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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.02248 |
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| _version_ | 1866910471969308672 |
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| author | Azmoodeh, Ehsan Beelders, Noah Mishura, Yuliya |
| author_facet | Azmoodeh, Ehsan Beelders, Noah Mishura, Yuliya |
| contents | In this short note, we introduce probabilistic Cauchy functional equations, specifically, functional equations of the following form: $$
f(X_1 + X_2) \stackrel{d}{=} f(X_1) + f(X_2),
$$ where $X_1$ and $X_2$ represent two independent identically distributed real-valued random variables governed by a distribution $μ$ having appropriate support on the real line. The symbol $\stackrel{d}{=}$ denotes equality in distribution. When $μ$ follows an exponential distribution, we provide sufficient (regularity) conditions on the function $f$ to ensure that the unique measurable solution to the above equation is solely linear. Furthermore, we present some partial results in the general case, establishing a connection to integrated Cauchy functional equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_02248 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Probabilistic Cauchy Functional Equations Azmoodeh, Ehsan Beelders, Noah Mishura, Yuliya Probability In this short note, we introduce probabilistic Cauchy functional equations, specifically, functional equations of the following form: $$ f(X_1 + X_2) \stackrel{d}{=} f(X_1) + f(X_2), $$ where $X_1$ and $X_2$ represent two independent identically distributed real-valued random variables governed by a distribution $μ$ having appropriate support on the real line. The symbol $\stackrel{d}{=}$ denotes equality in distribution. When $μ$ follows an exponential distribution, we provide sufficient (regularity) conditions on the function $f$ to ensure that the unique measurable solution to the above equation is solely linear. Furthermore, we present some partial results in the general case, establishing a connection to integrated Cauchy functional equations. |
| title | Probabilistic Cauchy Functional Equations |
| topic | Probability |
| url | https://arxiv.org/abs/2406.02248 |