Abstract
Let X be an N-dimensional RV taking values from \(\mathcal {X}\subseteq \mathbb {R}^N\).
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Notes
- 1.
We will be careless and omit a distinction between discrete and atomic measures; we can do so by assuming that our sigma algebras are sufficiently powerful such that only points can be atoms, cf. [Pin64, p. 25].
- 2.
One possible approach is to use [Gra90, Lemma 7.18] and show that our quantizer satisfies the conditions required there. Another approach is to recognize that \(\hat{X}^{(n)}\) is equivalent to a set of RVs—the integer part of X and the first n fractional bits of its binary expansion—and that this set grows with increasing n. Since moreover a number is equal to its binary expansion, we have \(\lim _{n\rightarrow \infty }\hat{X}^{(n)}=X\) \(P_{X}\)-a.s. We can thus apply [Gra90, Lemma 7.22] to obtain the result.
- 3.
The space \(\mathbb {R}^N\) is Hausdorff, so any two distinct points are separated by neighborhoods.
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Geiger, B.C., Kubin, G. (2018). General Input Distributions. In: Information Loss in Deterministic Signal Processing Systems. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-59533-7_3
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DOI: https://doi.org/10.1007/978-3-319-59533-7_3
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