New York Journal of Mathematics

New York J. Math. 4 (1998) 75–82.

Approximation of L_{2}-Processes by Gaussian Processes

M. A. Akcoglu, J. R. Baxter, D. M. Ha, and R. L. Jones

Abstract. Let T be an ergodic transformation of a nonatomic probability space, f an L_{2}-function, and K ≥ 1 an integer. It is shown that there is another L_{2}-function g, such that the joint distribution of T^{i}g, 1 ≤ i ≤ K, is nearly normal, and such that the corresponding inner products (T^{i}f, T^{j}f) and (T^{i}g, T^{j}g) are nearly the same for 1 ≤ i, j ≤ K. This result can be used to give a simpler and more transparent proof of an important special case of an earlier theorem [3], which was a refinement of Bourgain’s entropy theorem [9].

Contents

1. The Main Result 75 2. An Application 80 References 81

1. The Main Result

If a random vector Y = (Y_{1}, Y_{2}, …, Y_{K}) has the multivariate normal distribution, then the special structure allows one to make very precise statements about the set where sup_{1≤i≤K} Y_{i} > λ. Consequently, if one can reduce a general situation to the case of the multivariate normal, then the knowledge of the multivariate normal case can be used to obtain information about the general case. Thus, it is useful to find situations and techniques that allow us to make a reduction to the multivariate normal situation. In this paper we consider a family of operators that arise in ergodic theory, and show that such a reduction of the general case to the multivariate normal is possible by using a Rohlin tower argument.

Let T be an ergodic transformation of a nonatomic probability space, f an L_{2}-function, and K ≥ 1 an integer. We will show below that there is another L_{2}- function g, such that the joint distribution of T^{i}g, 1 ≤ i ≤ K, is nearly normal, and such that the corresponding inner products (T^{i}f, T^{j}f) and (T^{i}g, T^{j}g) are nearly the same for 1 ≤ i, j ≤ K. Using this approximation to the multivariate normal,

\begin{matrix}c \\ ◯ \end{matrix}1998 State University of New York

ISSN 1076-9803/98

Received March 4, 1998.

Mathematics Subject Classification. Primary: 28D99, Secondary: 60F99.

Key words and phrases. L_{2}-processes, Gaussian processes, Bourgain’s entropy theorem.

Research supported by NSERC and NSF Grants.

and its well understood properties, we will be able to obtain information about the sequence T^{i}f, 1 ≤ i ≤ K.

The particular application we have in mind is Bourgain’s celebrated entropy theorem [9], which establishes a connection between the pointwise and L_{2} behaviors of sequences of L_{2} contractions T_{n}, applied to L_{2} functions. Such a connection does exist, of course, if f ∈ L_{2} is such that the joint distribution of T_{i}f, 1 ≤ i ≤ K, is normal for each K. In this case all pointwise relations between these functions are determined by the L_{2} behavior of the sequence T_{n}f. The extension of this connection from the normal case to the general case is carried out in an existential way in Bourgain’s original proof, and also in the proof of a refined version of this theorem given in [3]. However, in many cases the passage from the normal case to the general case can be made constructively, in a simple and more transparent way, by approximating in the appropriate sense, the sequence T_{i}f, by a sequence with the multivariate normal distribution. Our main result, Theorem 1.6 below, has been obtained with this application in mind. In Section 2 we give a brief sketch of this application.

Let (X, F, μ) be a nonatomic probability space, T an ergodic transformation on X, f a function in L_{2}(μ), and K ≥ 1 an integer. We will show below (Theorem 1.6) that there exists another function g ∈ L_{2}(μ), such that the joint distribution of T^{i}g, 1 ≤ i ≤ K, is nearly normal, and such that the corresponding inner products (T^{i}f, T^{j}f) and (T^{i}g, T^{j}g) are nearly the same for 1 ≤ i, j ≤ K. Theorem 1.6 implies the following, stated formally as Corollary 1.11 below. Let A_{1}, …, A_{K} be K operators which are linear combinations of the iterates of T, or more generally are strong limits of such linear combinations. Then, given an f ∈ L_{2}, there is another g ∈ L_{2} such that the joint distribution of A_{i}g, 1 ≤ i ≤ K, is nearly normal, and such that the corresponding inner products (A_{i}f, A_{j}f) and (A_{i}g, A_{j}g) are nearly the same for 1 ≤ i, j ≤ K.

