Analysis | Other Applied Mathematics | Probability

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This paper presents an alternate proof of the divergence of the unique maximizer sequence {π‘₯βˆ— 𝑛} of a function sequence {𝐹𝑛(π‘₯)} that is derived from an adaptive algorithm based on the now classic optimal stopping problem, known by many names but here β€˜the secretary problem’. The alternate proof uses a result established by Nguyen, Xu, and Zhao (n.d.) regarding the uniqueness of maximizer points of a generalized function sequence {π‘†πœ‡,𝜎 𝑛 } and relies on the strict monotonicity of 𝐹𝑛(π‘₯) as 𝑛 increases in order to show divergence of {π‘₯βˆ— 𝑛}. Towards this, limits of the exponentiated Gaussian CDF are established as well as a closed form of 𝐹′ 𝑛 (π‘₯), the derivative of the sequence’s function. The proof is elementary but nontrivial. The result in Nguyen et al. (n.d.) relies heavily on a technical lemma but, here, the proof is more transparent and relies solely on fundamentals.