On optimal stationary couplings between stationary processes

El e c t ro nic J

o f

Pr

ob a bi l i t y

Electron. J. Probab.17(2012), no. 17, 1–20.

ISSN:1083-6489 DOI:10.1214/EJP.v17-1797

On optimal stationary couplings between stationary processes

Ludger Rüschendorf

Tomonari Sei

Abstract

By a classical result of [10] the %¯distance between stationary processes is iden- tified with an optimal stationary coupling problem of the corresponding stationary measures on the infinite product spaces. This is a modification of the optimal cou- pling problem from Monge–Kantorovich theory. In this paper we derive some general classes of examples of optimal stationary couplings which allow to calculate the%¯dis- tance in these cases in explicit form. We also extend the%¯distance to random fields and to general nonmetric distance functions and give a construction method for op- timal stationaryc¯-couplings. Our assumptions need in this case a geometric positive curvature condition.

Keywords:Optimal stationary couplings;%¯-distance; stationary processes; Monge–Kantorovich theory.

AMS MSC 2010:60E15; 60G10.

Submitted to EJP on May 30, 2011, final version accepted on February 2, 2012.

1 Introduction

[10] introduced the%¯distance between two stationary probability measuresµ,νon E^Z, where(E, %)is a separable, complete metric space (Polish space).

The%¯distance extends Ornstein’sd¯distance ([14]) and is applied to the information theoretic problem of source coding with a fidelity criterion, when the source statistics are incompletely known. The distance %¯is defined via the following steps. Let %n : Eⁿ×Eⁿ→Rdenote the average distance per component onEⁿ

%n(x, y) := 1 n

n−1

X

i=0

%(xi, yi), x= (x0, . . . , xn−1), y= (y0, . . . , yn−1). (1.1)

∗Supported by the leading-researcher program of Graduate School of Information Science and Technology, University of Tokyo, and by KAKENHI 19700258.

†Mathematische Stochastik, University of Freiburg, Eckerstr. 1, 79104 Freiburg, Germany.

E-mail:ruschen@stochastik.uni-freiburg.de

‡Department of Mathematics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan.

E-mail:sei@math.keio.ac.jp

Let%¯_ndenote the corresponding minimal`₁-metric also called Wasserstein distance or Kantorovich distance of the restrictions ofµ,νonEⁿ, i.e.

¯

%n(µ, ν) = inf Z

%n(x, y)dβ(x, y)|β∈M(µⁿ, νⁿ)

, (1.2)

whereµⁿ, νⁿ are the restrictions ofµ, ν onEⁿ, i.e. on the coordinates(x0, . . . , x_n−1) andM(µⁿ, νⁿ)is the Fréchet class of all measures on Eⁿ×Eⁿ with marginalsµⁿ,νⁿ. Then the%¯distance betweenµ,ν is defined as

¯

%(µ, ν) = sup

n∈N

¯

%n(µ, ν). (1.3)

In the original Ornstein version%was taken as discrete metric on a finite alphabet. It is known that%(µ, ν) = lim¯ _n→∞%¯_n(µ, ν)by Fekete’s lemma on superadditive sequences.

The%¯-distance has a natural interpretation as average distance per coordinate be- tween two stationary sources in an optimal coupling. This interpretation is further justified by the basic representation result (cp. [10, Theorem 1])

¯

%(µ, ν) = ¯%s(µ, ν) := inf

Γ∈Ms(µ,ν)

Z

%(x0, y0)dΓ(x, y) (1.4)

= inf{E[%(X0, Y0)]|(X, Y)∼Γ∈Ms(µ, ν)}. (1.5) HereMs(µ, ν)is the set of all jointly stationary (i.e. jointly shift invariant) measures on E^Z×E^Z with marginalsµ,ν and(X, Y)∼Γmeans thatΓis the distribution of(X, Y). Thus%(µ, ν)¯ can be seen as a Monge–Kantorovich problem onE^Zwith however a modi- fied Fréchet classM_s(µ, ν)⊂M(µ, ν). (1.5) states this as an optimal coupling problem between jointly stationary processesX,Y with marginalsµ,ν. A pair of jointly station- ary processes(X, Y)with distributionΓ∈Ms(µ, ν)is calledoptimal stationary coupling ofµ, νif it solves problem (1.5), i.e. it minimizes the stationary coupling distance%¯s.

By definition it is obvious (see [10]) that

¯

%1(µ, ν)≤%(µ, ν)¯ ≤ Z

%(x0, y0)dµ⁰(x0)dν⁰(y0), (1.6) the left hand side being the usual minimal`1-distance (Kantorovich distance) between the single componentsµ⁰,ν⁰.

As remarked in [10, Example 2] the main representation result in (1.4), (1.5) does not use the metric structure of%and%can be replaced by a general cost functioncon E×Eimplying then the generalized optimal stationary coupling problem

¯

c_s(µ, ν) = inf{E[c(X0, Y₀)]|(X, Y)∼Γ∈M_s(µ, ν)}. (1.7) Only in few cases information on this optimal coupling problem for %¯ resp. ¯c is given in the literature. [10] determine %¯for two i.i.d. binary sequences with success probabilitiesp₁,p₂. They also derive for quadratic costc(x₀, y₀) = (x₀−y₀)²upper and lower bounds for two stationary Gaussian time series in terms of their spectral densities.

We do not know of further explicit examples in the literature for the%¯distance. The aim of our paper is to derive optimal couplings and solutions for the %¯metric resp. the generalized¯cdistance.

The%¯resp. c¯distance is particularly adapted to stationary processes. One should note that from the general Monge–Kantorovich theory characterizations of optimal cou- plings for some classes of distancescare available and have been determined for time series and stochastic processes in some cases. For processes with values in a Hilbert space (like the weighted `2 or the weighted L² space) and for general cost functions

c, general criteria for optimal couplings have been given in [18] and [16]. For some examples and extensions to Banach spaces see also [2] and [17]. Some of these criteria have been further extended to measuresµ, ν in the Wiener space(W, H, µ)w.r.t. the squared distancec(x, y) =|x−y|²_Hby Feyel and Üstünel (2002, 2004) and [23]. All these results are also applicable to stationary measures and characterize optimal couplings between them. But they do not respect the special stationary structure as described in the representation result in (1.5), (1.7). In the following sections we want to determine optimal stationary couplings between stationary processes.

In Section 2 we consider the optimal stationary coupling of stationary processes on R^{and on} R^m with respect to squared distance. In Section 3 we give an extension to the case of random fields. Finally we consider in Section 4 an extension to general cost functions. We interpret an optimal coupling condition by a geometric curvature condition.