A special case of the main result, where T is an irrational rotation of the circle, has been proved in [2]. The general result has been stated in [3] without proof, observing that the special case in [2] would imply the general case with routine approximations, via Rohlin’s Lemma. As the proof in [2] is rather involved in details, however, a generalization of it turns out to be very complicated, even though rather routine in principle. In the present note we give a very simple and short direct proof of the general result, using essentially only Rohlin’s Lemma. This proof could have been made even shorter by replacing Lemmas 1.7 and 1.8 below by a reference to a general fact about Gauss processes. Indeed, the ergodicity of a Gauss process is equivalent to the continuity of its spectral measure (See, for example, pages 191 and 368 in [10]). This implies directly that a Gauss process can be approximated by an ergodic Gauss process, in the sense required for our proof. Lemmas 1.7 and 1.8 show that any L_{2}-process can be approximated, in the same sense, by an aperiodic L_{2}-process, which is enough for our purpose. To keep our presentation as elementary and self contained as possible, we give the simple proof of this fact.

Approximation of L_{2}-Processes by Gaussian Processes 77

Notation 1.1. By a space we mean a probability space and by a measure we mean a probability measure. All functions considered are measurable, either by assumption or construction. Let f_{n} and g_{n} be two finite or infinite sequences of real valued functions. Each sequence may be defined on a different space. These sequences will be called isomorphic if the joint distributions of finite sets of functions from one sequence is the same as the corresponding distributions from the other sequence. A transformation T on a space X = (X, F, μ) is a measure preserving point transformation, not necessarily invertible. A transformation T on X induces a transformation of functions on X, denoted by the same letter and defined by (T f)(x) = f(T x) for x ∈ X, where f is a function on X. Since T is not assumed to be invertible, given g : X → R, there need not be (measurable) f : X → R such that T f = g. If g is measurable with respect to T^{−1}F, however, then there is an f such that g = T f. If T^{i}x ≠ T^{j}x whenever i ≠ j, for almost all x, then T is called aperiodic. By a process (T, f) we mean a transformation T together with a function f : X → R. We identify a process (T, f) by the corresponding sequence f_{n} = T^{n}f. For each integer K ≥ 1 a process induces a measure ξ = ξ_{K} on R^{K}, as the distribution measure of the function (T^{1}f, …, T^{K}f) : X → R^{K}, which will be called the K-distribution measure of the process. Two processes (T, f) and (S, g) are called isomorphic if their K-distribution measures are the same for each K ≥ 1; or, equivalently, if the corresponding sequences T^{n}f and S^{n}g are isomorphic. If f is an L_{2} function, then (T, f) is called an L_{2}-process.

Definition 1.2 (L_{2}-equivalence). Let f_{n} and g_{n} be two (finite or infinite) sequences of L_{2}-functions, n ≥ 0. These sequences will be called L_{2}-equivalent if the corresponding inner products (f_{i}, f_{j}) and (g_{i}, g_{j}) are equal for all i, j ≥ 0. Let (T, f) and (S, g) be two L_{2} processes, where the associated transformations may or may not be on the same space. Then (T, f) and (S, g) will be called L_{2}-equivalent if the corresponding sequences f_{n} = T^{n}f and g_{n} = S^{n}g are L_{2}-equivalent. A sequence of L_{2} functions is called a Gauss (or normal) sequence if the joint distribution of any finite subsequence is normal. A process (T, f) is called a Gauss process if the corresponding sequence f_{n} = T^{n}f is a Gauss sequence. Although the following lemma is well known we will recall the simple proof. In this lemma the uniqueness is understood to be up to an isomorphism, as defined in 1.1.

Lemma 1.3. Any L_{2} sequence is L_{2}-equivalent to a unique Gauss sequence. In particular, any L_{2} process is L_{2}-equivalent to a unique Gauss process.

