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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
 
J Econom. Author manuscript; available in PMC 2010 May 1.
Published in final edited form as:
J Econom. 2009 May; 150(1): 86–98.
doi:  10.1016/j.jeconom.2009.02.009
PMCID: PMC2683026
NIHMSID: NIHMS100594

Central Limit Theorems and Uniform Laws of Large Numbers for Arrays of Random Fields

Abstract

Over the last decades, spatial-interaction models have been increasingly used in economics. However, the development of a sufficiently general asymptotic theory for nonlinear spatial models has been hampered by a lack of relevant central limit theorems (CLTs), uniform laws of large numbers (ULLNs) and pointwise laws of large numbers (LLNs). These limit theorems form the essential building blocks towards developing the asymptotic theory of M-estimators, including maximum likelihood and generalized method of moments estimators. The paper establishes a CLT, ULLN, and LLN for spatial processes or random fields that should be applicable to a broad range of data processes.

Keywords: Random field, spatial process, central limit theorem, uniform law of large numbers, law of large numbers

1 Introduction

Spatial-interaction models have a long tradition in geography, regional science and urban economics. For the last two decades, spatial-interaction models have also been increasingly considered in economics and the social sciences, in general. Applications range from their traditional use in agricultural, environmental, urban and regional economics to other branches of economics including international economics, industrial organization, labor and public economics, political economy, and macroeconomics.

The proliferation of spatial-interaction models in economics was accompanied by an upsurge in contributions to a rigorous theory of estimation and testing in spatial models. However, most of those contributions have focused on linear models of the Cliff-Ord type, cp. Cliff and Ord (1973, 1981), and Baltagi, Kelejian and Prucha (2007) for recent contributions. The development of a general asymptotic estimation theory for nonlinear spatial models under sets of assumptions that are both general and accessible for interpretation by applied researchers has been hampered by a lack of relevant central limit theorems (CLTs), uniform laws of large numbers (ULLNs), and pointwise laws of large numbers (LLNs) for dependent nonstationary spatial processes, also referred to as random fields. These limit theorems are the fundamental building blocks for the asymptotic theory of nonlinear spatial M-estimators, e.g., maximum likelihood and generalized method of moments estimators, and test statistics.1

Against this background, the aim of the paper is to establish a set of limit theorems under assumptions that are sufficiently general to accommodate a number of critical features frequently exhibited by processes in economic applications. Consequently, these limit theorems should allow for the development of a general asymptotic theory for parametric and non-parametric estimators of a wide range of linear and nonlinear spatial economic models. Many spatial processes in economics are nonstationary in that they are heteroskedastic and/or that the extent of dependence between variables may vary with locations. Furthermore, the processes may have asymptotically unbounded moments, in analogy with trending moments in the times series literature. For example, real estate prices often shoot up as one moves from the periphery towards the center of a megapolis; see, e.g., Bera and Simlai (2005) who report on sharp spikes in the variances of housing prices in Boston; for more examples, see also Cressie (1993). Spatial processes in economics are also typically not located on Zd, but on unevenly spaced lattices. Additionally, to cover some important classes of processes, e.g., Cliff-Ord type processes, where the random variables are also indexed by the sample size, it is critical to allow for a triangular array nature of the random field.

Towards these objectives, the paper derives a CLT, ULLN, and, for completeness, also an exemplary LLN for dependent random fields that (i) allow the random field to be nonstationary and even to exhibit asymptotically unbounded moments, (ii) allow for unevenly spaced locations and for general forms of sample regions, and (iii) allow the random variables to depend on the sample, i.e., to form a triangular array.

There exists an extensive literature on CLTs for mixing random fields. A comprehensive survey of this literature is provided in the important monographs by Bulinskii (1989), Nahapetian (1991), Doukhan (1994), Guyon (1995), Rio (2000) and Bradley (2007). As it turns out, none of the existing CLTs accommodates all of the above features essential for economic applications.

Our CLT for α-mixing random fields extends the Bolthausen (1982) and Guyon (1995) CLTs using Rio’s (1993) covariance inequality. In the time series literature, Rio’s inequality was employed by Doukhan, Massart and Rio (1994) to derive a CLT for stationary α-mixing processes under an optimal set of moment and mixing conditions. Using the same inequality, Dedecker (1998) obtained a CLT for stationary α-mixing random fields on Zd. Building on these results, we establish a CLT for nonstationary α-mixing random fields on unevenly spaced lattices. As in Doukhan, Massart and Rio (1994) and Dedecker (1998), Rio’s (1993) inequality enables us to prove a CLT from a mild set of moment and mixing conditions. A detailed comparison of the proposed CLT with the existing results is given in Section 3.

ULLNs are the key tools for establishing consistency of nonlinear estimators; cp., e.g., Gallant and White (1988), p. 19, and Pötscher and Prucha (1997), p. 17. Generic ULLN for time series processes have been introduced by Andrews (1987, 1992), Newey (1991) and Pötscher and Prucha (1989, 1994ab). These ULLNs are generic in the sense that they transform pointwise LLNs into ULLNs given some form of stochastic equicontinuity of the summands. ULLNs for time series processes, by their nature, assume evenly spaced observations on a line. They are not suitable for fields on unevenly spaced lattices. The generic ULLN for random fields introduced in this paper extends the one-dimensional ULLNs given in Pötscher and Prucha (1994a) and Andrews (1992). In addition to the generic ULLN, we provide low level sufficient conditions for stochastic equicontinuity that are relatively easy to check.2

For completeness, we also give a pointwise weak LLN, which is based on a subset of the assumptions maintained in our CLT. Thus, the trio of the results established in this paper can be used jointly in the proof of consistency and asymptotic normality of spatial estimators. Of course, the generic ULLN can also be combined with other LLNs.

The remainder of the paper is organized as follows. Section 2 introduces the requisite notation and definitions. The CLT for arrays of nonstationary α-and [var phi]-mixing random fields on irregular lattices is presented in Section 3. The generic ULLN, pointwise LLN and various sufficient conditions are discussed in Section 4. All proofs are relegated to the appendix. A longer version of the paper with additional discussions and more detailed proofs is available on the authors’ web pages.

2 Weak Dependence Concepts and Mixing Inequalities

In this section, we introduce the notation and definitions used throughout the paper. We consider spatial processes located on a (possibly) unevenly spaced lattice D [subset, dbl equals] Rd, d ≥ 1. It proves convenient to consider Rd as endowed with the metric ρ(i,j) = max1≤ld |jlil|, and the corresponding norm |i| = max1≤ld |il|, where il denotes the l-th component of i. The distance between any subsets U,V [set membership] D is defined as ρ(U,V) = inf {ρ(i,j): i [set membership] U and j [set membership] V}. Furthermore, let |U| denote the cardinality of a finite subset U [set membership] D.

The two basic asymptotic methods commonly used in the spatial literature are the so-called increasing domain and infill asymptotics, see, e.g., Cressie (1993), p. 480. Under increasing domain asymptotics, the growth of the sample is ensured by an unbounded expansion of the sample region. In contrast, under infill asymptotics, the sample region remains fixed, and the growth of the sample size is achieved by sampling points arbitrarily dense in the given region. In this paper, we employ increasing domain asymptotics, which is ensured by the following assumption on the lattice D.

Assumption 1

The lattice D [subset or is implied by] Rd, d ≥ 1, is infinite countable. All elements in D are located at distances of at least ρ0 > 0 from each other, i.e., ∀ i, j [set membership] D: ρ(i,j) ≥ ρ0; w.l.o.g. we assume that ρ0 > 1.

The assumption of a minimum distance has also been used by Conley (1999). It turns out that this single restriction on irregular lattices also provides sufficient structure for the index sets to permit the derivation of our limit results. In contrast to many CLTs in the literature, e.g., Neaderhouser (1978a,b), Nahapetian (1987), Bolthausen (1982), McElroy and Politis (2000), we do not impose any restrictions on the configuration and growth behavior of the index sets. Based on Assumption 1, Lemma A.1 in the Appendix gives bounds on the cardinalities of some basic sets in D that will be used in the proof of the limit theorems.

We now turn to the dependence concepts used in our theorems. Let {Xi,n; i [set membership] Dn, n [set membership] N} be a triangular array of real random variables defined on a probability space (Ω,An external file that holds a picture, illustration, etc.
Object name is nihms100594ig1.jpg,P), where Dn is a finite subset of D, and D satisfies Assumption 1. Further, let An external file that holds a picture, illustration, etc.
Object name is nihms100594ig2.jpg and An external file that holds a picture, illustration, etc.
Object name is nihms100594ig3.jpg be two sub-σ-algebras of An external file that holds a picture, illustration, etc.
Object name is nihms100594ig1.jpg. Two common concepts of dependence between An external file that holds a picture, illustration, etc.
Object name is nihms100594ig2.jpg and An external file that holds a picture, illustration, etc.
Object name is nihms100594ig3.jpg are α- and [var phi]-mixing, which have been introduced, respectively, by Rosenblatt and Ibragimov. The degree of dependence is measured in terms of the following α- and [var phi]-mixing coefficients:

α(U,B)=sup(P(AB)P(A)P(B),AU,BB),φ(U,B)=sup(P(AB)P(A),AU,BB,P(B)>0).

