LAUSR

In this erratum, we point out a mistake in a statement in Voln´yVoln´y and Wang [3, Section 6], published in Stochastic Processes and their Applications, 124(12):4012-4029, on the relation between Wu's physical dependence measure [4] and Hannan's condition [1] on stationary random fields. These conditions have been introduced in the past as easy-to-verify conditions for stationary random fields that lead to the central limit theorem and invariance principle. It had been known before that Wu's condition is strictly stronger than Hannan's condition in dimension one, as shown in Wu [4, Theorem 1]. However, it was stated in [3, p. 4026] that the argument for dimension one 'can be easily adapted to high dimension and the details are omitted'. This statement is not true. We are grateful to Davide Giraudo for having read carefully [3] and pointed out the mistake to us. It turned out that Wu and Hannan's condition are no longer comparable in dimension d ≥ 2. We have provided an example of stationary random field in [3, Proposition 6.1] that satisfies Hannan's condition, but not Wu's. Here, in Proposition 1 below, we provide another example in dimension d ≥ 2 that satisfies Wu's condition, but not Hannan's. Similar statements as the aforementioned one on the relationship between Wu and Hannan's conditions in [3] can also be found in [2]. The mistake does not affect other results in [2, 3], which are on limit theorems under Hannan's condition for stationary random fields. Recall that we consider stationary random fields in the form of (1) X i = f • T i (), i ∈ Z d where = { i } i∈Z d is a collection of i.i.d. random variables, f is a measurable function from R Z d to R, and T i given by [T i ()] j = i+j , i, j ∈ Z d , are the canonical shift operators on R Z d. For the sake of simplicity, we only recall the two conditions for d = 2 in (2) below, and refer to [3] for the case when d ≥ 3. Proposition 1. For every d ≥ 2, there exists a random field in the form of (1), such that Wu's condition holds, but Hannan's condition does not. Proof. We first recall the two conditions of interest for d = 2. The Hannan's condition is based on the so-called projection operators P i , i ∈ Z 2 defined as, for F i = σ(j : j ≤ i, j ∈ Z 2), P i g() = E(g() | F i) − E(g() | F i−(1,0)) − E(g() | F i−(0,1)) + E(g() |