2 Optimal couplings of stationary processes w.r.t. squared dis- tance

In this section we consider the optimal stationary coupling of stationary processes on the Euclidean space with respect to squared distance.

We first recall the classical result for optimal couplings onRⁿ. For two probability distributionsµ andν on R^{n let}Mn(µ, ν)be the set of joint distributions Γ of random variables X ∼ µand Y ∼ ν. Denote the Euclidean norm on R^{n by} k · k2. We call a joint distributionΓinM_n(µ, ν)an optimal coupling ifΓ attains the minimum ofR

kx− yk²₂Γ(dx, dy)overM_n(µ, ν).

Theorem 2.1 ([18] and [1]). For given measuresµand ν onRⁿ with existing second moments, there is an optimal couplingΓ∈Mn(µ, ν)and it is characterized by

Y ∈∂h(X) Γ-a.s. (2.1)

for some convex functionh, where the subgradient∂h(x)atxis defined by

∂h(x) ={y∈Rⁿ|h(z)−h(x)≥y·(z−x), ∀z∈Rⁿ}. (2.2) Furthermore, if µ is absolutely continuous with respect to the Lebesgue measure on Rⁿ, then the gradient ofhis essentially unique.

In the above theorem let µbe absolutely continuous and assume that µand ν are invariant under the map x = (x1, . . . , xn) 7→ Lnx = (xn, x1, . . . , x_n−1). Then, by the uniqueness result, the convex functionhin (2.1) must be invariant underLn. In addi- tion, ifhis differentiable, then the gradient∇hsatisfies(∇h)◦Ln =Ln◦(∇h). This identity motivates the following construction of optimal stationary coupling (see (2.3)).

Now we consider stationary processes. For simplicity, we first consider the one- dimensional caseE = R. The multi-dimensional case E = R^mis discussed later. Let Ω =E^Z =R^Zandc(x0, y0) = (x0−y0)². LetL: Ω→Ωdenote the left shift,(Lx)t=x_t−1. Then a pair of processes(X, Y)with values inΩ×Ωisjointly stationarywhen(X, Y)=^d (LX, LY) (=^d denotes equality in distribution). A Borel measurable map S : Ω → Ωis calledequivariant if

L◦S =S◦L. (2.3)

This notion is borrowed from the corresponding notion in statistics, where it is used in connection with statistical group models. The following lemma concerns some elemen- tary properties.

Lemma 2.2. a) A mapS: Ω→Ωis equivariant if and only ifS_t(x) =S₀(L^−tx)for any t, x.

b) IfX is a stationary process andSis equivariant then(X, S(X))is jointly stationary.

Proof. a) IfL◦S=S◦Lthen by inductionS=L^t◦S◦L^−tfor allt∈Z^{, and thus}S_t(x) = S₀(L^−tx). Conversely, ifS_t(x) =S₀(L^−tx), thenS_t−1(x) =S₀(L^−t+1x) =S_t(Lx). This impliesL(S(x)) =S(Lx).

b) SinceLX has the same law asX, it follows that(LX, L(S(X))) = (LX, S(LX)) = (I, S)(LX)= (I, S)(X^d ) = (X, S(X)), I denoting the identity.

For X ∼µ andS : Ω →Ωthe pair(X, S(X))is calledoptimal stationary coupling if it is an optimal stationary coupling w.r.t. µand ν := µ^S = S_#µ, i.e., whenν is the corresponding image (push-forward) measure.

To construct a class of optimal stationary couplings we define for a convex function f :Rⁿ→Ran equivariant mapS : Ω→Ω. Forx∈Ωlet

∂f(x) ={y∈Rⁿ|f(z)−f(x)≥y·(z−x), ∀z∈R} (2.4) denote the subgradient of f at x, where a·b denotes the standard inner product of vectorsaandb. By convexity∂f(x)6=∅. LetF(x) = (Fk(x))_{0≤k≤n−1}be measurable and F(x)∈∂f(x),x∈Rⁿ. The equivariant mapS is defined via Lemma 2.2 by

S0(x) =

n−1

X

k=0

Fk(x−k, . . . , x−k+n−1), St(x) =S0(L^−tx), x∈Ω. (2.5) For terminological reasons we write any map of the form (2.5) as

S0(x) =

n−1

X

k=0

∂kf(x−k, . . . , x−k+n−1), St(x) =S0(L^−tx), x∈Ω. (2.6) In particular for differentiable convexf the subgradient set coincides with the deriva- tive off,∂f(x) ={∇f(x)}and∂tf(x) =_∂x^∂

tf(x).

Remark 2.3. a) In information theory a map of the formS_t(x) =F(x_t−n+1, . . . , x_t+n−1) is called a sliding block code (see [10]). Thus our class of mapsS defined in (2.6) are particular sliding block codes.

b) [19, 20, 21] introduced so-called structural gradient models (SGM) for stationary time series, which are defined as{(Sϑ)^#Q|ϑ∈Θ}, whereQis the infinite product of the uniform distribution on[0,1], on[0,1]^Z,{S_ϑ |ϑ∈Θ}is a parametric family of transformations of the form given in (2.6)and S_ϑ^#Q denotes the pullback measure ofQbySϑ. It turns out that these models have nice statistical properties, e.g. they allow for simple likelihoods and allow the construction of flexible dependencies. The restriction to functions of the form (2.6)is well founded by an extended Poincaré lemma (see [21, Lemma 3]) saying in the case of differentiable f that these func- tions are the only ones with (the usual) symmetry and with an additional stationarity propertyS_t−1(x) =St(Lx)forx∈R^Z, which is related to our notion of equivariant mappings.

c) Even if a map S has a representation of the form (2.6), the inverse map S⁻¹ does not have the same form in general. We give an example. Let X = (Xt)t∈Z be a

real-valued stationary process with a spectral representationXt =R1

0 e^2πiλtM(dλ), whereM(dλ)is anL²-random measure. Define a processY = (Y_t)by

Yt=St(X) :=Xt+(Xt−1+Xt+1), 6= 0.

This is of the form (2.6) with a function f(x0, x1) = x²₀/4 +x0x1+x²₁/4 which is convex if||<1/2. Under this condition, the map X 7→Y is shown to be invertible as follows. The spectral representation ofY isN(dλ) := (1 +(e^2πiλ+ e^−2πiλ))M(dλ). Then we have the following inverse representation

X_t= Z 1

0

e^2πiλt

1 +(e^2πiλ+ e^−2πiλ)N(dλ) =X

s∈Z

b_sY_t−s,

where(bs)_s∈Z is defined by{1 +(e^2πiλ+ e^−2πiλ)}⁻¹ =P

s∈Zbse^−2πiλs. By standard complex analysis, the coefficients(bs)are explicitly obtained:

bs= z^|s|₊

(z+−z₋), z_±:= −1±√ 1−4²

2 .