The concepts of α- and [var phi]-mixing have been used extensively in the time series literature as measures of weak dependence. Recall that a time series process {Xt} is α-mixing if

limmsuptα(Ft,Ft+m+)=0

where Ft=σ(,Xt1,Xt) and Ft+m=σ(Xt+m,Xt+m+1,). This definition captures the basic idea of diminishing dependence between different events as the distance between them increases.

To generalize these concepts to random fields, one could resort to a direct analogy with the time-series literature and, for instance, define mixing coefficients over the σ-algebras generated by the half-spaces perpendicular to the coordinate axes. However, as demonstrated by Dobrushin (1968a,b), the resulting mixing conditions are generally restrictive for d > 1. They are violated even for simple two-state Markov chains on Z2. The problem with definitions of this ilk is that they neglect potential accumulation of dependence between σ-algebras σ(Xi; i [set membership] V1) and σ(Xi; i [set membership] V2) as the sets V1 and V2 expand while the distance between them is kept fixed. Given a fixed distance, it is natural to expect more dependence between two larger sets than between two smaller sets.

Thus, generalizing mixing concepts to random fields in a practically useful way requires accounting for the sizes of subsets on which σ-algebras reside. Mixing conditions that depend on subsets of the lattice date back to Dobrushin (1968a,b). They were further expanded by Bolthausen (1982) and Nahapetian (1987). Following these authors, we adopt the following definitions of mixing:

Definition 1

For U [subset, dbl equals] Dn and V [subset, dbl equals] Dn, let σn(U) = σ(Xi,n; i [set membership] U), αn(U,V) = α(σn(U),σn(V)) and [var phi]n(U,V) = [var phi]n(U),σn(V)). Then, the α- and [var phi]-mixing coefficients for the random field {Xi,n; i [set membership] Dn, n [set membership] N} are defined as follows:

αk,l,n(r)=sup(αn(U,V),Uk,Vl,ρ(U,V)r),φk,l,n(r)=sup(φn(U,V),Uk,Vl,ρ(U,V)r),

with k,l,r,n [set membership] N. To account for the dependence of the random field on n, we define furthermore:

α¯k,l(r)=supnαk,l,n(r),φ¯k,l(r)=supnφk,l,n(r).

As shown by Dobrushin (1968a,b), the weak dependence conditions based on the above mixing coefficients are satisfied by large classes of random fields including Gibbs fields. These mixing coefficients were also used by Doukhan (1994) and Guyon (1995), albeit without dependence on n. The α-mixing coefficients for arrays of random fields in McElroy and Politis (2000) are defined identically to [alpha]k,l(r). Doukhan (1994) provides an excellent overview of various mixing concepts.

A key role in establishing limit theorems for mixing processes is played by covariance inequalities. Early covariance inequalities for α- and [var phi]-mixing fields are collected, e.g., in Hall and Heyde (1980), pp. 277–280. In these inequalities, covariances are bounded by power moments. As such, they do not allow for “lower”, e.g., logarithmic, moments and are, therefore, not sharp. These inequalities were later improved by various authors including Herrndorf (1985), Bulinskii (1988), Bulinskii and Doukhan (1987), and Rio (1993). The proof of the α-mixing part of our CLT relies on Rio’s (1993) inequality. Instead of moments, it is based on upper-quantile functions, which makes explicit the relationship between the mixing coefficients and tail distribution of the process. To state this result and to formulate the assumptions of our CLT, we need the following definitions.

Definition 2

  1. For a random variable X, the “upper-tail” quantile function QX: (0,1) → [0, ∞) is defined as
    QX(u)=inf{t:P(X>t)u}.
  2. For the non-increasing sequence of the mixing coefficients {α¯1,1(m)}m=1, set [alpha]1,1(0) = 1 and define its “inverse” function αinv(u): (0,1) → N[union or logical sum]{0} as:
    αinv(u)=max{m0:α¯1,1(m)>u}.

Remark 1

For a detailed discussion of the “upper-tail” quantile function see, e.g., Bradley (2007), Vol. 1, pp. 318. It proves helpful to re-state some of those properties. For a random variable X, let FX(x) = P(X ≤ x) denote the cumulative distribution function and let FX1(u)=inf{x:FX(x)u} be the usual quantile function of X. Then

QX(u)=FX1(1u).

Clearly, QX(u) is non-increasing in (0,1). Furthermore, if U is a random variable that is uniformly distributed on [0,1], then the random variable QX(U) has the same distribution as X, and thus for any Borel function f: RR such that E|f(X)| < ∞, we have

Ef(X)=01f(QX(u))du.

If X and Y are random variables such that X ≤Y a.s., then for all u [set membership] (0,1), QX(u) ≤ QY(u). If X ≥ 0 a.s., then for all u [set membership] (0,1), QX(u) ≥ 0.

Using the upper-tail quantile function, Rio (1993) obtains a sharper covariance inequality for α-mixing variables. In deriving our CLT, we will use the following slightly weaker version of Rio’s inequality given in Bradley (2007, Vol. 1, p. 320): Suppose that X and Y are two real-valued random variables such that E |X| < ∞, E |Y| < ∞ and 01QX(u)QY(u)du<. Let α = α(σ(X),σ(Y)), then

Cov(X,Y)40αQX(u)QY(u)du.
(1)

3 Central Limit Theorem

In this section, we provide a CLT for random fields with (possibly) asymptotically unbounded moments. Let {Zi,n; i [set membership] Dn, n [set membership] N} be an array of zero-mean real random fields on a probability space (Ω,An external file that holds a picture, illustration, etc.
Object name is nihms100594ig1.jpg,P), where the index sets Dn are finite subsets of D [subset or is implied by] Rd, d ≥ 1, which is assumed to satisfy Assumption 1. In the following, let Sn = Σi[set membership]Dn Zi,n and σn2=Var(Sn).

The CLT focuses on α- and [var phi]-mixing fields which satisfy, respectively, the following sets of assumptions.

Assumption 2 (Uniform L2 integrability)

There exists an array of positive real constants {ci,n} such that

limksupnsupiDnE[Zi,n/ci,n21(Zi,n/ci,n>k)]=0,

where 1(·) is the indicator function.

To formulate the next assumption, let

Q¯i,n(k):=QZi,n/ci,n1(Zi,n/ci,n>k),

denote the “upper-tail” quantile function of Zi,n/ci,n, and let αinv(u) denote the inverse function of [alpha]1,1(m) as given in Definition 2(ii).

Assumption 3 (α-mixing)

The α-mixing coefficients satisfy:

  1. limksupnsupiDn01αinvd(u)(Q¯i,n(k)(u))2du=0,
  2. m=1md1α¯k,l(m)< for k + l ≤ 4,
  3. [alpha]1,∞(m) = O(m−d−ε) for some ε > 0.

Assumption 4 ([var phi]-mixing)

The [var phi]-mixing coefficients satisfy:

  1. m=1md1φ¯1,11/2(m)<,
  2. m=1md1φ¯k,l(m)< for k + l ≤ 4,
  3. [phi]1,∞(m) = O(m−d−ε) for some ε > 0.

Assumption 5

liminfnDn1Mn2σn2>0, where Mn= maxi[set membership]Dn ci,n.

Based on the above set of assumptions, we can now state the following CLT.

Theorem 1

Suppose {Dn} is a sequence of arbitrary finite subsets of D, satisfying Assumption 1, with |Dn| → ∞ as n → ∞. Suppose further that {Zi,n; i [set membership] Dn, n [set membership] N} is an array of zero-mean real-valued random variables satisfying Assumption 2 and where the random field is either

  1. α-mixing satisfying Assumption 3, or
  2. [var phi]-mixing satisfying Assumption 4.

If in addition Assumption 5 holds, then

σn1SnN(0,1).

Clearly, the CLT can be readily extended to vector-valued random fields using the standard Cramér-Wold device. Assumption 2 is a standard moment assumption seen in CLTs for time series processes with trending moments; see, e.g., de Jong (1997) and the references cited therein. It is implied by uniform Lr boundedness for some r > 2: supn,i[set membership]Dn E |Zi,n/ci,n|r < ∞, see Billingsley (1986), p. 219.

The nonrandom constants ci,n in Assumption 2 are scaling numbers that allow for processes with asymptotically unbounded (trending) second moments. As remarked in the Introduction, economic data are frequently nonstationary. They may even exhibit dramatic differences in the magnitudes of their second moments. As an example, Bera and Simlai (2005) report on sharp spikes in the variances of housing prices in Boston. Of course, in this context, one could also expect household incomes, household wealth, property taxes, etc., to show similar features. Empirical researchers may be reasonably concerned in such situations whether an asymptotic theory that assumes uniformly bounded moments would provide a good approximation of the small sample distribution of their estimators and test statistics. Towards establishing an asymptotic theory that is reasonably robust to such “irregular” behavior of second moments, Assumption 2 avoids the assumption that the second moments are uniformly bounded.

In the case of uniformly L2-bounded fields, the scaling numbers ci,n can be set to 1. In the case when there is no uniform bound on the second moments, they would typically be chosen as ci,n = max(vi,n, 1), where vi,n2=EZi,n2. A simple but illustrative example of a process with asymptotically unbounded second moments is the process Zi,n = ci,nXi, where the ci,n are nonrandom constants increasing in |i|, and {Xi} is a uniformly square integrable family. In the case d = 1, a CLT for this class of processes was established by Peligrad and Utev (2003), see Corollary 2.2.

