Volume 26, Issue 1
A Riesz Product Type Measure on the Cantor Group

Commun. Math. Res., 26 (2010), pp. 7-16.

Published online: 2021-05

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• Abstract

A Riesz type product as $$P_n = \prod\limits_{j=1}^n (1 + aω_j + bω_{j+1})$$ is studied, where $a, b$ are two real numbers with $|a| + |b| < 1$, and {$ω_j$} are independent random variables taking values in {−1, 1} with equal probability. Let d$ω$ be the normalized Haar measure on the Cantor group $Ω$ = {−1, 1}$^N$. The sequence of probability measures $\Big \{\frac{P_n{\rm d}ω}{E(P_n)} \Big \}$ is showed to converge weakly to a unique continuous measure on $Ω$, and the obtained measure is singular with respect to d$ω$.

• Keywords

Riesz product, Cantor group, weak topology, singularity of measure.

• AMS Subject Headings

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@Article{CMR-26-7, author = {Shi , Qiyan}, title = {A Riesz Product Type Measure on the Cantor Group}, journal = {Communications in Mathematical Research }, year = {2021}, volume = {26}, number = {1}, pages = {7--16}, abstract = {

A Riesz type product as $$P_n = \prod\limits_{j=1}^n (1 + aω_j + bω_{j+1})$$ is studied, where $a, b$ are two real numbers with $|a| + |b| < 1$, and {$ω_j$} are independent random variables taking values in {−1, 1} with equal probability. Let d$ω$ be the normalized Haar measure on the Cantor group $Ω$ = {−1, 1}$^N$. The sequence of probability measures $\Big \{\frac{P_n{\rm d}ω}{E(P_n)} \Big \}$ is showed to converge weakly to a unique continuous measure on $Ω$, and the obtained measure is singular with respect to d$ω$.

}, issn = {2707-8523}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cmr/19173.html} }
TY - JOUR T1 - A Riesz Product Type Measure on the Cantor Group AU - Shi , Qiyan JO - Communications in Mathematical Research VL - 1 SP - 7 EP - 16 PY - 2021 DA - 2021/05 SN - 26 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/cmr/19173.html KW - Riesz product, Cantor group, weak topology, singularity of measure. AB -

A Riesz type product as $$P_n = \prod\limits_{j=1}^n (1 + aω_j + bω_{j+1})$$ is studied, where $a, b$ are two real numbers with $|a| + |b| < 1$, and {$ω_j$} are independent random variables taking values in {−1, 1} with equal probability. Let d$ω$ be the normalized Haar measure on the Cantor group $Ω$ = {−1, 1}$^N$. The sequence of probability measures $\Big \{\frac{P_n{\rm d}ω}{E(P_n)} \Big \}$ is showed to converge weakly to a unique continuous measure on $Ω$, and the obtained measure is singular with respect to d$ω$.

QiyanShi. (2021). A Riesz Product Type Measure on the Cantor Group. Communications in Mathematical Research . 26 (1). 7-16. doi:
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