Journal of Statistical Planning and Inference 76 (1999) 1–17 On the bootstrap and the moving block bootstrap for the maximum of a stationary process Krishna B. Athreya 1 , Jun-ichiro Fukuchi ∗ , Soumendra N. Lahiri 2 Iowa State University, IA, USA Received 30 June 1997; accepted 22 June 1998 Abstract In this paper, asymptotic properties of bootstrap methods for the maximum of a stationary process are investigated. It is shown that the Efron’s bootstrap provides a valid approximation to the sampling distribution of the normalized maximum only in a restricted situation, but the moving block bootstrap is successful in a more general situation. c  1999 Elsevier Science B.V. All rights reserved. AMS classication: primary 62G09; 60G70; 60G10 Keywords: Bootstrap; Maximum; Moving block bootstrap; Stationary process 1. Introduction Efron (1979) introduced the bootstrap method of estimating the sampling distribu- tions of statistics. It is well known that when observations are independently and iden- tically distributed (i.i.d.), the Efron’s bootstrap (EB) provides a valid approximation to sampling distributions of a wide variety of statistics (the EB is said to be consis- tent in this case) including sample means, sample quantiles and Von Mises functionals (Bickel and Freedman, 1981). It is also known that, for some statistics, EB does not approximate their distributions at all. The maximum of random variables is one of the examples for which EB fails to be consistent. Recently, Swanepoel (1986), Deheuvels et al. (1993) and Athreya and Fukuchi (1994, 1997) (hereafter referred to as AF1, AF2, respectively) showed that one can make the EB consistent for extremes of i.i.d. random variables by making the resample size suitably smaller than the sample size. A natural question is: Is the bootstrap still consistent for the maximum of a stationary * Corresponding author; present address: Faculty of Economics, Hiroshima University, 1-2-1 Kagamiyama, Higashi-hiroshima 739 Japan. 1 Research supported in part by NSF Grant DMS 92-04938. 2 Research supported in part by NSF Grant DMS 95-05124. 0378-3758/99/$ – see front matter c  1999 Elsevier Science B.V. All rights reserved. PII: S0378-3758(98)00140-2