Assumption 5 is a counterpart to Assumptions 2, and may be viewed as an asymptotic negligibility condition, which ensures that no single summand dominates the sum. In the case of uniformly L2-bounded fields, it reduces to liminfnDn1σn2>0, which is the condition used by Guyon (1995); cp. also Bolthausen (1982).

To further illuminate some of the implications of Assumptions 2 and 5, we consider two special cases of the above illustrative example: Let {Zi, i [set membership] Dn [subset or is implied by] Zd} be an independent zero-mean random field with Dn = [1; n]d and |Dn| = nd. Suppose EZi2=iγ for some γ > 0. Then, as discussed above, ci2=iγ,Mn2=nγ and σn2n(γ+d). Clearly, in this case Assumptions 2 and 5 are satisfied for all γ > 0. Next, suppose that EZi2=2i. Then, ci2=2i,Mn2=2n and σn2nd12n, and hence, liminfnDn1Mn2σn2=0. Thus, Assumption 5 is violated in this case.

Assumption 3(a) reflects the trade-off between the conditions on the tail functions and mixing coefficients. To connect this assumption to the literature, we note that in the case of stationary square integrable random fields, Assumption 3(a) is implied by a somewhat simpler condition. More specifically, if

01αinvd(u)QZi2(u)du<,
(2)

then, limk01αinvd(u)(QZi(k)(u))2du=0, where i [set membership] Dn and QZi(k):=QZi1(Zi>k). This statement is proved in the appendix after the proof of the Corollary 1. For the case d = 1, condition (2) was used by Doukhan, Massart and Rio (1994); cp. also Dedecker (1998). In the proof of Theorem 1, we show conversely that under the maintained assumptions

supnsupiDn01αinvd(u)Qi,n2(u)du<.

In the nonstationary case, a sufficient condition for Assumption 3(a) is given in the following lemma. This condition involves uniform L2+δ integrability, which may be easier to verify in applications.

Corollary 1

Suppose the random field {Zi,n; i [set membership] Dn, n [set membership] N} satisfies the assumptions of the α-mixing part of Theorem 1, except that Assumptions 2 and 3(a) are replaced by: For for some δ > 0

limksupnsupiDnE[Zi,n/ci,n2+δ1(Zi,n/ci,n>k)]=0,
(3)

and

m=1α¯1,1(m)m[d(2+δ)/δ]1<
(4)

Then, Assumptions 2 and 3(a), and hence the conclusion of Theorem 1 hold.

Condition (3) is a typical moment assumption used in the CLTs for α-mixing processes. As shown in the proof of the Corollary 1, Condition (4) is weaker than the mixing condition m=1md1α¯1,1(m)δ/(2+δ)<, used in Bolthausen (1982).

We now relate Theorem 1 to existing results in the literature. In a seminal contribution, Bolthausen (1982) introduced a CLT for stationary α-mixing random fields on Zd, using Stein’s (1972) lemma. The proof of Theorem 1 is also based on Stein’s (1972) lemma, but Theorem 1 extends Bolthausen’s (1982) CLT in the following directions: (i) it allows for nonstationary random fields with asymptotically unbounded moments, (ii) it allows for unevenly spaced locations and relaxes restrictions on index sets, and (iii) allows for triangular arrays.

As discussed in the Introduction, there exists a vast literature on CLTs for mixing random fields. However, we are not aware of a CLT for random fields that accommodates all of the above crucial features, and/or contain Theorem 1 as a special case. Seminal and important contributions include Neader-houser (1978a,b), Nahapetian (1987), Bolthausen (1982), Guyon and Richardson (1984), Bulinskii (1988, 1989), Bradley (1992), Guyon (1995), Dedecker (1998), McElroy and Politis (2000), among others. All of these CLTs are for random fields on the evenly spaced lattice Zd, and the CLTs of Nahapetian (1987), Bolthausen (1982), Bradley (1992) and Dedecker (1998) furthermore maintain stationarity. The other papers permit nonstationarity but do not explicitly allow for processes with asymptotically unbounded moments. Also, most CLTs do not accommodate triangular arrays, and impose restrictions on the configuration and growth behavior of sample regions.

We next compare the moment and mixing conditions of some of the CLTs for nonstationary random fields with those maintained by Theorem 1. Guyon and Richardson (1984) and Guyon (1995), p. 11, consider random fields on Zd with uniformly bounded 2 + δ moments, while Theorem 1 assumes uniform L2 integrability. Moreover, Guyon and Richardson (1984) exploit mixing conditions for α∞,∞(r), which is somewhat restrictive, as discussed earlier.

Bulinskii (1988) establishes a covariance inequality for real random variables in Orlicz spaces, which allows him to derive a CLT for nonstationary α-mixing fields on Zd under a set of weak moment and mixing conditions. However, as shown by Rio (1993), given a moment condition, Bulinskii’s inequality results in slightly stronger mixing conditions than those implied by Rio’s (1993) inequality, which was used in deriving Theorem 1 and Corollary 1. For instance, in the case of finite 2 + δ moments, Bulinskii’s (1988) CLT would involve the mixing condition m=1md1α¯1,1(m)δ/(2+δ)<. As noted above, this condition is stronger than Condition (4) postulated in Corollary 1. Finally, Neaderhouser (1978a,b) and McElroy and Politis (2000) rely on more stringent moment and mixing conditions than Theorem 1.

4 Uniform Law of Large Numbers

Uniform laws of large numbers (ULLNs) are key tools for establishing consistency of nonlinear estimators. Suppose the true parameter of interest is θ0 [set membership] Θ, where Θ is the parameter space, and [theta w/ hat]n is a corresponding estimator defined as the maximizer of some real valued objective function Qn(θ) defined on Θ, where the dependence on the data is suppressed. Suppose further that EQn(θ) is maximized at θ0, and that θ0 is identifiably unique. Then for [theta w/ hat]n to be consistent for θ0, it suffices to show that Qn(θ) − EQn(θ) converge to zero uniformly over the parameter space; see, e.g., Gallant and White (1988), pp. 18, and Pötscher and Prucha (1997), pp. 16, for precise statements, which also allow the maximizers of EQn(θ) to depend on n. For many estimators the uniform convergence of Qn(θ) − EQn(θ) is established from a ULLN.

In the following, we give a generic ULLN for spatial processes. The ULLN is generic in the sense that it turns a pointwise LLN into the corresponding uniform LLN. This generic ULLN assumes (i) that the random functions are stochastically equicontinuous in the sense made precise below, and (ii) that the functions satisfy an LLN for a given parameter value. For stochastic processes this approach was taken by Newey (1991), Andrews (1992), and Pötscher and Prucha (1994a).3 Of course, to make the approach operational for random fields, we need an LLN, and therefore we also give an LLN for random fields. This LLN is exemplary, but has the convenient feature that it holds under a subset of the conditions maintained for the CLT. We also report on two sets of sufficient conditions for stochastic equicontinuity that are fairly easy to verify.

Just as for our CLT, we consider again arrays of random fields residing on a (possibly) unevenly spaced lattice D, where D [subset or is implied by] Rd, d ≥ 1, is assumed to satisfy Assumption 1. However, for the ULLN the array is not assumed to be real-valued. More specifically, in the following let {Zi,n; i [set membership] Dn, n [set membership] N}, with Dn a finite subset of D, denote a triangular array of random fields defined on a probability space (Ω,An external file that holds a picture, illustration, etc.
Object name is nihms100594ig1.jpg,P) and taking their values in Z, where (Z,An external file that holds a picture, illustration, etc.
Object name is nihms100594ig4.jpg) is a measurable space. In applications, Z will typically be a subset of Rs, i.e., Z [subset, dbl equals] Rs, and An external file that holds a picture, illustration, etc.
Object name is nihms100594ig4.jpg [subset, dbl equals] An external file that holds a picture, illustration, etc.
Object name is nihms100594ig3.jpg s, where An external file that holds a picture, illustration, etc.
Object name is nihms100594ig3.jpg s denotes the s-dimensional Borel σ-field. We remark, however, that it suffices for the ULLN below if (Z,An external file that holds a picture, illustration, etc.
Object name is nihms100594ig4.jpg) is only a measurable space. Further, in the following, let {fi,n(z,θ),i [set membership] Dn, n [set membership] N} and {qi,n(z,θ), i [set membership] Dn, n [set membership] N} be doubly-indexed families of real-valued functions defined on Z × Θ, i.e., fi,n: Z × Θ → R and qi,n: Z × Θ → R, where (Θ,ν) is a metric space with metric ν. Throughout the paper, the fi,n(·,θ) and qi,n(·,θ) are assumed An external file that holds a picture, illustration, etc.
Object name is nihms100594ig4.jpg/An external file that holds a picture, illustration, etc.
Object name is nihms100594ig3.jpg-measurable for each θ [set membership] Θ and for all i [set membership] Dn, n ≥ 1. Finally, let B(θ′,δ) be the open ball {θ [set membership] Θ: ν(θ′,θ) < δ.

