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Proposition 3.2. Under the assumptions of Proposition 2.2, and as- suming that F has finite support, as n →∞, we have

[nt] −2 −1 2 ′ sup sup Zn(s,t)+ dn [nt] (V (s,nt)) Jτ (s) Hτ (ηi)/τ! 0≤t≤1 0≤s≤1 i=1 X −τD τ 2τ = O(n L (n)(log log n) ) a.s.

Proof. By the same argument as in the end of the proof of Proposition 2.1, it suffices to show that the above bound is valid for t = 1. We have V (s − wn−1V (s,n),n) − V (s,n) = −V ′(s,n)wn−1V (s,n)+ O(V ′′(s,n)(wn−1V (s,n))2) a.s.

Thus, bearing in mind the definition of Zn(·, ·) [cf. (3.1)],

n 1 −2 −1 2 ′ sup Zn(s, 1) + 2dn n (V (s,n)) Jτ (s) Hτ (ηi)/τ! w dw 0≤s≤1 0 i=1 Z X n 4 −2 −2 ′′ 3 −2 −2 = O(dn n V (s,n)(V (s,n)) )= O dn n Hτ (ηi)/τ! ! ! Xi=1 = O(n−τDLτ (n)(log log n)2τ ) a.s., where, on assuming that F has finite support, we have made use of ′′ ∞ sup0≤s≤1 Jτ (s) < and the law of iterated logarithm for partial sums [cf. (2.7)]. This completes the proof. 

Concerning Section 4, Proposition 4.2 remains valid on assuming the con- dition (R), in addition to (i)–(iv) with (v) or (v′) for F . However, its almost sure bounds in (4.5) hold true only for γ ∈ (0, 1]. If γ > 1, the indicated bounds are valid only in probability. The reason for this is as follows. In [1] the following property of uniform order statistics was used in the i.i.d. case: 1+ε lim infn→∞ n(log n) U1,n = ∞ almost surely for all ε> 0. There is no such result available in the long-range dependent case. We note in passing that the approximation of the general quantile process by the uniform quantile process as stated in (4.7) remains valid under the conditions (i)–(iii) on F without assuming also (R). CORRECTION 3

The results for the general Bahadur–Kiefer process (Theorems 4.1 and 4.2) cannot be concluded from those for the uniform Bahadur–Kiefer process, as claimed in Remark 4.1 of the paper in view of (4.12). The reason for this is that the rate at which the uniform Bahadur–Kiefer process behaves (cf. Theorem 2.3) is, mutatis mutandis, the same as that of approximating the general quantile process ρn(y,t) by the uniform quantile process [cf. (4.7) in combination with (4.9) and (4.10)]. Moreover, in view of results in [2], we conjecture that the limiting process in Theorem 4.1 is to be changed to 1/2 times that of the claimed process, that is, to 1 J (y)J ′ (y)Y 2(t). (2 − τD)(1 − τD) τ τ τ The corresponding conjecture also applies to the remaining results of Theo- rems 4.1 and 4.2. In view of this comment and the scaling constants in the invariance principle for the uniform Bahadur–Kiefer process (cf. Theorem 2.3), we conclude that the big “O” almost sure rates in Proposition 4.2 can- not be replaced with “o.” In this regard, we note that Wu in [2] considered related approximations in case of nonsubordinated linear processes on [a, b], 0

Acknowledgments. The authors wish to thank Rafa l Kulik for bringing these points to their attention, for helping to correct their oversights and for reviewing their results in this regard.

REFERENCES [1] Csorg¨ o,˝ M. and Rev´ esz,´ P. (1978). Strong approximations of the quantile process. Ann. Statist. 6 882–894. MR0501290 [2] Wu, W. B. (2005). On the Bahadur representation of sample quantiles for dependent sequences. Ann. Statist. 33 1934–1963. MR2166566

M. Csorg¨ o˝ L. Wang B. Szyszkowicz Department of Mathematics School of Mathematics and Statistics Nanjing University Carleton University Nanjing 210093 1125 Colonel By Drive China Ottawa, Ontario E-mail: [email protected] Canada K1S 5B6 E-mail: [email protected] [email protected]