Note that |z+| <1 and |z₋| >1 since |2| < 1. Hencebs 6= 0for all s ∈ Z^{and the} inverse mapS⁻¹(Y) =P

sbsYsdoes not have a representation as in (2.6).

The following theorem implies that the class of equivariant maps defined in (2.6) gives a class of examples of optimal stationary couplings between stationary processes.

Theorem 2.4(Optimal stationary couplings of stationary processes onR^{). Let}f be a convex function on R^{n, let} S be the equivariant map defined in (2.6) and letX be a stationary process with lawµ. Assume that X0 and ∂kf(Xⁿ)(k = 0, . . . , n−1) are in L²(µ). Then(X, S(X))is an optimal stationary coupling w.r.t. squared distance between µandµ^S, i.e.

E[(X0−S0(X))²] = min

(X,Y)∼Γ∈Ms(µ,µ^S)E[(X0−Y0)²] = ¯cs(µ, µ^S),

Proof. Fix anyΓ∈Ms(µ, µ^S). By the gluing lemma (see Appendix A), we can construct a jointly stationary process(X, Y,X)˜ on a common probability space such thatX ∼µ, Y =S(X)and( ˜X, Y)∼Γ. From the definition ofY₀=S₀(X), we haveY₀∈L²(µ). Then by the assumption of identical marginals

A := 1

2E[(X0−Y0)²−( ˜X0−Y0)²]

= E[−X0Y0+ ˜X0Y0]

= E[( ˜X0−X0)S0(X)]

= E

"

( ˜X0−X0)

n−1

X

k=0

(∂kf)(X−k, . . . , X−k+n−1)

# .

Using the joint stationarity of(X,X)e we get withXⁿ= (X₀, . . . X_n−1),Xeⁿ = (Xe₀, . . . ,Xe_n−1) that

A = E

"_n−1 X

k=0

( ˜X_k−X_k)(∂_kf)(X₀, . . . , X_n−1)

#

≤ E[f( ˜Xⁿ)−f(Xⁿ)]

= 0,

the inequality is a consequence of convexity off. This implies optimality of(X, Y). We note that the last equality uses integrability off(Xⁿ), which comes from convexity off and theL²-assumptions. This completes the proof.

Theorem 2.4 allows to determine explicit optimal stationary couplings for a large class of examples. Note that – at least in principle – the¯cdistance can be calculated in explicit form for this class of examples.

The construction of Theorem 2.4 can be extended to multivariate stationary se- quences in the following way. Let(Xt)t∈Z be a stationary process, Xt ∈ R^{m and let} f : (R^m)ⁿ →Rbe a convex function on(R^m)ⁿ. Define an equivariant mapS : (R^m)^Z→ (R^m)^Zby

S0(x) =

n−1

X

k=0

∂kf(x_−k, . . . , x_−k+n−1) S_t(x) =S₀(L^−tx), x∈Ω = (R^m)^Z

(2.7)

whereL^−toperates on each component ofxand∂`f is (a representative of) the subgra- dient off w.r.t. the`-th component. Thus for differentiablef we obtain

S0(x) =

n−1

X

k=0

∇kf(x_−k, . . . , x_−k+n−1) (2.8) where∇`f is the gradient off w.r.t. the`-th component.

Then the following theorem is proved similarly to Theorem 2.4.

Theorem 2.5(Optimal stationary couplings of stationary processes onR^{m). Let}f be a convex function on(R^m)ⁿ and letS be the equivariant map onΩ = (R^m)^Z defined in (2.7). LetX be a stationary process onR^mwith distributionµand assume thatX₀and

∂_kf(Xⁿ),0≤k≤n−1, are square integrable. Then(X, S(X))is an optimal stationary coupling betweenµandµ^S =S#µw.r.t. squared distance, i.e.

E[kX0−S0(X)k²₂] = inf{E[kX0−Y0k²₂]|(X, Y)∼Γ∈Ms(µ, µ^S)}= ¯cs(µ, µ^S). (2.9) Remark 2.6. Multivariate optimal coupling results as in Theorem 2.5 for the squared distance or later in Theorem 4.1 for general distance allow to compare higher dimen- sional marginals of two real stationary processes. For this purpose we consider a lifting of one-dimensional processes to multi-dimensional processes as follows. For fixedmwe define an injective mapqfromR^Zto(R^m)^Zbyq(x) = (q_k(x))_k∈Z= ((x_k, . . . , x_k+m−1))_k∈Z. Note that q satisfies the equivariant condition (2.3). For one-dimensional processes X = (Xk) ∼ µ and Y = (Yk) ∼ ν define m-dimensional processes X˜ = q(X) and Y˜ = q(Y)and denote their distributions by µ˜ and ν˜, respectively. Let c^(m) be a cost function onR^m×R^m. Then we have the optimal coupling problems betweenµ˜andν˜as

¯

c^(m)_s (˜µ,ν) = inf{E[c˜ ^(m)( ˜X0,Y˜0)]|( ˜X,Y˜)∼Γ˜ ∈Ms(˜µ,ν˜)}

= inf{E[c^(m)(q₀(X), q₀(Y))]|(X, Y)∼Γ∈M_s(µ, ν)},

where the second equality follows from the fact that any ( ˜X,Y˜) ∼ Γ˜ ∈ Ms(˜µ,ν)˜ is supported onq(R^Z)×q(R^Z), andq⁻¹( ˜X)∼µ, q⁻¹( ˜Y)∼ν. If the cost functionc^(m)is the squared distance as in Theorem 2.5, then we can solve the lifted problem immediately when we solve the casem= 1since

¯

c^(m)_s (˜µ,ν) = inf{Ek˜ X˜0−Y˜0k²₂|( ˜X,Y˜)∼Γ˜ ∈Ms(˜µ,ν˜)}

= inf{E[m(X0−Y0)²]|(X, Y)∼Γ∈Ms(µ, ν)}=m¯c⁽¹⁾_s (µ, ν).

For generalcnot written as sum of one-dimensional cost functions the quantityc¯^(m)s (˜µ,ν)˜ has a meaning different from one-dimensional ones.

3 Optimal stationary couplings of random fields

In the first part of this section we introduce the%¯distance defined on a product space in the case of countable groups and establish an extension of the [10] representation result to random fields. In a second step we extend this result to amenable groups on a Polish function space. This motivates the consideration of the optimal stationary coupling result as in Section 2.