4.1 Generic Uniform Law of Large Numbers

The literature contains various definitions of stochastic equicontinuity. For a discussion of different stochastic equicontinuity concepts see, e.g., Andrews (1992) and Pötscher and Prucha (1994a). We note that apart from differences in the mode of convergence, the essential differences in those definitions relate to the degree of uniformity. We shall employ the following definition.4

Definition 3

Consider the array of random functions {fi,n(Zi,n, θ), i [set membership] Dn, n ≥ 1}. Then fi,n is said to be

  1. L0 stochastically equicontinuous on Θ iff for every ε > 0
    limsupn1DniDnP(supθΘsupθB(θ,δ)fi,n(Zi,n,θ)fi,n(Zi,n,θ)>ε)0asδ0;
  2. Lp stochastically equicontinuous, p > 0, on Θ iff
    limsupn1DniDnE(supθΘsupθB(θ,δ)fi,n(Zi,n,θ)fi,n(Zi,n,θ)p)0asδ0;
  3. a.s. stochastically equicontinuous on Θ iff
    limsupn1DniDnsupθΘsupθB(θ,δ)fi,n(Zi,n,θ)fi,n(Zi,n,θ)0a.s.asδ0.

Stochastic equicontinuity-type concepts have been used widely in the statistics and probability literature; see, e.g., Pollard (1984). Andrews (1992), within the context of one-dimensional processes, refers to L0 stochastic equicontinuity as termwise stochastic equicontinuity. Pötscher and Prucha (1994a) refer to the stochastic equicontinuity concepts in Definition 3(a) [(b)], [[(c)]] as asymptotic Cesàro L0 [Lp], [[a.s.]] uniform equicontinuity, and adopt the abbreviations ACL0UEC [ACLpUEC], [[a.s.ACUEC]]. The following relationships among the equicontinuity concepts are immediate: ACLpUEC [implies] ACL0UEC [is implied by] a.s.ACUEC.

In formulating our ULLN, we will allow again for trending moments. We will employ the following domination condition.

Assumption 6 (Domination Condition)

There exists an array of positive real constants {ci,n} such that for some p ≥ 1:

limsupn1DniDnE(di,np1(di,n>k))0ask

where di,n(ω) = supθ[set membership]Θ |qi,n(Zi,n(ω), θ)|/ci,n.

We now have the following generic ULLN.

Theorem 2

Suppose {Dn} is a sequence of arbitrary finite subsets of D, satisfying Assumption 1, with |Dn| → ∞ as n → ∞. Let (Θ,ν) be a totally bounded metric space, and suppose {qi,n(z, θ), i [set membership] Dn, n [set membership] N} is a doubly-indexed family of real-valued functions defined on Z × Θ satisfying Assumption 6. Suppose further that the qi,n(Zi,n, θ)/ci,n are L0 stochastically equicontinuous on Θ, and that for all θ [set membership] Θ0, where Θ0 is a dense subset of Θ, the stochastic functions qi,n(Zi,n, θ) satisfy a pointwise LLN in the sense that

1MnDniDn[qi,n(Zi,n,θ)Eqi,n(Zi,n,θ)]0i.p.[a.s.]asn,
(5)

where Mn = maxi[set membership]Dn. Let Qn(θ) = [Mn |Dn|]−1 Σ i[set membership]Dn qi,n(Zi,n, θ), then

  1. supθΘQn(θ)EQn(θ)0i.p.[a.s.]asn
    (6)
  2. Qn(θ) = EQn(θ) is uniformly equicontinuous in the sense that
    limsupnsupθΘsupθB(θ,δ)Q¯n(θ)Q¯n(θ)0asδ0.
    (7)

The above ULLN adapts Corollary 4.3 in Pötscher and Prucha (1994a) to arrays of random fields, and also allows for asymptotically unbounded moments. The case of uniformly bounded moments is covered as a special case with ci,n = 1 and Mn = 1.

The ULLN allows for infinite-dimensional parameter spaces. It only maintains that the parameter space is totally bounded rather than compact. (Recall that a set of a metric space is totally bounded if for each ε > 0 it can be covered by a finite number of ε-balls). If the parameter space Θ is a finite-dimensional Euclidian space, then total boundedness is equivalent to boundedness, and compactness is equivalent to boundedness and closedness. By assuming only that the parameter space is totally bounded, the ULLN covers situations where the parameter space is not closed, as is frequently the case in applications.

Assumption 6 is implied by uniform integrability of individual terms, di,np, i.e., limksupnsupiDnE(di,np1(di,n>k))=0, which, in turn, follows from their uniform Lr-boundedness for some r > p, i.e., supn supi[set membership]Dn ||di,n||r < ∞.

Sufficient conditions for the pointwise LLN and the maintained L0 stochastic equicontinuity of the normalized function qi,n(Zi,n, θ)/ci,n are given in the next two subsections. The theorem only requires the pointwise LLN (5) to hold on a dense subset Θ0, but, of course, also covers the case where Θ0 = Θ.

As it will be seen from the proof, L0 stochastic equicontinuity of qi,n(Zi,n, θ)/ci,n and the Domination Assumption 6 jointly imply that qi,n(Zi,n, θ)/ci,n is Lp stochastic equicontinuous for p ≥ 1, which in turn implies uniform convergence of Qn(θ) provided that a pointwise LLN is satisfied. Therefore, the weak part of ULLN will continue to hold if L0 stochastic equicontinuity and Assumption 6 are replaced by the single assumption of Lp stochastic equicontinuity for some p ≥ 1.

4.2 Pointwise Law of Large Numbers

The generic ULLN is modular in the sense that it assumes a pointwise LLN for the stochastic functions qi,n(Zi,n; θ) for fixed θ [set membership] Θ. Given this feature, a ULLN can be obtained by combining the generic ULLN with available LLNs. In the following, we give an exemplary LLN for arrays of real random fields {Zi,n; i [set membership] Dn, n [set membership] N} taking values in Z = R with possibly asymptotically unbounded moments, which can in turn be used to establish a LLN for qi,n(Zi,n; θ). The LLN below has the convenient feature that it holds under a subset of assumptions of the CLT, Theorem 1, which simplifies their joint application.

The CLT was derived under the assumption that the random field was uniformly L2 integrable. As expected, for the LLN it suffices to assume uniform L1 integrability.

Assumption 2 * (Uniform L1 integrability)

There exists an array of positive real constants {ci,n} such that

limksupnsupiDnE[Zi,n/ci,n1(Zi,n/ci,n>k)]=0,

where 1(·) is the indicator function.

A sufficient condition for Assumption 2* is supn supi[set membership]Dn E|Zi,n/ci,n|1+η < ∞ for some η > 0. We now have the following LLN.

Theorem 3

Suppose {Dn} is a sequence of arbitrary finite subsets of D, satisfying Assumption 1, with |Dn| → ∞ as n → ∞. Suppose further that {Zi,n; i [set membership] Dn, n [set membership] N} is an array of real random fields satisfying Assumptions 2* and where the random field is either

  1. α-mixing satisfying Assumption 3(b) with k = l = 1, or
  2. [var phi]-mixing satisfying Assumption 4(b) with k = l = 1.

Then

1MnDniDn(Zi,nEZi,n)L10,

where Mn = maxi[set membership]Dn ci,n.

The existence of first moments is assured by the uniform L1 integrability assumption. Of course, L1-convergence implies convergence in probability, and thus the Zi,n also satisfies a weak law of large numbers. Comparing the LLN with the CLT reveals that not only the moment conditions employed in the former are weaker than those in the latter, but also the dependence conditions in the LNN are only a subset of the mixing assumptions maintained for the CLT.

There is a vast literature on weak LLNs for time series processes. Most recent contributions include Andrews (1988) and Davidson (1993), among others. Andrews (1988) established an L1-law for triangular arrays of L1-mixingales. Davidson (1993) extended the latter result to L1-mixingale arrays with trending moments. Both results are based on the uniform integrability condition. In fact, our moment assumption is identical to that of Davidson (1993). The mixingale concept, which exploits the natural order and structure of the time line, is formally weaker than that of mixing. It allows these authors to circumvent restrictions on the sizes of mixingale coefficients, i.e., rates at which dependence decays. In contrast, the above LLN maintains assumptions on the rates of decay of the mixing coefficients.

The above LLN can be readily used to establish a pointwise LLN for stochastic functions qi,n(Zi,n; θ) under the α- and [var phi]-mixing conditions on Zi,n postulated in the theorem. For instance, suppose that qi,n(·,θ) is An external file that holds a picture, illustration, etc.
Object name is nihms100594ig4.jpg/An external file that holds a picture, illustration, etc.
Object name is nihms100594ig3.jpg-measurable and supn supi[set membership]Dn E|qi,n(Zi,n; θ)/ci,n|1+η < ∞ for each θ [set membership] Θ and some η > 0, then qi,n(Zi,n; θ)/ci,n is uniformly L1 integrable for each θ [set membership] Θ. Recalling that the α- and [var phi]-mixing conditions are preserved under measurable transformation, we see that qi,n(Zi,n; θ) also satisfies an LNN for a given parameter value θ.

4.3 Stochastic Equicontinuity: Sufficient Conditions

In the previous sections, we saw that stochastic equicontinuity is a key ingredient of a ULLN. In this section, we explore various sufficient conditions for L0 and a.s. stochastic equicontinuity of functions fi,n(Zi,n, θ) as in Definition 3. These conditions place smoothness requirement on fi,n(Zi,n, θ) with respect to the parameter and/or data. In the following, we will present two sets of sufficient conditions. The first set of conditions represent Lipschitz-type conditions, and only requires smoothness of fi,n(Zi,n, θ) in the parameter θ. The second set requires less smoothness in the parameter, but maintains joint continuity of fi,n both in the parameter and data. These conditions should cover a wide range of applications and are relatively simple to verify. Lipschitz-type conditions for one-dimensional processes were proposed by Andrews (1987, 1992) and Newey (1991). Joint continuity-type conditions for one-dimensional processes were introduced by Pötscher and Prucha (1989). In the following, we adapt those conditions to random fields.

