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In this paper, we study some basic analytic properties of a sequence of functions $\{S^{\mu,σ}_n\}$ that is directly derived in an adaptive algorithm originating from the classical score-based secretary problem. More specifically, we show that: 1. the uniqueness of maximum points of the function sequence $\{S^{\mu,σ}_n\};$ 2. the maximum point sequence of $\{S^{\mu,σ}_n\}$ monotone increases to infinity as $n$ tends to infinity. All of the proofs are elementary but nontrivial.
}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v57n4.24.05}, url = {http://global-sci.org/intro/article_detail/jms/23713.html} }In this paper, we study some basic analytic properties of a sequence of functions $\{S^{\mu,σ}_n\}$ that is directly derived in an adaptive algorithm originating from the classical score-based secretary problem. More specifically, we show that: 1. the uniqueness of maximum points of the function sequence $\{S^{\mu,σ}_n\};$ 2. the maximum point sequence of $\{S^{\mu,σ}_n\}$ monotone increases to infinity as $n$ tends to infinity. All of the proofs are elementary but nontrivial.