Asymptotic exponentiality of the first exit time of the Shiryaev–Roberts diffusion with constant positive drift. Issue 3 (3rd July 2017)

• Record Type: Journal Article
• Authors: Polunchenko, Aleksey S.
• Is Part Of: Sequential analysis. Volume 36:Issue 3(2017)
• Page Start: 370
• Page End: 383
• Publication Date: 2017-07-03
• Subjects: Convergence of probability measures -- first exit time -- Generalized Shiryaev–Roberts procedure -- Laplace transform -- Markov diffusion -- moment generating function -- quickest change-point detection -- Whittaker functions; 62L10 -- 60G40 -- 60J60; Sequential analysis -- Periodicals; 519.54
• DOI: 10.1080/07474946.2017.1360089
• Languages: English
• ISSNs: 0747-4946

Abstract

ABSTRACT: We consider the first exit time of a Shiryaev–Roberts diffusion with constant positive drift from the interval [0, A ] with A >0 fixed. We show that the moment generating function (Laplace transform) of a suitably standardized version of the first exit time converges to that of the unit-mean exponential distribution as A →+∞. The proof is explicit in that the moment generating function of the first exit time is first expressed analytically and in a closed form and then the desired limit as A →+∞ is evaluated directly. The result is of importance in the area of quickest change-point detection, and its discrete-time counterpart has been previously established—although in a different manner—by Pollak and Tartakovsky (2009a ).