Proof. Let Ω = R^{Z} be the shift space. Points in Ω are denoted by ω = (ω_{i})_{i∈Z}, with coordinate functions ω_{i} : Ω → R. The shift transformation S : Ω → Ω is defined as (Sω)_{i} = ω_{i−1}. If f_{i}, 1 ≤ i ≤ K, is a finite L_{2} sequence, then there is a unique Gauss measure on R^{K} such that the inner product of the coordinate functions ω_{i} and ω_{j} with respect to this measure is equal to the corresponding inner product of f_{i} and f_{j}, for all 1 ≤ i, j ≤ K. If f_{n} is an infinite sequence then the Gauss measures corresponding to finite segments are compatible and define a unique measure on R^{N}. If f_{n} = T^{n}f, then this measure on R^{N} is invariant under the restriction of the shift transformation to R^{N}, defined in an obvious way. Hence it has a unique extension to a shift invariant measure on Ω. Then we see that (T, f) is L_{2}-equivalent to the Gauss process (S^{−1}, ω_{0}). □

We will need two concepts of “closeness”. In the first case, say that two sequences of functions are L^{2} close if their corresponding inner products are close. To be precise we give the following definition.

Definition 1.4 (L_{2}-closeness). Let K ≥ 1 be an integer and ε > 0. Two L_{2} sequences f_{n} and g_{n} are called L_{2}-close within (K, ε) if

|(f_{i}, f_{j}) − (g_{i}, g_{j})| < ε

for 0 ≤ i, j ≤ K − 1. Two L_{2} processes (T, f) and (S, g) will be called L_{2}-close within (K, ε), if the corresponding sequences f_{n} = T^{n}f and g_{n} = S^{n}g are L_{2}-close within (K, ε).

We will also need a second measure of closeness. The idea is that two measures are weakly close if the results of integration against continuous functions are close. Two finite sequences of functions will be called weakly close if their distribution measures are weakly close. The formal definition is as follows.

Definition 1.5 (Weak-closeness). Let K ≥ 1 and U ≥ 1 be integers, φ_{1}, …, φ_{U} bounded continuous functions R^{K} → R, and ε > 0. Two sequences of measurable functions f_{n} and g_{n}, n ≥ 0, will be called weakly close within (K, ε, φ_{1}, …, φ_{U}) if

|∫_{R^{K}} φ_{u}dξ − ∫_{R^{K}} φ_{u}dη| < ε

for 1 ≤ u ≤ U, where ξ and η denote the joint distribution measures of the R^{K}- valued functions (f_{0}, …, f_{K−1}) and (g_{0}, …, g_{K−1}). To simplify the notation we will also say that these sequences are weakly close within (K, ε), the choice of a finite number of bounded continuous functions R^{K} → R being understood implicitly. Two processes (T, f) and (S, g) will be called weakly close within (K, ε), if the corresponding sequences are weakly close within (K, ε).

Theorem 1.6. Let T be an aperiodic transformation and f ∈ L_{2}(X). Let (V, h) be the Gauss process which is L_{2}-equivalent to (T, f). Let

(K, ε) = (K, ε, φ_{1}, …, φ_{U})

be as in the definition 1.5 of weak-closeness. Then there is a function g ∈ L_{2}(X), such that (V, h) and (T, g) are both weakly and L_{2}-close within (K, ε).

Proof. The passage from (V, h) to (T, g) will be accomplished in two steps, through an intermediary process (V′, h′). These two passages will be justified by the Lemmas 1.8 and 1.9 to be obtained below. In the first step, the Gauss process (V, h) is replaced by a process (V′, h′) such that V′ is aperiodic and (V, h) and (V′, h′) are both weakly and L_{2}-close within (K, ε/2). This step is justified by Lemma 1.8, which states that any L_{2}-process can be approximated by an aperiodic L_{2}-process, both in weak- and L_{2}-closeness sense. Next, we find an L_{2}-function g on X so that (V′, h′) and (T, g) are both weakly and L_{2}-close within (K, ε/2). This step is justified by Lemma 1.9, which is a consequence of Rohlin’s Lemma for aperiodic transformations. It is clear that (V, h) and (T, g) are both weakly and L_{2}-close within (K, ε). □

We now give the details of these lemmas. In what follows

(K, ε) = (K, ε, φ_{1}, …, φ_{U})

is as specified before.

