Extending the V-sequence reduction to arbitrary independent distributions
Abstract
Let X1,…,Xn be independent random variables with arbitrary known distributions, where n≥2. The optimal probability Vn(X1,…,Xn) of choosing an overall maximum satisfies Vn(X1,…,Xn)≥(1−1/n)n−1, and this bound is attained. We obtain this known sharp bound by extending the reduction lemma of Allaart and Islas (2016) from continuous to arbitrary distributions. We construct a V-sequence whose optimal win probability does not exceed that of the original sequence, and then apply the bound for V-sequences. The construction first replaces atoms by uniform observations on strictly ordered intervals and applies the original continuous reduction lemma. The reduced distributions are determined by a common critical level and a common probability for splitting a possible atom at that level.
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BibTeX
@misc{islas2026extending,
title = {{Extending the V-sequence reduction to arbitrary independent distributions}},
author = {José A. Islas},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35600},
eprint = {2610.35600},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35600},
}
RIS
TY - UNPB
TI - Extending the V-sequence reduction to arbitrary independent distributions
AU - José A. Islas
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35600
UR - https://arxiv.org/abs/2610.35600
AB - Let X_1,…,X_n be independent random variables with arbitrary known distributions, where n≥2. The optimal probability V_n(X_1,…,X_n) of choosing an overall maximum satisfies V_n(X_1,…,X_n)≥(1−1/n)^n−1, and this bound is attained. We obtain this known sharp bound by extending the reduction lemma of Allaart and Islas (2016) from continuous to arbitrary distributions. We construct a V-sequence whose optimal win probability does not exceed that of the original sequence, and then apply the bound for V-sequences. The construction first replaces atoms by uniform observations on strictly ordered intervals and applies the original continuous reduction lemma. The reduced distributions are determined by a common critical level and a common probability for splitting a possible atom at that level.
ER -