We consider stationary real random fields on an abstract groupG. Section 2 was concerned with the case of stationary discrete time processes, whereG=Z. Interest- ing extensions concern the case of stationary random fields on latticesG=Z^{d or the} case of stationary continuous time stochastic processes withG=R^orG=R^d.

To state the most general version of the representation result, we prepare some notations and definitions. Let(G,G) be a topological group with the neutral element e. LetB be a Polish space equipped with a continuous and non-negative cost function c(x, y), x, y ∈ B. We assume that the group G continuously acts on B on the left:

(gh).x = g.(h.x), e.x = x and the map x 7→ g.x is continuous. A Borel probability measureµon B is called stationary if µ^g = µfor everyg ∈ G, whereµ^g is the push- forward measure ofµbyg.

Example 3.1. If Gis countable, an example of B is the product spaceΩ = E^G of a Polish spaceE(e.g. E=R) equipped with the product topology. The left group action ofGonΩis defined by(g.x)h=x_g⁻¹_h. Indeed,

((gh).x)k =x(gh)⁻¹k=xh⁻¹g⁻¹k = (h.x)g⁻¹k= (g.(h.x))k. It is easy to see thate.x=xand the functionx7→g.xis continuous.

If Gis not countable, thenΩ = E^G is not Polish. One can consider a Polish space B ⊂ Ω such that the projection B → E, x 7→ xe, is measurable and g.B = B. For example, letG=R^,E =R^andB be the set of all continuous functions onG=R^with the compact-open topology, that is, definef_n →f inB ifsup_x∈K|f_n(x)−f(x)| →0 for each compactK. Then all the requirements are satisfied.

We assume thatGis an amenable group, i.e. there exists a sequenceλ_n of asymp- totically right invariant probability measures onGsuch that

sup

A∈G

|λn(Ag)−λn(A)| →0 whenn→ ∞. (3.1) The hypothesis of amenability is central for example in the theory of invariant tests.

Many of the standard transformation groups are amenable. A typical exception is the free group of two generators. The Ornstein distance can be extended to this class of stationary random fields as follows. Define the average distance w.r.t. λ_n by

cn(x, y) :=

Z

c(g⁻¹.x, g⁻¹.y)λn(dg). (3.2) For example, if B = E^G and c(x, y) depends only on(xe, ye), say c(xe, ye), then cn is given by

c_n(x, y) = Z

c(x_g, y_g)λ_n(dg). (3.3)

Letµandνbe stationary probability measures onB. The induced minimal probabil- ity metric is given by

¯

cn(µ, ν) = inf{E[cn(X, Y)]|(X, Y)∼Γ∈M(µ, ν)}. (3.4)

Finally, the natural extension of the¯cmetric of [10] is defined as

¯

c(µ, ν) = sup

n

¯

cn(µ, ν). (3.5)

The optimal stationary coupling problem is introduced similarly as in Section 2 by

¯

c_s(µ, ν) = inf{E[c(X, Y)]|(X, Y)∼Γ∈M_s(µ, ν)} (3.6) whereMs(µ, ν) = {Γ ∈ M(µ, ν) | Γ^(g,g) = Γ, ∀g ∈ G} is the class of jointly stationary measures with marginalsµand ν. We use the notationΓ(c) = E[c(X, Y)]andΓ(cn) = E[cn(X, Y)]forΓ∈M(µ, ν).

We now can state an extension of the Gray–Neuhoff–Shields representation result for the¯cdistance of stationary random fields to amenable groups.

Theorem 3.2 (General representation result for¯cdistance). LetGbe an amenable group acting on a Polish spaceB and cbe a non-negative continuous cost function on B×B. Letµ,νbe stationary probability measures onB. Assume that forX ∼µ(resp.

ν), Ec(X, y) < ∞ fory ∈ B. Then the extended Ornstein distance c¯defined in (3.5) coincides with the optimal stationary coupling distance¯cs,

¯

c(µ, ν) = ¯c_s(µ, ν).

In particular,¯cdoes not depend on choice ofλn.

Proof. To prove that ¯c(µ, ν) ≤ c¯s(µ, ν) let for ε > 0 given Γ ∈ Ms(µ, ν) be such that Γ(c) ≤ ¯cs(µ, ν) +ε. Then using the integrability assumption and stationary of Γ we obtain for alln∈N

¯

cn(µ, ν) ≤ Γ(cn) = E[

Z

c(g⁻¹.X, g⁻¹.Y)λn(dg)]

= Z

E[c(g⁻¹.X, g⁻¹.Y)]λn(dg) = Γ(c)≤¯cs(µ, ν) +ε.

This implies that¯c(µ, ν)≤¯c_s(µ, ν).

For the converse direction we choose for fixed ε > 0 and n ≥ 0 an element Γ_n ∈ M(µ, ν)such thatΓn(cn)≤c¯n(µ, ν) +ε. We define probability measures{¯Γn}by

Γ¯_n(A) :=

Z

G

Γ_n(g.A)λ_n(dg). (3.7)

Note thatΓ¯n(c) = Γn(cn). Indeed, Γ¯n(c) =

Z

c(x, y)¯Γn(dx, dy) = Z Z

c(x, y)Γn(g.dx, g.dy)λn(dg)

= Z Z

c(g⁻¹.x, g⁻¹.y)Γn(dx, dy)λn(dg) = Z

cn(x, y)Γn(dx, dy) = Γn(cn).

Using Fubini’s theorem we obtain that Γ¯n(h.A)−Γ¯n(A) =

Z

G

(Γn(gh.A)−Γn(g.A))λn(dg)

= Z

B×B

(λn(Cx,yh⁻¹)−λn(Cx,y))Γn(dxdy), (3.8) whereC_x,y ={g∈G|(x, y)∈g.A}. By amenability (3.1) ofGwe have

|¯Γ_n(h.A)−Γ¯_n(A)| ≤ Z

B×B

|λ_n(C_x,yh⁻¹)−λ_n(C_x,y)|Γ_n(dxdy)→0 (3.9)

asn→ ∞, i.e. Γ¯_n is asymptotically left invariant onB×B.

We haveΓ¯_n∈M(µ, ν)since projections on finite components ofΓ¯_nare Γ¯n(A1×Ω) =

Z

G

Γn(g.A1×Ω)λn(dg)

= Z

G

µ(g.A1)λn(dg) =µ(A1)

sinceµis stationary. Using tightness of{Γ¯n}we get a weakly converging subsequence of {Γ¯n}. Without loss of generality we assume that{Γ¯n} converges weakly to some probability measureΓ¯ onB×B. In consequence by (3.9) we getΓ¯∈M_s(µ, ν). Finally,

¯

cs(µ, ν) ≤ ¯Γ(c)≤lim sup ¯Γn(c) = lim sup Γn(cn)

≤ lim sup ¯c_n(µ, ν) +ε≤¯c(µ, ν) +ε for allε >0which concludes the proof.