Lemma 1.7. Let X = [0, 1) be the unit circle with Lebesgue measure. Let α_{j} be a sequence in X converging to α, with the corresponding rotations T_{j} and T. Then, for any f ∈ L_{2}, and for sufficiently large j, the processes (T_{j}, f) and (T, f) are both weakly and L_{2}-close within (K, ε).

Proof. For any fixed n the sequence T_{j}^{n}f converges to T^{n}f in L_{2} norm. Hence, (T_{j}, f) and (T, f) are L_{2}-close within (K, ε) for all sufficiently large j. Let Q_{j} : X → R^{K} be defined as

Q_{j}(x) = (f(x), T_{j}f(x), …, T_{j}^{K−1}f(x))

and let Q : X → R^{K} be the similar mapping defined in terms of T instead of T_{j}. We see that Q_{j} converges to Q in measure. This shows that (T_{j}, f) and (T, f) are also weakly close within (K, ε), for all sufficiently large j. □

Lemma 1.8. For any L_{2} process (T, f) there is an aperiodic process (S, g) such that these two processes are both weakly and L_{2}-close within (K, ε).

Proof. If T is a periodic transformation of period n, then (T, f) is isomorphic (as defined in 1.1) to a process for which the underlying space is the unit circle and the transformation is the rotation by 1/n. Approximating 1/n by irrational numbers and applying the previous lemma, we see that in this case there is, in fact, an ergodic process (S, g) satisfying our requirements. In the general case, partition the underlying space X for T into the T -invariant sets X_{1}, X_{2}, …, X_{∞}, where, for 1 ≤ k < ∞, X_{k} is the set of all x ∈ X such that T^{k}x = x but T^{i}x ≠ x if 1 ≤ i < k, and X_{∞} = X − ∪_{1≤k<∞}X_{k}. The restriction of f to ∪_{N<n<∞}X_{n} goes to zero both in L_{1} and L_{2} norms, as N → ∞. Hence we will assume, without loss of generality, that f vanishes on ∪_{N<n<∞}X_{n}, for some N. Change X to Y, by replacing each X_{n} by a circle Y_{n}, with the same measure as X_{n}, and leaving X_{∞} = Y_{∞} unchanged. Then (T, f) is isomorphic to a process (R, g), where the restriction of R to Y_{n} is the rotation by 1/n, and the restriction to Y_{∞} is equal to the restriction of T to X_{∞}. Then the required aperiodic transformation S will be obtained by replacing each rotation by an irrational rotation. By choosing the rotation α_{n} on Y_{n}, for 1 ≤ n ≤ N, sufficiently close to 1/n, we see that this process (S, g) satisfies our requirements. □

Lemma 1.9. Let T and S be two aperiodic transformations on X and Y, respectively. Then, given an f ∈ L_{2}(X), there is a g ∈ L_{2}(Y) such that (T, f) and (S, g) are both weakly and L_{2} close within (K, ε).

Proof. We will show that, given any δ > 0 there is a function g, with the same distribution as f, and two sets X_{0} ⊂ X and Y_{0} ⊂ Y, with measures greater than 1 − 2δ, such that the joint distributions of (f, T f, …, T^{K−1}f) on X_{0} is the same as the joint distribution of (S^{0}g, …, S^{K−1}g) on Y_{0}. This will complete the proof. In fact, note that all the functions we are considering, T^{i}f and S^{j}g, have the same distribution. Hence, if δ is sufficiently small, we see that the processes (T, f) and (S, g) are both weakly and L_{2} close within (K, ε).

To construct such a g, find a nonnegative integer R such that K/(R + 1) < δ. In what follows r ranges over the integers {0, 1, …, R}. Use Rohlin’s Lemma to find F ⊂ X and G ⊂ Y, with measures equal to (1 − δ)/(R + 1), such that both of the families of sets T^{−r}F and S^{−r}G are pairwise disjoint in their respective spaces.

