Living Reference Work Entry

Handbook of Scan Statistics

pp 1-28

Date: Latest Version

Martingale Methods

  • Vladimir PozdnyakovAffiliated withDepartment of Statistics, University of Connecticut Email author 
  • , J. Michael SteeleAffiliated withDepartment of Statistics, The Wharton School, University of Pennsylvania


We survey the ways that martingales and the method of gambling teams can be used to obtain otherwise hard-to-get information for the moments and distributions of waiting times for the occurrence of simple or compound patterns in an independent or a Markov sequence. We also survey how such methods can be used to provide moments and distribution approximations for a variety of scan statistics, including variable length scan statistics. Each of the general problems considered here is accompanied by one or more concrete examples that illustrate the computational tractability of the methods.


Scan Pattern Martingale Waiting time Gambling teams Shifted exponential distribution