We continue to maintain the setup defined at the beginning of the section.

4.3.1 Lipschitz in Parameter

Condition 1

The array fi,n (Zi,n, θ) satisfies for all θ,θ′ [set membership] Θ and i [set membership]Dn, n ≥ 1 the following condition:

fi,n(Zi,n,θ)fi,n(Zi,n,θ)Bi,nh(ν(θ,θ))a.s.,

where h is a nonrandom function such that h(x) ↓ 0 as x ↓ 0, and Bi,n are random variables that do not depend on θ such that for some p > 0

limsupnDn1iDnEBi,np<[limsupnDn1iDnBi,n<a.s.]

Clearly, each of the above conditions on the Cesàro sums of Bi,n is implied by the respective condition on the individual terms, i.e., supnsupiDnEBi,np< [supn supi[set membership]Dn Bi,n < ∞ a.s.]

Proposition 1

Under Condition 1, fi,n(Zi,n, θ) is L0 [a.s.] stochastically equicontinuous on Θ.

4.3.2 Continuous in Parameter and Data

In this subsection, we assume additionally that Z is a metric space with metric τ and An external file that holds a picture, illustration, etc.
Object name is nihms100594ig4.jpg is the corresponding Borel σ-field. Also, let BΘ (θ, δ) and BZ(z, δ) denote δ-balls respectively in Θ and Z.

We consider functions of the form:

fi,n(Zi,n,θ)=k=1Krki,n(Zi,n)ski,n(Zin,θ),
(8)

where rki,n: Z → R and ski,n(·, θ): ZR are real-valued functions, which are An external file that holds a picture, illustration, etc.
Object name is nihms100594ig4.jpg/An external file that holds a picture, illustration, etc.
Object name is nihms100594ig3.jpg-measurable for all θ [set membership] Θ, 1 ≤ kK, i [set membership] Dn, n ≥ 1. We maintain the following assumptions.

Condition 2

The random functions fi,n(Zi,n, θ) defined in (8) satisfy the following conditions:

  1. For all 1 ≤ k ≤ K
    limsupn1DniDnErki,n(Zi,n)<.
  2. For a sequence of sets {Km} with Km [set membership] An external file that holds a picture, illustration, etc.
Object name is nihms100594ig4.jpg, the family of nonrandom functions ski,n(z, ·), 1 ≤ k ≤ K, satisfy the following uniform equicontinuity-type condition: For each m [set membership] N,
    supnsupiDnsupzKmsupθΘsupθB(θ,δ)ski,n(z,θ)ski,n(z,θ)0asδ0.
  3. Also, for the sequence of sets {Km}
    limmlimsupn1DniDnP(Zi,nKm)=0.

We now have the following proposition, which extends parts of Theorem 4.5 in Pötscher and Prucha (1994a) to arrays of random fields.

Proposition 2

Under Condition 2, fi,n(Zi,n, θ) is L0 stochastically equicontinuous on Θ.

We next discuss the assumptions of the above proposition and provide further sufficient conditions. We note that the fi,n are composed of two parts, rki,n and ski,n, with the continuity conditions imposed only on the second part. Condition 2 allows for discontinuities in rki,n with respect to the data. For example, the rki,n could be indicator functions. A sufficient condition for Condition 2(a) is the uniform L1 boundedness of rki,n, i.e., supn supi[set membership]Dn E |rki,n(Zi,n)| < ∞.

Condition 2(b) requires the nonrandom functions ski,n to be equicontinuous with respect to θ uniformly for all z [set membership] Km. This assumption will be satisfied if the functions ski,n(z, θ), restricted to Km × Θ, are equicontinuous jointly in z and θ. More specifically, define the distance between the points (z, θ) and (z′, θ′) in the product space Z × Θ by r((z, θ); (z′, θ′)) = max ν(θ, θ′), τ(z, z′). This metric induces the product topology on Z × Θ. Under this product topology, let B((z′, θ′), δ) be the open ball with center (z′, θ′) and radius δ in Km × Θ. It is now easy to see that Condition 2(b) is implied by the following condition for each 1 ≤ kK

supnsupiDnsup(z,θ)Km×Θsup(z,θ)B((z,θ),δ)ski,n(z,θ)ski,n(z,θ)0asδ0,

i.e., the family of nonrandom functions {ski,n(z,θ)}, restricted to Km × Θ, is uniformly equicontinuous on Km×Θ. Obviously, if both Θand Km are compact, the uniform equicontinuity is equivalent to equicontinuity, i.e.,

supnsupiDnsup(z,θ)B((z,θ),δ)ski,n(z,θ)ski,n(z,θ)0asδ0.

Of course, if the functions furthermore do not depend on i and n, then the condition reduces to continuity on Km × Θ. Clearly, if any of the above conditions holds on Z × Θ, then it also holds on Km × Θ.

Finally, if the sets Km can be chosen to be compact, then Condition 2(c) is an asymptotic tightness condition for the average of the marginal distributions of Zin. Condition 2(c) can frequently be implied by a mild moment condition. In particular, the following is sufficient for Condition 2(c) in the case Z = Rs: KmRs is a sequence of Borel measurable convex sets (e.g., a sequence of open or closed balls), and lim supn→∞ |Dn|−1Σi[set membership]Dn Eh(Zin) < ∞ where h: [0, ∞) → [0, ∞) is a monotone function such that limx→∞ h(x) = ∞; for example, h(x) = xp with p > 0. The claim follows from Lemma A4 in Pötscher and Prucha (1994b) with obvious modification to the proof.

We note that, in contrast to Condition 1, Condition 2 will generally not cover random fields with trending moments since in this case part (c) would typically not hold.

5 Concluding Remarks

The paper derives a CLT, ULLN, and, for completeness, also an exemplary LLN for spatial processes. In particular, the limit theorems (i) allow the random field to be nonstationary and to exhibit asymptotically unbounded moments, (ii) allow for locations on unevenly spaced lattices in Rd and for general forms of sample regions, and (iii) allow the random variables to form a triangular array.

Spatial data processes encountered in empirical work are frequently not located on evenly spaced lattices, are nonstationary, and may even exhibit spikes in their moments. Random variables generated by the important class of Cliff-Ord type spatial processes form triangular arrays. The catalogues of assumptions maintained by the limit theorems developed in this paper are intended to accommodate all these features in order to make these theorems applicable to a broad range of data processes in economics.

CLTs, ULLNs and LLNs are the fundamental building blocks for the asymptotic theory of nonlinear spatial M-estimators, e.g., maximum likelihood and generalized method of moments estimators, and test statistics. An interesting direction for future research would be to generalize the above limit theorems to random fields that are not mixing, but can be approximated by mixing fields. This could be achieved, for example, by introducing the concept of near-epoch dependent random fields similar to the one used in the time-series literature. We are currently working in this direction.

Acknowledgments

We are grateful to B. M. Pötscher, three anonymous referees and the editor P. M. Robinson for their helpful comments and suggestions. We also thank the participants of the First World Conference of the Spatial Econometrics Association, Cambridge, July 2007, as well as the seminar participants at the Humbolt University, Institute for Advanced Studies in Vienna, New York University, Duke University, North Carolina State University, Ohio State University, SUNY Albany, and the University of Maryland for helpful discussions. This research benefitted from a University of Maryland Ann G. Wylie Dissertation Fellowship for the first author and from financial support from the National Institute of Health through SBIR grants R43 AG027622 and R44 AG027622 for the second author.

A Appendix: Cardinalities of Basic Sets on Irregular Lattices in Rd

The following lemma establishes bounds on the cardinalities of basic sets in D that will be used in the proof of the limit theorems. Its proof is elementary and is therefore omitted.

Lemma A.1

Suppose that Assumption 1 holds. Let Bi(r) be the closed ball of the radius r centered in i [set membership] Rd. Then,

  1. The ball Bi(1/2) with i [set membership] Rd contains at most one element of D, i.e., |Bi(1/2) ∩D| ≤ 1.
  2. There exists a constant C < ∞ such that for h ≥ 1
    supiRdBi(h)DChd,
    i.e., the number of elements of D contained in a ball of radius h centered at i [set membership] Rd is O(hd) uniformly in i.
  3. For m ≥ 1 and i [set membership] Rd let
    Ni(1,1,m)={jD:mρ(i,j)<m+1}
    be the number of all elements of D located at any distance h [set membership] [m,m+1) from i. Then, there exists a constant C < ∞ such that
    supiRdNi(1,1,m)Cmd1.
  4. Let U and V be some finite disjoint subsets of D. For m ≥ 1 and i [set membership] U let
    Ni(2,2,m)={(A,B):A=2,B=2,AUwithiA,BVandjBwithmρ(i,j)<m+1}
    be the number of all different combinations of subsets of U composed of two elements, one of which is i, and subsets of V composed of two elements, where for at least one of the elements, say j, we have m ≤ ρ(i,j) < m+ 1. Then there exists a constant C < ∞ such that
    supiUNi(2,2,m)Cmd1UV.
  5. Let V be some finite subset of D. For m ≥ 1 and i [set membership] Rd let
    Ni(1,3,m)={B:B=3,BVandjBwithmρ(i,j)<m+1}
    be the number of the subsets of V composed of three elements, at least one of which is located at a distance h [set membership] [m,m + 1) from i. Then there exists a constant C < ∞ such that
    supiRdNi(1,3,m)Cmd1V2.

