Statistics and Probability Letters 93 (2014) 134–142 Contents lists available at ScienceDirect Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro Bootstrapping the empirical distribution of a linear process Farid El Ktaibi a,∗ , B. Gail Ivanoff a , Neville C. Weber b a Department of Mathematics & Statistics, University of Ottawa, 585 King Edward, Ottawa ON K1N 6N5, Canada b School of Mathematics & Statistics, University of Sydney, NSW 2006, Australia article info Article history: Received 5 December 2013 Received in revised form 13 June 2014 Accepted 21 June 2014 Available online 2 July 2014 MSC: primary 62G09 secondary 62G30 60G10 62M10 Keywords: Causal linear process Empirical process Moving block bootstrap Goodness of fit abstract The validity of the moving block bootstrap for the empirical distribution of a short memory causal linear process is established under simple conditions that do not involve mixing or association. Sufficient conditions can be expressed in terms of the existence of moments of the innovations and summability of the coefficients of the linear model. Applications to one and two sample tests are discussed. © 2014 Elsevier B.V. All rights reserved. 1. Introduction There is a vast literature on the asymptotic behaviour of the empirical process generated by a stationary sequence (X i , i ∈ Z) of random variables and the validity of associated bootstrap techniques. Almost always, conditions on mixing or association of the sequence are imposed. As noted in recent papers by Sharipov and Wendler (2012) and Radulović (2012), mixing conditions can be difficult to verify and can exclude linear processes. In this paper we establish the validity of the moving block bootstrap (MBB) for the empirical process associated with short memory causal linear processes. In the case of causal linear processes X i =  j≥0 a j ξ i−j , i ∈ Z, (1) where (ξ j : j ∈ Z) is a sequence of independent and identically distributed (i.i.d.) random variables and (a j : j ∈ N) is a sequence of constants. If the innovations have finite variance, the process is said to have short memory if  ∞ j=0 |a j | < ∞. This model includes a wide range of stationary ARMA time series models, many of which are not mixing. A classic example due to Ibragimov is the following: let the ξ i ’s be i.i.d. N (0, 1) and let a i be the coefficient of z i in the power series expansion of the function h(z ) = (1 − z ) p , where p > 4 is non-integer. In this case, |a i |= O(i 1−p ) but (X i ) is not strong mixing (cf. Gorodetskii, 1977). Therefore, a different approach is needed to establish empirical and bootstrap empirical central limit theorems for linear processes. ∗ Corresponding author. E-mail addresses: felkt081@uottawa.ca (F. El Ktaibi), givanoff@uottawa.ca (B. Gail Ivanoff), neville.weber@sydney.edu.au (N.C. Weber). http://dx.doi.org/10.1016/j.spl.2014.06.019 0167-7152/© 2014 Elsevier B.V. All rights reserved.