A central limit theorem for the optimal selection process for monotone subsequences of maximum expected length

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• 作者：Bruss ; F. Thomas ; Delbaen ; Freddy
• 关键词：60G40 ; Martingales ; Normalization ; Optimality principle ; Online selection ; Poisson process ; Intensities ; Dynkin's theorem ; Squared variation ; Predictable process ; Compensator ; Infinitesimal generator ; Bracket process ; Covariance of processes ; Functional C
• 刊名：Stochastic Processes and their Applications
• 出版年：2004
• 出版时间：December, 2004
• 年：2004
• 卷：114
• 期：2
• 页码：287-311
• 全文大小：319 K

文摘

This article provides a refinement of the main results for the monotone subsequence selection problem, previously obtained by Bruss and Delbaen (Stoch. Proc. Appl. 96 (2001) 313). Let Formula Not Shown be a Poisson process with intensity 1 defined on the positive half-line. Let Formula Not Shown be the corresponding occurrence times, and let Formula Not Shown be a sequence of i.i.d. uniform random variables on [0,1], independent of the Formula Not Shown 's. We observe the Formula Not Shown sequentially. Call Formula Not Shown the observed value at time Formula Not Shown . For a given horizon Formula Not Shown, consider the objective to select in sequential order, without recall on preceding observations, a subsequence of monotone increasing values of maximal expected length. Let Formula Not Shown be the random number of selected values under the optimal strategy. Extending the objective of our first paper the main goal of the present paper is to understand the whole process Formula Not Shown, where the random variable Formula Not Shown denotes the number of the selected values up to time Formula Not Shown under the Formula Not Shown -optimal strategy. We show that this process obeys, under suitable normalization, a Central Limit Theorem. In particular, we show that this holds in a more complete sense than one would expect. The problem of interdependence of this process with two other processes studied before is overcome by the simultaneous study of three associated martingales. This analysis is based on refined martingale methods, and a non-negligible level of technical sophistication seems unavoidable. But then, the results are rewarding. We not only get the “right” functional Central Limit Theorem for Formula Not Shown tending to infinity but also the (singular) covariance matrix of the three-dimensional process summarizing the interacting processes. We feel there is no other way to understand these interactions, and believe that this adds value to our approach.