Example 3.3. Let G be countable andλ_n = _|F¹

n|

P

g∈Fnε_g for some increasing class of finite sets Fn ⊂ Gwith G= ∪nFn, whereεg denotes the point-mass measure atg. Amenability ofGcorresponds to the condition thatFnis asymptotically right invariant in the sense that

|Fn∩(Fnh)|/|Fn| →1, ∀h∈G. (3.10) For example, the group G = Z is amenable because F_n = {b−n/2c, . . . ,bn/2c −1}

satisfies the above conditions. In the optimal coupling problem, we can take F_n⁰ = {0, . . . , n−1}instead ofF_n becauseµandν are stationary, althoughF_n⁰ does not cover Z^.

Now take the product spaceB =E^Gand assume thatc(x, y)depends only on(xe, ye) and is denoted as c(x, y) = c(xe, ye). Then we obtaincn(x, y) = _|F¹

n|

P

g∈Fnc(xg, yg) =:

cn(xF_n, yF_n), wherexF_n = (xg)_g∈Fn. We now show that¯cn(µ, ν)in (3.4) is equal to inf{E[cn(XF_n, YF_n)]|(XF_n, YF_n)∼ΓF_n ∈M(µF_n, νF_n)} (3.11) withX_Fn =π_Fn(X), Y_Fn =π_Fn(Y), µ_Fn =µ^πFn andν_Fn =ν^πFn, where π_Fn is defined by π_Fn(x) = x_Fn. The equation (3.11) follows from the general extension property of probability measures with given marginals, that is, we can construct

Γ(dx, dy) = ΓF_n(dxF_n, dyF_n)µ_G\Fn(dx_G\Fn|xF_n)ν_G\Fn(dy_G\Fn|yF_n)

from anyΓ_Fn ∈M(µ_Fn, ν_Fn)(see also Appendix A). Finally, the original representation result (1.4) follows from Theorem 3.2 withG=Zsince (3.11) is consistent with (1.2).

Motivated by the representation results in Theorem 3.2 we now consider the op- timal stationary coupling problem for general groups G acting on Ω = R^{G and the} squared distancec(x, y) = (x₀−y₀)². LetF be a finite subset ofGand letf :R^F →R be a convex function. The function f is naturally identified with a function on Ω by f(x) = f((xg)_g∈F). As in Section 2 any choice of the subgradient off is denoted by ((∂gf)(x))g∈F. Define an equivariant Borel measurable function S : Ω → Ω by the shifted sum of gradients

S_e(x) =X

g∈F

(∂_gf)(gx) and S_h(x) =S_e(h⁻¹x), h∈G. (3.12) Note thatS_e(x)depends only on(x_g)_g∈G(F), whereG(F)is the subgroup generated by F inG. We haveS◦g=g◦S for anyg∈Gbecause

Sh(gx) =Se(h⁻¹gx) =S_g⁻¹_h(x) = (gS(x))h.

Hence ifX is a stationary random field, then(X, S(X))is a jointly stationary random field.

We obtain the following theorem.

Theorem 3.4. Let µ be a stationary probability measure onΩ = R^G with respect to a general group of measurable transformations G. Let S be an equivariant map as defined in(3.12)with a convex functionf. LetX be a real stationary random field with lawµand assume thatXeand(∂gf(X))g∈F are inL²(µ). Then(X, S(X))is an optimal stationary coupling w.r.t. squared distance betweenµandµ^S, i.e.

E[(X_e−S_e(X))²] = min

(X,Y)∼Γ∈Ms(µ,µ^S)

E[(X_e−Y_e)²] = ¯c_s(µ, µ^S).

Proof. The construction of the equivariant mapping in (3.12) and the following remark allow us to transfer the proof of Theorem 2.5 to the class of random field models. Fix Γ∈M_s(µ, µ^S). LetG(F)be the subgroup generated byF inG. ThenG(F)is countable (or finite). We denote the restricted measure ofµ onR^{G(F) by} µ|_G(F). By the gluing lemma, we can consider a jointly stationary random field(Xg, Yg,X˜g)_g∈G(F)on a com- mon probability space such that(X_g)_g∈G(F)∼µ|_G(F),Y_g =S_g(X)and( ˜X_g, Y_g)_g∈G(F)∼ Γ|_G(F). Then we have

1

2E[(Xe−Se(X))²−( ˜Xe−Se(X))²] = E[Se(X)( ˜Xe−Xe)]

= X

g∈F

E

(∂gf)(gX)

( ˜Xe−Xe)

= X

g∈F

E

(∂gf)(X)

( ˜Xg−Xg)

≤E[f( ˜X)−f(X)]

= 0.

This implies that(X, S(X)) is an optimal stationary coupling w.r.t. squared distance between the random fieldsµandµ^S =S#µ.

The generalization to the multi-dimensional caseE=R^mis now obvious and omit- ted.

4 Optimal stationary couplings for general cost functions

We consider general cost functionscon general spaces other than the squared dis- tance on R^m. The Monge–Kantorovich problem and the related characterization of optimal couplings have been generalized to general cost functions c(x, y)in [16, 17], while [13] extended the squared loss case to manifolds; see also the surveys in [15]

and [24, 25]. Based on these developments we will extend the optimal stationary cou- pling results in Sections 2, 3 to more general classes of distance functions. Some of the relevant notions from transportation theory are collected in the Appendix B. We will restrict to the case of time parameterZ. As in Section 3 an extension to random fields with generaltime parameter is straightforward.

LetE₁, E₂ be Polish spaces. and letc:E₁×E₂ →Rbe a measurable cost function.

Forf :E₁→R^andx₀∈E₁let

∂^cf(x0) =

y0∈E2|c(x0, y0)−f(x0) = inf

z0∈E1

{c(z0, y0)−f(z0)} (4.1) denote the set ofc-supergradients off inx0.

A function ϕ : E₁ → R∪ {−∞} is called c-concave if there exists a function ψ:E₂→R∪ {−∞}such that

ϕ(x) = inf

y∈E₂(c(x, y)−ψ(y)), ∀x∈E₁. (4.2) Ifϕ(x) = c(x, y₀)−ψ(y₀), then y₀ is ac-supergradient ofϕat x. For squared distance c(x, y) = kx−yk²₂ inR^m = E1 =E2 c-concavity ofϕis equivalent to the concavity of x7→ϕ(x)− kxk²₂/2.

