On the maximal correlation of some stochastic processes
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912993802977280 |
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| author | Chang, Yinshan Chen, Qinwei |
| author_facet | Chang, Yinshan Chen, Qinwei |
| contents | We study the maximal correlation coefficient $R(X,Y)$ between two stochastic processes $X$ and $Y$. In the case when $(X,Y)$ is a random walk, we find $R(X,Y)$ using the Csáki-Fischer identity and the lower semicontinuity of the map $\text{Law}(X,Y) \to R(X,Y)$. When $(X,Y)$ is a two-dimensional Lévy process, we express $R(X,Y)$ in terms of the Lévy measure of the process and the covariance matrix of the diffusion part of the process. Consequently, for a two-dimensional $α$-stable random vector $(X,Y)$ with $0<α<2$, we express $R(X,Y)$ in terms of $α$ and the spectral measure $τ$ of the $α$-stable distribution. We also establish analogs and extensions of the Dembo-Kagan-Shepp-Yu inequality and the Madiman-Barron inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17109 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the maximal correlation of some stochastic processes Chang, Yinshan Chen, Qinwei Probability Statistics Theory 60G51, 62J10, 62H20 We study the maximal correlation coefficient $R(X,Y)$ between two stochastic processes $X$ and $Y$. In the case when $(X,Y)$ is a random walk, we find $R(X,Y)$ using the Csáki-Fischer identity and the lower semicontinuity of the map $\text{Law}(X,Y) \to R(X,Y)$. When $(X,Y)$ is a two-dimensional Lévy process, we express $R(X,Y)$ in terms of the Lévy measure of the process and the covariance matrix of the diffusion part of the process. Consequently, for a two-dimensional $α$-stable random vector $(X,Y)$ with $0<α<2$, we express $R(X,Y)$ in terms of $α$ and the spectral measure $τ$ of the $α$-stable distribution. We also establish analogs and extensions of the Dembo-Kagan-Shepp-Yu inequality and the Madiman-Barron inequality. |
| title | On the maximal correlation of some stochastic processes |
| topic | Probability Statistics Theory 60G51, 62J10, 62H20 |
| url | https://arxiv.org/abs/2411.17109 |