---|
| author | Hediger, Michael |
| author_facet | Hediger, Michael |
| contents | Let $(X,\mathcal{B},P)$ be a probability space and $\mathit{a}$ be a sub $σ$-field that is generated by an increasing sequence of sub $σ$-fields $(\mathit{a}_{n})_{n \in \mathbb{N}}$. Given $θ\in Θ$, where $Θ$ is some set, let $(X_{n}^θ)_{n \in \mathbb{N}}$ be a martingale adapted to $(\mathit{a}_{n})_{n \in \mathbb{N}}$. Martin (1969) provides sufficient conditions to show that $(X_{n}^θ)_{n \in \mathbb{N}}$ converges a.s. uniformly on $Θ$ to a random variable $X^θ$. His results are based on the assumption that there exists an integer $n$ s.t. the conditional entropy given $\mathit{a}_{n}$ is uniformly bounded over the set of finite partitions of $X$ with atoms from $\mathit{a}$. This study complements Martin's results by studying the latter assumption on the maximal conditional entropy in the context of measurable partitions of Lebesgue spaces. We provide conditions under which $\mathit{a}$ conveys too much information for the maximal conditional entropy to be finite. As an example, we consider the space of continuous functions with a compact support, equipped with the Borel $σ$-field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_05546 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A note on maximal conditional entropy on Lebesgue spaces Hediger, Michael Probability 28D20, 94A17, 60B05, 60G42 Let $(X,\mathcal{B},P)$ be a probability space and $\mathit{a}$ be a sub $σ$-field that is generated by an increasing sequence of sub $σ$-fields $(\mathit{a}_{n})_{n \in \mathbb{N}}$. Given $θ\in Θ$, where $Θ$ is some set, let $(X_{n}^θ)_{n \in \mathbb{N}}$ be a martingale adapted to $(\mathit{a}_{n})_{n \in \mathbb{N}}$. Martin (1969) provides sufficient conditions to show that $(X_{n}^θ)_{n \in \mathbb{N}}$ converges a.s. uniformly on $Θ$ to a random variable $X^θ$. His results are based on the assumption that there exists an integer $n$ s.t. the conditional entropy given $\mathit{a}_{n}$ is uniformly bounded over the set of finite partitions of $X$ with atoms from $\mathit{a}$. This study complements Martin's results by studying the latter assumption on the maximal conditional entropy in the context of measurable partitions of Lebesgue spaces. We provide conditions under which $\mathit{a}$ conveys too much information for the maximal conditional entropy to be finite. As an example, we consider the space of continuous functions with a compact support, equipped with the Borel $σ$-field. |
| title | A note on maximal conditional entropy on Lebesgue spaces |
| topic | Probability 28D20, 94A17, 60B05, 60G42 |
| url | https://arxiv.org/abs/2310.05546 |