B Appendix: Proofs of CLT

The proof of Theorem 1 adapts the strategy employed by Bolthausen (1982) in proving his CLT for stationary random fields on regular lattices.

B.1 Some Useful Lemmata

Lemma B.1

(Bradley, 2007, Vol. 1, pp. 326, for q = 1) Let α(m), m = 1,2,… be a non-increasing sequence such that 0 ≤ α(m) ≤1 and α(m) → 0 as m → ∞. Set α(0) = 1 and define the “inverse function” α−1: (0,1) → N [union or logical sum]{0} as

α1(u)=max{m0:α(m)>u}foru(0,1).

Let f: (0,1) → [0, ∞) be a Borel function, then for q ≥ 1:

  1. m=1mq10α(m)f(u)du01[α1(u)]qf(u)du,
  2. 01[α1(u)]qduqm=1α(m)mq1, for any q ≥ 1.

Proof of Lemma B.1

The proof is similar to Bradley (2007), Vol. 1, pp. 326, and is available on the authors’ webpages.

Lemma B.2

Let Y ≥ 0 be some non-negative random variable, let FY and QY be the c.d.f. and the upper-tail quantile function of Y, and for some k > 0 let FY(k) and QY(k) be the c.d.f. and the upper-tail quantile function of Y 1(Y > k). Furthermore, define

u(k)=P(Y>k)=1FY(k),

then QY(u) ≤ k for u [set membership] [u(k),1), and

QY(k)(u)={QY(u)foru(0,u(k))0foru(u(k),1).

Proof of Lemma B.2

In light of Remark 1 for u [set membership] (0,1):

QY(u)=inf{y:FY(y)1u},QY(k)(u)=inf{y:FYk(y)1u}.

Furthermore, observe that

FYk(y)=P(Y1(Y>k)y)={0y<0FY(k)0ykFY(y)y>k

since P (Y 1(Y > k) = 0) = P(Yk) = FY (k).

Let u [set membership] [u(k),1): Then 1 − u(k) ≥ 1 − u, and thus

QY(u)=inf{y:FY(y)1u}inf{y:FY(y)1u(k)}=inf{y:FY(y)FY(k)}k,
(B.1)

and

QY(k)(u)=inf{y:FY(k)(y)1u}inf{y:FY(k)(y)1u(k)}=inf{y:FY(k)(y)FY(k)}=0,
(B.2)

since FY(k)(y)FY(k) for all y ≥ 0.

Now let u [set membership] (0,u(k)): Then 1 − u(k) < 1 − u and

{y:FY(k)(y)1u}{y:FY(k)(y)>1u(k)}={y:FY(k)(y)>FY(k)}.

Consequently, {y:FY(k)(y)1u}{y:y>k}. Observing that for y > k we have FY(k)(y)=FY(y), it follows that

QY(k)(u)=inf{y:FY(y)1u}=QY(u).
(B.3)

The claims of the lemma now follow from (B.1), (B.2) and (B.3).

B.2 Proof of Theorem and Corollaries

Proof of Theorem 1

We give the proof for α-mixing fields. The argument for [var phi]-mixing fields is similar. The proof is lengthy, and for readability we break it up into several steps.

1. Notation and Reformulation

Define

Xi,n=Zi,n/Mn

where Mn = maxi[set membership]Dn ci,n is as in Assumption 5. Let σn,Z2=Var[iDnZi,n] and σn,X2=Var[iDnXi,n]=Mn2σn,Z2. Since

σn,X1iDnXi,n=σn,Z1iDnZi,n,

to prove the theorem, it suffices to show that σn,X1iDnXi,nN(0,1). In this light, it proves convenient to switch notation from the text and to define

Sn=iDnXi,n,σn2=Var(iDnXi,n).

That is, in the following, Sn denotes Σi[set membership]Dn Xi,n rather than Σi[set membership]Dn Zi,n, and σn2 denotes the variance of Σi[set membership]Dn Xi,n rather than of Σi[set membership]Dn Zi,n.

We next establish the moment and mixing conditions for Xi,n implied by the assumptions of the CLT. Observe that by definition of Mn, we have |Xi,n| ≤ |Zi,n/ci,n| and hence E[|Xi,n|2 1(|Xi,n| > k)] ≤ E[|Zi,n/ci,n|2 1(|Zi,n/ci,n| > k)]. Thus, in light of Assumption 2,

limksupn·supiDnE[Xi,n21(Xi,n>k)]=0,
(B.4)

i.e., the Xi,n are uniformly L2 integrable. This further implies that

X22=supnsupiDnEXi,n2<.
(B.5)

Clearly, the mixing coefficients for Xi,n and Zi,n are identical, and thus the mixing conditions postulated in Assumption 3 for Zi,n also apply to Xi,n.

Using the new notation, Assumption 5 further implies:

liminfnDn1σn2>0.
(B.6)

2. Truncated Random Variables

In the following, we will consider truncated versions of the Xi,n. For k > 0 we define the following random variables

Xi,n(k)=Xi,n1(Xi,nk),Xi,n(k)=Xi,n1(Xi,n>k),
(B.7)

and the corresponding variances as

σn,k2=Var[iDnXi,n(k)],σn,k2=Var[iDnXi,n(k)].

We note that

σnσn,kσn,k.
(B.8)

To see this, define

Sn,k=iDnXi,n(k)EXi,n(k),Sn,k=iDnXi,n(k)EXi,n(k),

and observe that Sn = Sn,k + Sn,k, σn = ||Sn||2, σn,k =||Sn.k||2 and [sigma with tilde]n,k=||Sn,k||2. The inequality in (B.8) is now readily established using Minkowski’s inequality.

In the following, let Fi,n(x) and Qi,n be the c.d.f. and the upper-tail quantile function of |Xi,n|, let Fi,n(k) and Qi,n(k) be the c.d.f. and the upper-tail quantile function of Xi,n(k), respectively, and let

ui,n(k)=P(Xi,n>k)=1Fi,n(k).

Next, we deduce from Assumption 3 some basic properties of the upper-tail quantile function of |Xi,n| and Xi,n(k) that will be utilized in the proof below.

We first establish that

limksupnsupiDn01αinvd(u)(Qi,n(k)(u))2du=0,
(B.9)

where αinv(u) is the inverse of [alpha]1,1(m) given in Definition 2. Since |Xi,n| 1(|Xi,n| > k) ≤ |Zi,n/ci,n| 1(|Zi,n/ci,n| > k), in light of Remark 1, we have

Qi,n(k)(u)QZi,n/ci,n1(Zi,n/ci,n>k)(u).

Property (B.9) now follows immediately from Assumption 3(a).

Next, we establish that

supnsupiDn01αinvd(u)Qi,n2(u)du=K1<.
(B.10)

In light of (B.9), for any finite ε > 0 there exists a k < ∞ (which may depend on ε) such that

supnsupiDn01αinvd(u)(Qi,n(k)(u))2duε.

By Lemma B.2,

Qi,n2(u){(Qi,n(k)(u))2foru(0;ui,n(k))k2foru(ui,n(k);1).

Hence, using the above inequality and Part (b) of Lemma B.1 with q = d, it is readily seen that

01αinvd(u)Qi,n2(u)duε+dk2m=1α¯1,1(m)md1<

since m=1α¯1,1(m)md1< by Assumption 3(b). This verifies property (B.10).

3. Bounds for Variances

In the following, we will use the weaker version of Rio’s covariance inequality given in (1). Using this inequality gives

cov(Xi,n,Xj,n)40α¯1,1(ρ(i,j))Qi,n(u)Qj,n(u)du.
(B.11)

Also, in light of the convention α1,1(0) = 1, Remark 1, and (B.5) we have:

Var(Xi,n)=01Qi,n2(u)du=0α¯1,1(0)Qi,n2(u)dusupnsupiDnEXi,n2=X22<.

We next establish a bound on σn2. Using Lemma A.1(iii) and (B.11) yields:

σn2i,jDncov(Xi,n,Xj,n)4i,jDn0α¯1,1(ρ(i,j))Qi,n(u)Qj,n(u)du2iDnjDn0α¯1,1(ρ(i,j))[Qi,n2(u)+Qj,n2(u)]du4iDnm=0jDn:ρ(i,j)[m,m+1]0α¯1,1(ρ(i,j))Qi,n2(u)du4iDnm=0Ni(1,1,m)0α¯1,1(m)Qi,n2(u)du4CDnX22+4CDnsupn,iDnm=1md10α¯1,1(m)Qi,n2(u)du4CDnX22+4CDnsupn,iDn01αinvd(u)Qi,n2(u)duDnB2,
(B.12)

with B2=4C(X22+K1)<. The first inequality in the last line follows from Part (a) of Lemma B.1 by setting α(m) =[alpha]1,1(m) and f(u)=Qi,n2(u). The last inequality follows from (B.5) and (B.10).

Thus, supnDn1σn2<. By condition (B.6), there exists an N* and B1 > 0 such that for all nN*, we have B1Dnσn2. Combining this inequality with (B.12) yields for nN*:

0<B1Dnσn2B2Dn,
(B.13)

where 0 < B1B2 < ∞.