Let X_{1} = ∪_{r=0}^{R}T^{−r}F and Y_{1} = ∪_{r=0}^{R}S^{−r}G. Let B = T^{−R}F and C = S^{−R}G. Let f_{r} : B → R be the restriction of T^{r}f to B. If Y = (Y, G, ν), note that C ∈ S^{−R}G. Consider C as a measure space with the restriction of ν to C ∩ S^{−R}G. Then C is a nonatomic measure space with the total measure equal to μ(B). Hence there are R + 1 functions g_{r} : C → R with the same joint distribution as the functions f_{r}. Furthermore, since these functions are S^{−R}G-measurable, there are w_{r} : S^{−R+r}G → R such that S^{r}w_{r} = g_{r}. We then define g on Y_{1} as the function whose restriction to S^{−R+r}G is equal to w_{r}. Then g restricted to Y_{1} has the same distribution as f restricted to X_{1}. Define g on Y − Y_{1} in such a way that g and f have the same distributions. We then let X_{0} = ∪_{r=K}^{R}T^{−r}F and Y_{0} = ∪_{r=K}^{R}S^{−r}G and see that all the requirements are satisfied. □

Notation 1.10. Let T be a transformation. Let L = L(T) be the class of operators on functions that are linear combinations of the iterates of T. Let A = A(T) be the class of bounded L_{2} operators A for which the following is true. For each f ∈ L_{2} and for each ε > 0 there is a B ∈ L such that ∥Af − Bf∥_{2} < ε.

Corollary 1.11. Let T be an aperiodic transformation on a nonatomic probability space X. Let (K, ε, φ_{1}, …, φ_{U}) be as in the definition of weak-closeness. Given K operators A_{i}, 1 ≤ i ≤ K, in A and f ∈ L_{2}(X), let q_{i} be the Gauss sequence which is L_{2}-equivalent to the sequence A_{i}f. Then there is a g ∈ L_{2}(X), such that the sequences q_{i} and A_{i}g are both weakly and L_{2}-close within (K, ε).

Proof. We will assume that each A_{i} belongs to L. This is not a loss of generality, since we can find B_{i} ∈ L such that, if q′_{i} is the Gauss sequence which is L_{2}- equivalent to the sequence B_{i}f, then the sequences q_{i} and q′_{i} are both weakly and L_{2}-close within (K, ε/2). Hence we assume that each A_{i} is of the form A_{i} = ∑_{j=0}^{N} α_{ij}T^{j}, 1 ≤ i ≤ K, with real coefficients α_{ij}. Let ψ : R^{N+1} → R^{K} be defined as (ψ(x))_{i} = ∑_{j=0}^{N} α_{ij}x_{j}, where 1 ≤ i ≤ K, 0 ≤ j ≤ N, and x = (x_{j}) ∈ R^{N+1}. Let M = ∑ |α_{ij}α_{kl}|, where the summation is over 1 ≤ i, k ≤ K and 0 ≤ j, l ≤ N. Let (V, h) be the Gauss process which is L_{2}-equivalent to (T, f). Use Theorem 1.6 to find a g ∈ L_{2}(X) such that the processes (V, h) and (T, g) are both weakly and L_{2}-close within (N + 1, ε/M, φ_{1} ∘ ψ, …, φ_{U} ∘ ψ). This g satisfies the required condition. □

2. An Application

As mentioned at the beginning of this note, we will now apply Corollary 1.11 to give a simple proof of a result in [3], in an important special case. The following two definitions, taken from [3], give a type of L_{2} behaviour and a type of pointwise behaviour for a finite sequence of functions. The theorem to be proved establishes a connection between these behaviours.

Definition 2.1 (δ-Spanning sequences). Let 0 < δ ≤ 1. A sequence of L_{2} functions (f_{1}, …, , f_{K}) will be called a δ-spanning sequence if ∥f_{k}∥_{2} ≤ 1 and ∥f_{k} − Q_{k−1}f_{k}∥_{2} ≥ δ for each k = 1, …, K, where Q_{0} = 0 and Q_{k} is the orthogonal projection on the subspace spanned by (f_{1}, …, f_{k}). Let (T_{1}, …, T_{K}) be finitely many L_{2} operators and f be an L_{2} function. Then f is called a δ-spanning function (for (T_{1}, …, T_{K}) ) if ∥f∥_{2} = 1 and if (T_{1}f, …, T_{K}f) is a δ-spanning sequence.

