Benford Behavior in Stick Fragmentation Problems
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908500295155712 |
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| author | Fang, Bruce Irons, Ava Lippelman, Ella Miller, Steven J. |
| author_facet | Fang, Bruce Irons, Ava Lippelman, Ella Miller, Steven J. |
| contents | Benford's law is the statement that in many real-world data sets, the probability of having digit \(d\) in base \(B\), where \(1 \leq d \leq B\), as the first digit is \(\log_{B}\left(\tfrac{d+1}{d}\right)\). We sometimes refer to this as weak Benford behavior, and we say that a data set exhibits strong Benford behavior in base \(B\) if the probability of having significand at most \(s\), where \(s \in [1,B)\), is \(\log_{B}(s)\). We examine Benford behaviors in the stick fragmentation model. Building on the work on the 1-dimensional stick fragmentation model, we employ combinatorial identities on multinomial coefficients to reduce the high-dimensional stick fragmentation model to the 1-dimensional model and provide a necessary and sufficient condition for the lengths of the stick fragments to converge to strong Benford behavior. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_17360 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Benford Behavior in Stick Fragmentation Problems Fang, Bruce Irons, Ava Lippelman, Ella Miller, Steven J. Probability 60A10, 11K06 (primary), 60E10 (secondary) Benford's law is the statement that in many real-world data sets, the probability of having digit \(d\) in base \(B\), where \(1 \leq d \leq B\), as the first digit is \(\log_{B}\left(\tfrac{d+1}{d}\right)\). We sometimes refer to this as weak Benford behavior, and we say that a data set exhibits strong Benford behavior in base \(B\) if the probability of having significand at most \(s\), where \(s \in [1,B)\), is \(\log_{B}(s)\). We examine Benford behaviors in the stick fragmentation model. Building on the work on the 1-dimensional stick fragmentation model, we employ combinatorial identities on multinomial coefficients to reduce the high-dimensional stick fragmentation model to the 1-dimensional model and provide a necessary and sufficient condition for the lengths of the stick fragments to converge to strong Benford behavior. |
| title | Benford Behavior in Stick Fragmentation Problems |
| topic | Probability 60A10, 11K06 (primary), 60E10 (secondary) |
| url | https://arxiv.org/abs/2508.17360 |