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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2303.14731 |
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| _version_ | 1866908240122478592 |
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| author | Priyadarshi, Amit Roychowdhury, Mrinal K. Verma, Manuj |
| author_facet | Priyadarshi, Amit Roychowdhury, Mrinal K. Verma, Manuj |
| contents | Let $ν$ be a Borel probability measure on a $d$-dimensional Euclidean space $\mathbb{R}^d$, $d\geq 1$, with a compact support, and let $(p_0, p_1, p_2, \ldots, p_N)$ be a probability vector with $p_j>0$ for $0\leq j\leq N$. Let $\{S_j: 1\leq j\leq N\}$ be a set of contractive mappings on $\mathbb{R}^d$. Then, a Borel probability measure $μ$ on $\mathbb R^d$ such that $μ=\sum_{j=1}^N p_jμ\circ S_j^{-1}+p_0ν$ is called an inhomogeneous measure, also known as a condensation measure on $\mathbb{R}^d$. For a given $r\in (0, +\infty)$, the quantization dimension of order $r$, if it exists, denoted by $D_r(μ)$, of a Borel probability measure $μ$ on $\mathbb{R}^d$ represents the speed at which the $n$th quantization error of order $r$ approaches to zero as the number of elements $n$ in an optimal set of $n$-means for $μ$ tends to infinity. In this paper, we investigate the quantization dimension for such a condensation measure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_14731 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quantization dimensions for inhomogeneous bi-Lipschitz Iterated Function Systems Priyadarshi, Amit Roychowdhury, Mrinal K. Verma, Manuj Probability 37A50, 28A80, 94A34 Let $ν$ be a Borel probability measure on a $d$-dimensional Euclidean space $\mathbb{R}^d$, $d\geq 1$, with a compact support, and let $(p_0, p_1, p_2, \ldots, p_N)$ be a probability vector with $p_j>0$ for $0\leq j\leq N$. Let $\{S_j: 1\leq j\leq N\}$ be a set of contractive mappings on $\mathbb{R}^d$. Then, a Borel probability measure $μ$ on $\mathbb R^d$ such that $μ=\sum_{j=1}^N p_jμ\circ S_j^{-1}+p_0ν$ is called an inhomogeneous measure, also known as a condensation measure on $\mathbb{R}^d$. For a given $r\in (0, +\infty)$, the quantization dimension of order $r$, if it exists, denoted by $D_r(μ)$, of a Borel probability measure $μ$ on $\mathbb{R}^d$ represents the speed at which the $n$th quantization error of order $r$ approaches to zero as the number of elements $n$ in an optimal set of $n$-means for $μ$ tends to infinity. In this paper, we investigate the quantization dimension for such a condensation measure. |
| title | Quantization dimensions for inhomogeneous bi-Lipschitz Iterated Function Systems |
| topic | Probability 37A50, 28A80, 94A34 |
| url | https://arxiv.org/abs/2303.14731 |