Consider E1 =E2 = R^m. The characterization of optimal couplingsT(x) ∈∂^cϕ(x) for somec-concave functionϕleads for regular ϕto a differential characterization of c-optimal coupling functionsT

(∇xc)(x, T(x)) =∇ϕ(x). (4.3)

In case (4.3) has a unique solution inT(x)this equation describes optimalc-coupling functionsTin terms of differentials ofc-concave functionsϕand the set ofc-supergradients

∂^cϕ(x)reduces to just one element

∂^cϕ(x) ={∇xc^∗(x, ϕ(x))}. (4.4) Herec^∗is the Legendre transform ofc(x,·)and∇xc(x,·)is invertible and(∇xc)⁻¹(x, ϕ(x))

=∇xc^∗(x, ϕ(x))(see [16, 15] and [24, 25]). For functionsϕwhich are notc-concave, the supergradient∂^cϕ(x)is empty at some pointx. Even ifϕisc-concave, the supergradi- ent may be empty. Ifc(x, y) =h(x−y)with a superlinear strictly convex functionhon R^m, the existence of supergradients and regularity ofc-concave functions are proved in the appendix of [9].

The construction of optimal stationary c-couplings of stationary processes can be pursued in the following way. Define the average distance per componentcn : E₁ⁿ× E₂ⁿ→R^by

cn(x, y) = 1 n

n−1

X

t=0

c(xt, yt) (4.5)

and assume that for some functionf : E₁ⁿ →R, there exists a functionFⁿ : Eⁿ₁ →E₂ⁿ such that

Fⁿ(x) = (Fk(x))_{0≤k≤n−1}∈∂^cnf(x), x∈E₁ⁿ. (4.6) Note that (4.6) needs to be satisfied only on the support of (the projection of) the station- ary measureµ. In general we can expect∂^cnf(x)6=∅,∀x∈E₁ⁿ only iff iscn-concave.

For fixedy0, . . . , y_n−1∈E2we introduce the functionhc(x0) =_n¹Pn−1

k=0c(x0, yk),x0∈E1. hc(x)describes the average distance ofx0to thenpointsy0, . . . , yn−1inE2. We define an equivariant mapS:E₁^Z→E₂^Z by

S0(x)∈∂^c(hc(x0))|_{yk=Fk(x−k,...,x−k+n−1),0≤k≤n−1}

S_t(x) =S₀(L^−tx), S(x) = (S_t(x))_t∈Z. ^(4.7) Here thec-supergradient is taken for the functionhc(x0)and the formula is evaluated atyk =Fk(x_−k, . . . , x_−k+n−1),0 ≤k≤n−1. After these preparations we can state the following theorem.

Theorem 4.1 (Optimal stationary c-couplings of stationary processes). Let X = (Xt)t∈Z be a stationary process with values in E1 and with distributionµ, let c : E1× E₂ →Rbe a measurable distance function onE₁×E₂and letf :E₁ⁿ →R^{be measur-} ablec_n-concave. IfS is the equivariant map induced byf in (4.7)and ifc(X₀, S₀(X)), {c(Xk, Fk(Xⁿ))}ⁿ⁻¹_k=0 and f(Xⁿ)are integrable, then (X, S(X))is an optimal stationary c-coupling of the stationary measuresµ,µ^S i.e.

E[c(X0, S0(X))] = inf{E[c(Y0, Z0)]|(Y, Z)∼Γ∈Ms(µ, µ^S)}= ¯cs(µ, µ^S). (4.8)

Proof. The construction of the equivariant function in (4.7) allows us to extend the basic idea of the proof of Theorem 2.4 to the case of general cost function. Fix anyΓ∈ Ms(µ, µ^S). By the gluing lemma, we can consider a jointly stationary process(X, Y,X)˜ on a common probability space with propertiesX ∼µ,Y =S(X)and( ˜X, Y)∼Γ. Then we have by construction in (4.7) and using joint stationarity of(X,X˜)

E[c(X0, S0(X))−c( ˜X0, S0(X))]

≤ E

"

n⁻¹

n−1

X

k=0

{c(X0, y_k)−c( ˜X₀, y_k)}

_y

k=F_k(X−k,...,X−k+n−1)

#

= E

"

n⁻¹

n−1

X

k=0

{c(Xk, yk)−c( ˜Xk, yk)}

_y

k=F_k(X₀,...,Xn−1)

#

= Eh

c_n(Xⁿ, Fⁿ(Xⁿ))−c_n( ˜Xⁿ, Fⁿ(Xⁿ))i

≤ E[f(Xⁿ)−f( ˜Xⁿ)]

= 0.

The first inequality is a consequence ofS0∈∂^c(hc)(x0). The last inequality follows from cn-concavity off while the last equality is a consequence of the assumption thatX= ˜^d X. As consequence we obtain that(X, S(X))is an optimal stationaryc-coupling.

The conditions in the construction (4.7) of optimal stationary couplings in Theorem 4.1 (conditions (4.6), (4.7)) simplify essentially in the casen= 1. In this case we get as corollary of Theorem 4.1

Corollary 4.2. LetX= (Xt)_t∈U be a stationary process with values inE1and distribu- tionµand letc:E1×E2 →Rbe a cost function as in Theorem 4.1. Letf :E1 →R^be measurablec-concave and define

S0(x)∈∂^cf(x0), St(x) =S0(L^−tx)∈∂^cf(xt), S(x) = (St(x))t∈Z. (4.9) Then(X, S(X))is an optimal stationaryc-coupling of the stationary measuresµ,µ^S.

Thus the equivariant componentwise transformation of a stationary process by su- pergradients of ac-concave function is an optimal stationary coupling. In particular in the case thatE₁ =R^k several examples ofc-optimal transformations are given in [17]

resp. [15] which can be used to apply Corollary 4.2.

In case n ≥ 1 conditions (4.6), (4.7) are in general not obvious. In some cases cn-convexity of a functionf :Eⁿ₁ →Ris however easy to see.

Lemma 4.3. Let f(x) = Pn−1

k=0f_k(x_k), f_k : E₁ → R^, 0 ≤ k ≤ n−1. If the f_k’s are c-concave,0≤k≤n−1, thenf iscn-concave and

∂^cnf(x) =

n−1

X

k=0

∂^cf(x_k). (4.10)

Proof. Let yk ∈ ∂^cfk(xk), 0 ≤ k ≤ n−1, then with y = (yk)_{0≤k≤n−1} by definition of c-supergradients

c_n(x, y)−f(x) = 1 n

X

k

(c(x_k, y_k)−f_k(x_k)) = inf{c_n(z, y)−f(z);z∈E₁ⁿ} and thusy∈∂^cnf(x). The converse inclusion is obvious.

