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Diffusion approximations in the online increasing subsequence problem
Stochastic Processes and their Applications ( IF 1.4 ) Pub Date : 2021-06-08 , DOI: 10.1016/j.spa.2021.06.001
Alexander Gnedin , Amirlan Seksenbayev

The online increasing subsequence problem is a stochastic optimisation task with the objective to maximise the expected length of subsequence chosen from a random series by means of a nonanticipating decision strategy. We study the structure of optimal and near-optimal subsequences in a standardised planar Poisson framework. Following a long-standing suggestion by Bruss and Delbaen (2004), we prove a joint functional limit theorem for the transversal fluctuations about the diagonal of the running maximum and the length processes. The limit is identified explicitly with a Gaussian time-inhomogeneous diffusion. In particular, the running maximum converges to a Brownian bridge, and the length process has another explicit non-Markovian limit.



中文翻译:

在线递增子序列问题中的扩散近似

在线增加子序列问题是一项随机优化任务,其目标是通过非预期决策策略使从随机序列中选择的子序列的预期长度最大化。我们研究了标准化平面泊松框架中最优和接近最优子序列的结构。根据Bruss 和Delbaen (2004) 的长期建议,我们证明了关于运行最大值和长度过程对角线的横向涨落的联合泛函极限定理。该限制明确标识为高斯时间非均匀扩散。特别是,运行最大值收敛到布朗桥,长度过程有另一个明确的非马尔可夫限制。

更新日期:2021-06-09
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