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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.20489 |
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| _version_ | 1866912295210188800 |
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| author | Sariev, Hristo |
| author_facet | Sariev, Hristo |
| contents | Although regular conditional distributions (r.c.d.) are well-defined and widely used measure-theoretic objects, they can violate our intuition from the classical definition of a conditional probability given an event. For that purpose, the notion of a proper r.c.d. has been introduced. Here, we study how properness, viewed as a property of probability kernels in general, is related to stationarity, compatibility, reversibility and totality, revealing the effects these properties have on the structure of probability kernels. As a further development, we consider the inverse problem of characterizing certain classes of r.c.d.s in terms of the above properties. In particular, we derive necessary and sufficient conditions under which, for a given probability kernel, there exists a unique (in some sense) sub-$σ$-algebra such that the probability kernel is a proper r.c.d. given that sub-$σ$-algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_20489 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On some properties of probability kernels Sariev, Hristo Probability 60A10, 60A05, 28A99 Although regular conditional distributions (r.c.d.) are well-defined and widely used measure-theoretic objects, they can violate our intuition from the classical definition of a conditional probability given an event. For that purpose, the notion of a proper r.c.d. has been introduced. Here, we study how properness, viewed as a property of probability kernels in general, is related to stationarity, compatibility, reversibility and totality, revealing the effects these properties have on the structure of probability kernels. As a further development, we consider the inverse problem of characterizing certain classes of r.c.d.s in terms of the above properties. In particular, we derive necessary and sufficient conditions under which, for a given probability kernel, there exists a unique (in some sense) sub-$σ$-algebra such that the probability kernel is a proper r.c.d. given that sub-$σ$-algebra. |
| title | On some properties of probability kernels |
| topic | Probability 60A10, 60A05, 28A99 |
| url | https://arxiv.org/abs/2503.20489 |