Lemma 4.3 allows to construct some examples of functionsFⁿ satisfying condition (4.5). Forn >1non-emptiness of thec-supergradient ofh_c(x₀) = ¹_nPn−1

k=0c(x₀, y_k)has to be established. The conditionu0∈∂^chc(x0)is equivalent to

c(x₀, u₀)−h_c(x₀) = inf

z (c(z, u₀)−h_c(z)). (4.11) In the differentiable case (4.11) implies the necessary condition

∇xc(x0, u0) =∇xhc(x0) = 1 n

n−1

X

k=0

∇xc(x0, yk). (4.12) If the mapu→ ∇xc(x0, u)is invertible then equation (4.12) implies

u0= (∇xc)⁻¹(x0,·) 1 n

n−1

X

k=0

∇xc(x0, yk)

!

(4.13) (see (4.4)). Thus in case that (4.11) has a solution, it is given by (4.13).

Lemma 4.4. Suppose that for somex0∈E1,∂^chc(x0)6=∅, and that the mapE2→E2: u7→ ∇xc(x0, u)is one to one, then∂^chc(x0)is reduced to the single pointu0defined by

u0= (∇xc)⁻¹(x0,·) 1 n

n−1

X

k=0

∇xc(x0, yk)

!

. (4.14)

Example 4.5. Ifc(x, y) = H(x−y)for a superlinear strictly convex functionH, then

∇_xc(x,·)is invertible and we can construct the necessaryc-supergradients ofh_c. The c-concavity ofhc is not discussed here. If for examplec(x, y) =kx−yk²₂, wherek · k2is the Euclidean norm, then we get for anyx0∈R^m,

u0=u0(x0) = 1 n

n−1

X

k=0

yk =y (4.15)

is independent ofx0and

y∈∂^ch_c(x₀), ∀x0∈R^m. (4.16) Ifc(x, y) =kx−yk^p₂,p >1, then we get forx0∈R^m

u₀=u₀(x₀) =x₀+ka(x₀)k

p−11

2

a(x0) ka(x0)k2

, (4.17)

wherea(x0) = _n¹Pn−1

k=0kx0−ykk^p−1_{2 kx}^x0−yk

0−ykk2. For this and related further examples see [17] and [9].

Thec-concavity ofh_chas a geometrical interpretation. u₀∈∂^ch_c(x₀)if the difference of the distance ofz0inE1tou0inE2and the average distance ofz0to the given points y0, . . . , y_n−1 in E2 is minimized in x0. The c-concavity of hc can be interpreted as a positive curvature condition for the distancec. To handle this condition we introduce the notion of convex stability.

Definition 4.6. The cost functioncis calledconvex stableof indexn≥1if∂^ch_c(x₀)6=∅ for anyx₀∈E₁andy∈E₂ⁿ, where

hc(x0) = 1 n

n−1

X

k=0

c(x0, yk), x0∈E1. (4.18) The costcis called convex stable if it is convex stable of indexnfor alln≥1.

Example 4.7. Let E₁ = E₂ = H be a Hilbert space, as for example H = R^{m, let} c(x, y) =kx−yk²₂/2and fixy∈Hⁿ, then

hc(x0) = 1 n

n−1

X

k=0

c(x0, yk)

=c(x0,y) +¯ 1 n

n−1

X

k=0

c(¯y, yk), (4.19)

wherey¯= _n¹Pn−1

k=0yk Thus by definition (4.2)hc isc-concave and a c-supergradient of h_c is given byy¯independent ofx₀, i.e.

¯

y∈∂^ch_c(x₀), ∀x0∈H. (4.20) Thus the squared distancecis convex stable.

The property of a cost function to be convex stable is closely connected with the geometric property of non-negative cross curvature. LetE1andE2be open connected subsets inR^m(m≥1) with coordinatesx= (xⁱ)^m_i=1andy= (y^j)^m_j=1. Letc:E1×E2→R beC^2,2, i.e. cis two times differentiable in each variable. Denote the cross derivatives byc_ij,k=∂³c/∂xⁱ∂x^j∂y^kand so on. Definec_x(x, y) = (∂c/∂xⁱ)^m_i=1,c_y(x, y) = (∂c/∂y^j)^m_j=1, U_x={c_x(x, y)|y∈E₂} ⊂R^m,V_y ={c_y(x, y)|x∈E₁} ⊂R^m. Assume the following two conditions.

[B1] The mapsc_x(x,·) :E₂→U_xandc_y(·, y) :E₁→V_y are diffeomorphic, i.e., they are injective and the matrix(c_i,j(x, y))is invertible everywhere.

[B2] The setsU andV are convex.

The conditions [B1] and [B2] are called bi-twist and bi-convex conditions, respectively.

Now we define thecross curvatureσ(x, y;u, v)inx∈E₁,y∈E₂,u∈R^mandv∈R^mby

σ(x, y;u, v) := X

i,j,k,l

−cij,kl+X

p,q

cij,qc^p,qcp,kl

!

uⁱu^jv^kv^l (4.21)

where(c^i,j)denotes the inverse matrix of(ci,j).

The following result is given by [11]. Note that these authors use the terminology time-convex sliding-mountaininstead of the notion convex-stability as used in this pa- per.

Proposition 4.8. Assume the conditions [B1] and [B2]. Thenc is convex stable if and only if the cross curvature is non-negative, i.e.,

σ(x, y;u, v)≥0, ∀x, y, u, v. (4.22) The cross-curvature is related to the Ma-Trudinger-Wang tensor ([12]), which is the restriction ofσ(x, y;u, v)toP

i,juⁱv^jc_i,j = 0. Known examples that have non-negative cross-curvature are then-sphere with the squared Riemannian distance ([11], [7]), its perturbation ([3], [8]), their tensorial product and their Riemannian submersion.

IfE1, E2 ⊂R, then the conditions [B1] and [B2] are implied from a single condition in casecx,y =∂²c(x, y)/∂x∂y 6= 0. Hence we have the following result as a corollary. A selfcontained simplified proof of this result is given in Appendix C.

Proposition 4.9. Let E1, E2 be open intervals inR^{and let}c ∈C^2,2,c :E1×E2 →R^. Assume thatcx,y 6= 0for allx, y. Thencis convex stable if and only ifσ(x, y) :=−cxx,yy+ cxx,ycx,yy/cx,y ≥0.