Definition 2.2 ((δ, ε)-sweeping). Let 0 < ε < δ ≤ 1. A sequence of functions (h_{1}, …, h_{K}) will be called a (δ, ε)-sweeping sequence if ∥h_{k}∥_{1} < ε for each k = 1, …, K, and if max_{1≤k≤K} |h_{k}| > δ − ε on a set of measure greater than 1 − ε. Let (T_{1}, …, T_{K}) be finitely many operators and h be a function. Then h is called a (δ, ε)-sweeping function (for (T_{1}, …, T_{K}) ) if ∥h∥_{∞} ≤ 1, ∥h∥_{1} < ε, and if (T_{1}h, …, T_{K}h) is a (δ, ε)-sweeping sequence.

Remark 2.3. The significance of (δ, ε)-sweeping is as follows. Let T_{n} be a sequence of L_{∞} contractions and δ > 0 fixed. If for each ε, 0 < ε < δ there are arbitrarily large Ks for which the segment (T_{1}, …, T_{K}) has a (δ, ε)-sweeping function, then one can show that there are functions h for which the sequence T_{n}h diverges a.e. Also, its degree of divergence can be characterized by “δ-sweeping”, as defined in [4]. Also, see [1], [5], [6], [7], [8], and [11] for more details.

Theorem 2.4. Let 0 < ε < δ ≤ 1. Then there is an integer K = K(δ, ε) ≥ 1 with the following property. Let (T_{1}, …, T_{K}) be K contractions in L_{2}. If there is a δ-spanning function f for (T_{1}, …, T_{K}) such that the joint distribution of (f, T_{1}f, …, T_{K}f) is normal, then there is an M > 0 such that

h = (1/M)((f ∧ M) ∨ (−M))

is a (δ, ε)-sweeping function for (T_{1}, …, T_{K}).

This is proved in [3]. As mentioned earlier, it relates a certain type of pointwise behaviour of K functions with a joint normal distribution to the L_{2} behaviour of these functions. The following theorem removes the normality assumption for certain types of contractions. A more general version of it was also proved in [3], in a longer and nonconstructive way. An application of Corollary 1.11 gives a shorter and more transparent proof for the important special case below. Recall the definition of A(T) given in 1.10.

Theorem 2.5. Let T be an ergodic transformation in a nonatomic probability space. Let 0 < ε < δ ≤ 1. Then there is an integer K = K(δ, ε) ≥ 1 with the following property. Let (A_{1}, …, A_{K}) be K L_{2} contractions in A(T). If there is a δ-spanning function f for (A_{1}, …, A_{K}), then there is a function g, and an M > 0 such that h = (1/M)((g ∧ M) ∨ (−M)) is a (δ, ε)-sweeping function for (A_{1}, …, A_{K}).

Sketch of the Proof. The δ-spanning property is preserved under L_{2} closeness and the (δ, ε)-sweeping property is preserved under weak closeness. If h_{i} is the Gauss sequence which is L_{2}-equivalent to A_{i}f, there is g ∈ L_{2}(X) such that A_{i}g which is as close to h_{i} as we want, both in the weak and in the L_{2} sense. This follows from the Corollary 1.11. Using Theorem 2.4, we see that this g satisfies the desired condition. □

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MAA: Department of Mathematics, University of Toronto, Toronto, Ontario M5S 1A1, Canada

akcoglu@math.toronto.edu

JRB: School of Mathematics, University of Minneapolis, Minneapolis, MN 55455, USA

baxter@math.umn.edu

DMH: Department of Mathematics, Middle East Technical Unversity, 06531 Ankara, Turkey

hadzung@arf.math.metu.edu.tr

RLJ: Department of Mathematics, DePaul University, 2219 N. Kenmore, Chicago, IL 60614

rjones@condor.depaul.edu http://www.depaul.edu/˜ rjones/

This paper is available via http://nyjm.albany.edu:8000/j/1998/4-6.html.