Since Xi,n(k) is a measurable function of Xi,n, clearly α(σ(Xi,n(k)),σ(Xj,n(k)))α(σ(Xi,n),σ(Xj,n)). Using analogous arguments as above, it is readily seen that for each k > 0:

σn,k24CDnsupn,iDn{E(Xi,n(k))2+01αinvd(u)(Qi,n(k)(u))2du}.
(B.14)

In light of the l.h.s. inequality (B.13) and inequality (B.14), we have

limksupnNσn,k2σn24CB1limksupnsupiDnE(Xi,n(k))2+4CB1limksupnsupiDn01αinvd(u)(Qi,n(k)(u))2du.

Observe that E(Xi,n(k))2=EXi,n21(Xi,n>k). It now follows from (B.4) and (B.9) that

limksupnNσn,k2σn2=0,
(B.15)

and furthermore, utilizing (B.8),

limksupnN1σn,kσnlimksupnNσn,kσn=0.
(B.16)

4. Reduction to Bounded Variables

We would like to thank Benedikt Pötscher for helpful discussions on this step of the proof. The proof employs a truncation argument in conjunction with Proposition 6.3.9 of Brockwell and Davis (1991). For k > 0 consider the decomposition

Yn=σn1iDnXi,n=Vnk+(YnVnk)

with

Vnk=σn1iDn(Xi,n(k)EXi,n(k)),YnVnk=σn1iDn(Xi,n(k)EXi,n(k)),

and let V ~ N(0,1). We next show that Yn [implies] N(0,1) if

σn,k1iDn(Xi,n(k)EXi,n(k))N(0,1)
(B.17)

for each k = 1,2,… We note that (B.17) will be verified in subsequent steps.

To show that Yn [implies] N(0,1) given (B.17) holds, we first verify condition (iii) of Proposition 6.3.9 in Brockwell and Davis (1991). By Markov’s inequality

P(YnVnk>ε)=P(σn1iDn(Xi,n(k)EXi,n(k))>ε)σn,k2ε2σn2.

In light of (B.15)

limklimsupnP(YnVnk>ε)limklimsupnσn,k2ε2σn2=0,

which verifies the condition.

Next, observe that

Vnk=σn,kσn[σn,k1iDn(Xi,n(k)EXi,n(k))].

Suppose r(k) = limn→∞ σn,k/σn exists, then Vnk [implies] Vk ~ N(0,r2(k)) in light of (B.17). If furthermore, limk→∞ r(k) → 1, then Vk [implies] V ~ N(0,1), and the claim that Yn [implies] N(0,1) would follow by Proposition 6.3.9 of Brockwell and Davis (1991). However, in the case of nonstationary variables limn→∞ σn,kn need not exist, and therefore, we have to use a different argument to show that Yn [implies] V ~ N(0,1). We shall prove it by contradiction.

Let x2133 be the set of all probability measures on (R,An external file that holds a picture, illustration, etc.
Object name is nihms100594ig3.jpg). Observe that we can metrize x2133 by, e.g., the Prokhorov distance, say d(.,.). Let μn and μ be the probability measures corresponding to Yn and V, respectively, then μn [implies] μ iff d(μn,μ) → 0 as n → ∞. Now suppose that Yn does not converge to V. Then for some ε > 0 there exists a subsequence {n(m)} such that d(μn(m),μ) > ε for all n(m). Observe that by (B.13) and (B.14) we have 0≤ σn,k/σnB2/B1 < ∞ for all k > 0 and all nN*, where N* does not depend on k. W. l.o.g. assume that with n(m) ≥ N*, and hence 0 ≤ σn(m),k/σn(m)B2/B1 < ∞ for all k > 0 and all n(m). Consequently, for k = 1 there exists a subsubsequence {n(m(l1))} such that σn(m(l1)),1/σn(m(l1)) → r(1) as l1 → ∞. For k = 2 there exists a subsubsubsequence {n(m(l1(l2)))} such that σn(m(l1(l2))),2/σn(m(l1(l2))), → r(2) as l2 → ∞. The argument can be repeated for k = 3,4….. Now construct a subsequence {nl} such that n1 corresponds to the first element of {n(m(l1))}, n2 corresponds to the second element of {n(m(l1(l2)))}, and so on, then for k = 1,2, …, we have:

limlσnl,kσnl=r(k).

Moreover, it follows from (B.16) that

limkr(k)1limklimlr(k)σnl,kσnl+limksupnNσn,kσn1=0.

Given (B.17), it follows that Vnlk =[implies] Vk ~ N(0,r2(k)). Then, by Proposition 6.3.9 of Brockwell and Davis (1991), Ynl [implies] V ~ N(0,1) as l → ∞. Since {nl} [subset, dbl equals] {n(m)}, this contradicts the hypothesis that d(μn(m),μ) > ε for all n(m).

Thus, we have shown that Yn [implies] N(0,1) if (B.17) holds. In light of this, it suffices to prove the CLT for bounded variables. Thus, in the following, we will assume that |Xi,n| ≥ CX < ∞.

5. Renormalization

Since |Dn| → ∞ and [alpha]1,(mn) = O(mdε), it is readily seen that we can choose a sequence mn such that

α¯1,(mn)Dn1/20
(B.18)

and

mndDn1/20
(B.19)

as n → ∞. Now, for such mn define:

an=i,jDn,ρ(i,j)mnE(Xi,nXj,n).

Using arguments similar to those employed in derivation of (B.12), it can be easily shown that sufficiently large n, say nN**N*:

σn2=an+o(Dn)=an(1+o(1)).
(B.20)

For nN**, define

S¯n=an1/2Sn=an1/2iDnXi,n.

To demonstrate that σn1SnN(0,1), it therefore suffices to show that Sn [implies] N(0,1).

6. Limiting Distribution of Sn

From the above discussion supnNES¯n2<. In light of Stein’s Lemma (see, e.g., Lemma 2, Bolthausen, 1982) to establish that Sn [implies] N(0,1), it suffices to show that

limnE[(iλS¯n)exp(iλS¯n)]=0

In the following, we take nN**, but will not indicate that explicitly for notational simplicity. Define

Sj,n=iDn,ρ(i,j)mnXi,nandS¯j,n=an1/2Sj,n,

then

(iλS¯n)exp(iλS¯n)=A1,nA2,nA3,n,

with

A1,n=iλeiλS¯n(1an1jDnXj,nSj,n),A2,n=an1/2eiλS¯njDnXj,n[1iλS¯j,neiλS¯j,n],A3,n=an1/2jDnXj,neiλ(S¯nS¯j,n).

To complete the proof, it suffices to show that E|Ak,n| → 0 as n → ∞ for k = 1,2,3. The latter can be verified using Lemma A.1 and arguments analogous to those in Guyon (1995), pp. 112–113. A detailed proof of these statements is available on the authors’ webpages. This completes the proof of the theorem.

Proof of Corollary 1

As in Theorem 1, let Q¯i,n(k):=QZi,n/ci,n1(Zi,n/ci,n>k) and let αinv(u) be the inverse of [alpha]1,1(m) as given in Definition 2. By Hölder’s inequality,

limksupnsupiDn01αinvd(u)(Q¯i,n(k)(u))2du[01αinvd(2+δ)/δdu]δ/(2+δ)×[limksupnsupiDn01(Q¯i,n(k)(u))2+δdu]2/(2+δ)
(B.21)

In light of Remark 1 and condition (3) maintained by the lemma, we have

limksupnsupiDn01(Q¯i,n(k)(u))2+δdu=limksupnsupiDnE[Zi,n/ci,n2+δ1(Zi,n/ci,n>k)]=0.

Hence to complete the proof, it suffices to show that the first term on the r.h.s. of (B.21) is finite. To see this, observe that by Part (b) of Lemma B.1 with α(m) = [alpha]1,1(m) and q = d(2 + δ)/δ we have

01αinvd(2+δ)/δ(u)dud(2+δ)δm=1α¯1,1(m)m[d(2+δ)/δ]1<,

where the r.h.s. is finite by condition (4) maintained by the lemma.

Finally, we verify the claim made in the discussion of Corollary 1 that the mixing condition

m=1α¯1,1(m)m[d(2+δ)/δ]1<
(B.22)

is weaker than the condition

m=1md1α¯1,1(m)δ/(2+δ)<
(B.23)

used in the previous version of the CLT. To see this, observe that condition (B.23) implies that

mdα¯1,1(m)δ/(2+δ)0.
(B.24)

Next, note that the ratio of the summands in (B.22) and (B.23) equals [md[alpha]1,1(m)δ/(2+δ)]2 and therefore, tends to zero as m → ∞. Hence, condition (B.23) indeed implies (B.22).

Proof of Claim before Corollary 1

By Lemma B.2, QZi(k)(u)QZi(u) for each k > 0 and all u [set membership] (0,1). Consequently, for all k > 0:

01αinvd(u)(QZi(k)(u))2du01αinvd(u)QZi2(u)du<.

Furthermore, by Lemma B.2, we have limkQZi(k)(u)=0. Therefore, by the Dominated Convergence Theorem:

limk01αinvd(u)(QZi(k)(u))2du=0,

as required.