Example 4.10. LetE₁, E₂⊂Rbe open intervals and letE₁∩E₂=∅. Considerc(x, y) =

1

p|x−y|^pwithp≥2orp <1. Thencis convex stable. In factc_x,y=−(p−1)|x−y|^p−26= 0 for allx, yandσ(x, y) = (p−1)(p−2)|x−y|^p−4≥0for allx, y. Asp→0, we also have a convex stable costc(x, y) = log|x−y|.

If the cost functioncis a metric then the optimal coupling in the caseE₁=E₂=R can be reduced to the case ofE₁∩E₂ = ∅ as in the classical Kantorovich–Rubinstein theorem. This is done by subtracting (and renormalizing) from the marginalsµ0,ν0the lattice infimum, i.e. defining

µ⁰₀:=1

a(µ0−µ0∧ν0), ν₀⁰ := 1

a(ν0−µ0∧ν0). (4.23) The new probability measures live on disjoint subsets to which the previous proposition can be applied.

Some classes of optimalc-couplings for various distance functionsc have been dis- cussed in [17], see also [15]. The examples discussed in these papers can be used to establishc_n-concavity off in some cases. This is an assumption used in Theorem 4.1 for the construction of the optimal stationary couplings. Note thatcn is convex-stable if cis convex-stable. Therefore the following proposition due to [6] (partially [22]) is also useful to construct acn-concave functionf.

Proposition 4.11. Assume [B1] and [B2]. Then c satisfies the non-negative cross curvature condition if and only if the space of c-concave functions is convex, that is, (1−λ)f+λgisc-concave as long asf andgarec-concave andλ∈[0,1].

Example 4.12. Consider Example 4.10 again. LetE₁= (0,1),E₂= (−∞,0),c(x₁, y₁) = p⁻¹(x1−y1)^p (p≥2) andcn(x, y) = (np)⁻¹Pn−1

k=0(xk−yk)^p. An example ofcn-concave functions of the formf(x) =Pn−1

k=0fk(xk)with suitable real functionsfk is given in [17]

Example 1 (b). We add a further example here. Putx¯=n⁻¹Pn−1

k=0xkand letf(x) =A(¯x) with a real functionA. We provef(x)isc_n-concave ifA⁰ ≥1andA⁰⁰≤0. For example, A(ξ) =ξ+√

ξsatisfies this condition. Equation (4.3) becomes

n⁻¹(xi−yi)^p−1=n⁻¹A⁰(¯x) (4.24) which uniquely determinesyi∈E2sinceA⁰≥1andxi∈E1. To provecn-concavity off, it is sufficient to show convexity ofx7→cn(x, y)−f(x)for eachy. Indeed, the Hessian is

δ_ijn⁻¹(p−1)(x_i−y_i)^p−2−n⁻²A⁰⁰(¯x) −n⁻²A⁰⁰(¯x)0

in matrix sense. Note that the set of functions A satisfying A⁰ ≥ 1 and A⁰⁰ ≤ 0 is convex, which is consistent with Proposition 4.11. Therefore, any convex combination of A(¯x) and the cn-concave function P

kfk(xk)discussed above is also cn-concave by Proposition 4.11.

Appendix

A Gluing lemma for stationary measures

The gluing lemma is a well known construction of joint distributions. We repeat this construction in order to derive an extension to the gluing of jointly stationary processes.

For given probability measures P and Qon some measurable spaces E1 and E2, we denote the set of joint probability measures on E1×E2 with marginals P and Q by M(P, Q).

Lemma A.1 (Gluing lemma). Let P₁, P₂, P₃ be Borel probability measures on Polish spacesE₁, E₂, E₃, respectively. LetP₁₂ ∈M(P₁, P₂)andP₂₃ ∈M(P₂, P₃). Then there exists a probability measureP123 onE1×E2×E3 with marginalsP12 onE1×E2 and P23onE2×E3.

Proof. LetP_1|2(·|·)be the regular conditional probability measure such that P12(A1×A2) =

Z

A2

P_1|2(A1|x)P2(dx) andP_3|2(·|·)be the regular conditional probability measure such that

P₂₃(A₂×A₃) = Z

A₂

P_3|2(A₃|x)P₂(dx).

Then a measureP123uniquely defined by P₁₂₃(A₁×A₂×A₃) :=

Z

A2

P_1|2(A₁|x)P_3|2(A₃|x)P₂(dx) (A.1) satisfies the required condition.

Next we consider an extension of the gluing lemma to stationary processes. We note that even if a measureP123onE₁^Z×E₂^Z×E₃^Zhas stationary marginalsP12onE₁^Z×E₂^Zand P23 onE₂^Z×E₃^Z, it is not necessarily true thatP is stationary. For example, consider the {−1,1}-valued fair coin processes X = (Xt)t∈Z and Y = (Yt)t∈Z independently, and letZ_t = (−1)^tX_tY_t. Then(X, Y)and(Y, Z)have stationary marginal distributions respectively, but(X, Y, Z)is not jointly stationary becauseX_tY_tZ_t= (−1)^t.

For given stationary measuresP andQon some product spaces, letMs(P, Q)be the jointly stationary measures with marginal distributionsP andQon the corresponding product spaces.

Lemma A.2. LetE₁, E₂, E₃be Polish spaces. LetP₁, P₂, P₃be stationary measures on E₁^Z, E₂^Z, E₃^Z, respectively. LetP12∈Ms(P1, P2)andP23 ∈Ms(P2, P3). Then there exists a jointly stationary measureP123onE^Z₁ ×E₂^Z×E₃^Zwith marginalsP12andP23.

Proof. We define P₁₂₃ by (A.1) and check joint stationarity ofP₁₂₃. First, since P₁₂ is stationary, the conditional probabilityP_1|2is stationary in the sense thatP_1|2(LA₁|Lx) = P_1|2(A1|x)for anyA1andx(P2-a.s.). Indeed, for anyA1andA2,

Z

A2

P_1|2(A1|x)P2(dx) =P12(A1×A2)

=P12(LA1×LA2)

= Z

LA2

P_1|2(LA1|x)P2(dx)

= Z

A₂

P_1|2(LA1|Lx)P2(dx),

where the second and last equality is due to stationarity of P12 and P2, respectively.

Now joint stationarity ofP123follows from (A.1) and stationarity ofP_1|2,P_3|2andP2.

B c -concave function

We review some basic results onc-concavity. See [16, 17, 15, 24, 25] for details.

LetE1andE2be two Polish spaces andc:E1×E2→Rbe a measurable function.