C Appendix: Proofs of ULLN and LLN

Proof of Theorem 2

In the following we use the abbreviations ACL0UEC [ACLpUEC] [[a.s.ACUEC]] for L0 [Lp], [[a.s.]] stochastic equicontinuity as defined in Definition 3. We first show that ACL0UEC and the Domination Assumptions 6 for gi,n(Zi,n,θ) = qi,n(Zi,n,θ)/ci,n jointly imply that the gi,n(Zi,n,θ) is ACLpUEC, p ≥ 1.

Given ε > 0, it follows from Assumption 6 that we can choose some k =k(ε) < ∞ such that

limsupn1DniDnE(di,np1(di,n>k))<ε3·2p.
(C.1)

Let

Yi,n(δ)=supθΘsupθB(θ,δ)gi,n(Zi,n,θ)gi,n(Zi,n,θ)p,

and observe that Yi,n(δ)2pdi,np, then

E[Yi,n(δ)]ε/3+EYi,n(δ)1(Yi,n(δ)>ε/3,di,n>k)+EYi,n(δ)1(Yi,n(δ)>ε/3,di,nk)ε/3+2pEdi,np1(di,n>k)+2pkpP(Yi,n(δ)>ε/3)
(C.2)

From the assumption that the gi,n(Zi,n) is ACL0UEC, it follows that we can find some δ = δ(ε) such that

limsupn1DniDnP(Yi,n(δ)>ε)=limsupn1DniDnP(supθΘsupθB(θ,δ)gi,n(Zi,n,θ)gi,n(Zi,n,θ)>ε1p)ε3(2k)p
(C.3)

It now follows from (C.1), (C.2) and (C.3) that for δ = δ(ε),

limsup1DniDnEYi,n(δ)ε/3+2plimsupn1DniDnEdi,np1(di,n>k)+2pkplimsupn1DniDnP(Yi,n(δ)>ε/3)ε,

which implies that gi,n(Zi,n) is ACLpUEC, p ≥ 1.

We next show that this in turn implies that Qn(θ) is ALpUEC, p ≥ 1, as defined in Pötscher and Prucha (1994a), i.e., we show that

limsupnE{supθΘsupθB(θ,δ)Qn(θ)Qn(θ)p}0asδ0.

To see this, observe that

EsupθΘsupθB(θ,δ)Qn(θ)Qn(θ)p1DniDnEsupθΘsupθB(θ,δ)qi,n(Zi,n,θ)qi,n(Zi,n,θ)p/ci,np=1DniDnEYi,n(δ)

where we have used inequality (1.4.3) in Bierens (1994). The claim now follows since the lim sup of the last term goes to zero as δ → 0, as demonstrated above. Moreover, by Theorem 2.1 in Pötscher and Prucha (1994a), Qn(θ) is also AL0UEC, i.e., for every ε > 0

limsupnP{supθΘsupθB(θ,δ)Qn(θ)Qn(θ)>ε}0asδ0.

Given the assumed weak pointwise LLN for Qn(θ), the i.p. portion of part (a) of the theorem now follows directly from Theorem 3.1(a) of Pötscher and Prucha (1994a).

For the a.s. portion of the theorem, note that by the triangle inequality

limsupnsupθΘsupθB(θ,δ)Qn(θ)Qn(θ)limsupn1DniDnsupθΘsupθB(θ,δ)gi,n(Zi,n,θ)gi,n(Zi,n,θ).

The r.h.s. of the last inequality goes to zero as δ → 0, since gi,n is a.s.ACUEC by assumption. Therefore,

limsupnsupθΘsupθB(θ,δ)Qn(θ)Qn(θ)0asδ0a.s.

i.e., Qn is a.s.AUEC, as defined in Pötscher and Prucha (1994a). Given the assumed strong pointwise LLN for Qn(θ) the a.s. portion of part (a) of the theorem now follows from Theorem 3.1(a) of Pötscher and Prucha (1994a).

Next observe that since a.s.ACUEC [implies] ACL0UEC we have that Qn(θ) is ALpUEC, p ≥ 1, both under the i.p. and a.s. assumptions of the theorem. This in turn implies that Qn(θ) = EQn(θ) is AUEC, by Theorem 3.3 in Pötscher and Prucha (1994a), which proves part (b) of the theorem.

Proof of Theorem 3

Define Xi,n = Zi,n/Mn, and observe that

[DnMn]1iDn(Zi,nEZi,n)=Dn1iDn(Xi,nEXi,n).

Hence, it suffices to prove the LLN for Xi,n.

We first establish mixing and moment conditions for Xi,n from those for Zi,n. Clearly, if Zi,n is α-mixing [[var phi]-mixing], then Xi,n is also α-mixing [[var phi]-mixing] with the same coefficients. Thus, Xi,n satisfies Assumption 3(b) with k = l = 1 [Assumption 4(b) with k = l = 1]. Furthermore, since Zi,n/ci,n is uniformly L1 integrable, Xi,n is also uniformly L1 integrable, i.e.,

limksupnsupiDnE[Xi,n1(Xi,n>k)]=0.
(C.4)

In proving the LLN we consider truncated versions of Xi,n. For 0 < k < ∞ let

Xi,n(k)=Xi,n1(Xi,nk),Xi,n(k)=Xi,nXi,n(k)=Xi,n1(Xi,n>k).

In light of (C.4)

limksupnsupiDnEXi,n(k)=0.
(C.5)

Clearly, Xi,n(k) is a measurable function of Xi,n, and thus Xi,n(k) is also α-mixing [[var phi]-mixing] with mixing coefficients not exceeding those of Xi,n.

By Minkowski’s inequality

EiDn(Xi,nEXi,n)2EiDnXi,n(k)+EiDn(Xi,n(k)EXi,n(k))
(C.6)

and thus

limnDn1iDn(Xi,nEXi,n)12limksupnsupiDnEXi,n(k)+limklimnDn1iDn(Xi,n(k)EXi,n(k))1
(C.7)

where ||.||1 denotes the L1-norm. The first term on the r.h.s. of (C.7) goes to zero in light of (C.5). To complete the proof, we now demonstrate that also the second term converges to zero. To that effect, it suffices to show that Xi,n(k) satisfies an L1-norm LLN for fixed k.

Let σn,k2=Var[iDnXi,n(k)], then by Lyapunov’s inequality

Dn1iDn(Xi,n(k)EXi,n(k))1Dn1σn,k.
(C.8)

Using Lemma A.1(iii), the mixing inequality of Thereom A.5 of Hall and Heyde (1980) and arguments as in Step 2 of the proof of the CLT, we have in the α-mixing case:

σn,k24Dn(k2+CKk2).

with C < ∞, and K=m=1md1α¯1,1(m)< by Assumption 3(b). Consequently, the r.h.s. of (C.8) is seen to go to zero as n → ∞, which establishes that the Xi,n(k) satisfies an L1-norm LLN for fixed k. The proof for the [var phi]-mixing case is analogous. This completes the proof.

Proof of Proposition 1

Define the modulus of continuity of fi,n(Zi,n) as

w(fi,n,Zi,n,δ)=supθΘsupθB(θ,δ)fi,n(Zi,n,θ)fi,n(Zi,n,θ).

Further observe that {ω: w(fi,n,Zi,n) > ε} [subset, dbl equals] {ω: Bi,nh(δ) > ε}. By Markov’s inequality and the i.p. part of Condition 1, we have

limsupn1DniDnP[w(fi,n,Zi,n,δ)>ε]limsupn1DniDnP[Bi,n>εh(δ)][h(δ)ε]plimsupn1DniDnEBi,npC1[h(δ)ε]p0asδ0

for some C1 < ∞, which establishes the i.p. part of the theorem. For the a.s. part, observe that by the a.s. part of Condition 1 we have a.s.

limsupn1DniDnw(fi,n,Zi,n,δ)h(δ)limsupn1DniDnBi,nC2h(δ)0asδ0

for some C2 < ∞, which establishes the a.s. part of the theorem.

Proof of Proposition 2

The proof is analogous to the first part of the proof of Theorem 4.5 in Pötscher and Prucha (1994a), and is therefore omitted.

Footnotes

JEL Classification: C10, C21, C31

1Conley (1999) makes an important contribution towards developing an asymptotic theory of GMM estimators for spatial processes. However, in deriving the limiting distribution of his estimator, he assumes stationarity which allows him to utilize Bolthausen’s (1982) CLT for stationary random fields on Zd.

2The existing literature on the estimation of nonlinear spatial models has maintained high-level assumptions such as first moment continuity to imply uniform convergence; cp., e.g., Conley (1999). The results in this paper are intended to be more accessible, and in allowing, e.g., for nonstationarity, to cover larger classes of processes.

3We note that the uniform convergence results of Bierens (1981), Andrews (1987), and Pötscher and Prucha (1989, 1994b) were obtained from closely related approach by verifying the so-called first moment continuity condition and from local laws of large numbers for certain bracketing functions. For a detailed discussion of similarities and differences, see Pötscher and Prucha (1994a).

4All suprema and infima over subsets of Θ of random functions used below are assumed to be P-a.s. measurable. For sufficient conditions see, e.g., Pollard (1984), Appendix C, or Pötscher and Prucha (1994b), Lemma 2.

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Contributor Information

Nazgul Jenish, Department of Economics, New York University, 19 W. 4th Street, 6FL, New York, NY 10012.

Ingmar R. Prucha, Department of Economics, University of Maryland, Tydings Hall, Room 3147A, College Park, MD 20742.

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