Ergodic quasi-exchangeable stationary processes are isomorphic to Bernoulli processes

TL;DR

This paper proves that ergodic quasi-exchangeable stationary processes are isomorphic to Bernoulli processes and extends De Finetti's theorem to mixtures of Bernoulli processes under uniform integrability conditions.

Contribution

It establishes the isomorphism between ergodic quasi-exchangeable stationary processes and Bernoulli processes, and generalizes De Finetti's theorem for mixtures.

Findings

Ergodic quasi-exchangeable stationary processes are isomorphic to Bernoulli processes.

Processes with uniformly integrable Radon-Nikodym derivatives are mixtures of Bernoulli processes.

Application to determinantal processes demonstrates the theoretical results.

Abstract

{\abstract{\textwidth=4,5 in} A discrete time process, with law $\mu$, is quasi-exchangeable if for any finite permutation $\sigma$ of time indices, the law $\mu_\sigma$ of the resulting process is equivalent to $\mu$. For a quasi-exchangeable stationary process we prove mainly (1) that if the process is ergodic then it is isomorphic to a Bernoulli process and (2) that if the family of all Radon-Nikodym derivatives $\{{d\mu_\sigma\over d\mu}\}$ is uniformly integrable then the process is a mixture of Bernoulli processes, which generalizes De Finetti's Theorem. We give application of (1) to some determinantal processes. }

Equations

$$P(X_1\in A_1,...,X_n\in A_n)=P(X_{\tau (1)}\in A_1,...,X_{\tau (n)}\in A_n),$$

$$P(X_1\in A_1,...,X_n\in A_n)=P(X_1\in A_{\sigma (1)},...,X_n\in A_{\sigma (n)}).$$

$$\tau (k)=k+1,\text{for}1\le k\le n-1,\text{and}\tau (n)=1,$$

$$P(X_1\in A_1,...,X_{n-1}\in A_{n-1})=P(X_2\in A_1,...,X_n\in A_{n-1})(\text{shift-inv}).$$

$$\Omega :=K^{\mathbb{Z}},\text{ endowed with the product sigma algebra}\mathcal{B}:={\mathcal{F}}^{\otimes \mathbb{Z}},\text{ and}$$

$$X_n^{\prime }(\omega )={\omega }_n,\text{ for all integer }n\in \mathbb{Z}\text{and all}\omega \in \Omega .$$

$$\mu (X_{-l}^{\prime }\in A_{-l},...,X_k^{\prime }\in A_k):=P(X_1\in A_{-l},...,X_{k+l+1}\in A_k).$$

$$\mu (X_{-l}^{\prime }\in A_{-l},...,X_k^{\prime }\in A_k)=P(X_p\in A_{-l},...,X_{k+l+p}\in A_k).(\text{shift-inv}{)}^{\prime }.$$

$$\sigma \in H⟺\sigma \in G,\text{and}\exists N,\sigma (n)=n,\forall n,\mid n\mid \ge N.$$

$$(T_{\tau }(\omega ){)}_n={\omega }_{\tau (n)},\forall n\in \mathbb{Z}.$$

$$T_{\tau \circ \sigma }=T_{\sigma }\circ T_{\tau },$$

$$T_{\sigma }^{-1}=T_{{\sigma }^{-1}}.$$

$$\mu (X_{\sigma (-N)}^{\prime }\in A_1,...,X_{\sigma (N)}^{\prime }\in A_{2N+1})=\mu (X_{-N}^{\prime }\in A_1,...,X_N^{\prime }\in A_{2N+1})(\text{exchang}).$$

$$\mu \circ T_{\sigma }^{-1}(X_{-N}^{\prime }\in A_1,...,X_N^{\prime }\in A_{2N+1})=\mu (X_{-N}^{\prime }\in A_1,...,X_N^{\prime }\in A_{2N+1}).$$

$${\varphi }_{\sigma }:=\frac{d\mu \circ T_{\sigma }^{-1}}{d\mu }.$$

$$\Pi (A_{-L},...,A_0):=\{\omega \in \Omega :{\omega }_{-L}\in A_{-L},...,{\omega }_0\in A_0\}$$

$$F(A_1,...,A_s):=\{\omega \in \Omega :{\omega }_1\in A_1,...,{\omega }_s\in A_s\}.$$

$${\mathcal{A}}_I:=\text{the smallest algebra containing the sets}\{\omega \in \Omega :{\omega }_j\in A\},j\in I,A\in \mathcal{F},$$

$$\text{and}{\mathcal{B}}_I:=\text{the sigma algebra generated by }{\mathcal{A}}_I.$$

$$I=\{n\in \mathbb{Z}:n\le 0\}{\mathcal{A}}_I\text{ is denoted}{\mathcal{A}}_{\le 0}$$

$$I=\{n\in \mathbb{Z}:n\ge p\}{\mathcal{A}}_I\text{ is denoted}{\mathcal{A}}_{\ge p}$$

$$I=\mathbb{Z}{\mathcal{A}}_I\text{ is denoted}\mathcal{A}.$$

$$\sigma (j)=P+j,\text{and}\sigma (P+j)=j,\text{if}1\le j\le k$$

$$\sigma (n)=n\text{if}n\notin \{1,...,k\}\cup \{P+j:j=1,...,k\},$$

$$T_{\sigma }(\omega {)}_n={\omega }_n,\text{if}n\notin \{1,...,k\}\cup \{P+j:j=1,...,k\}$$

$$T_{\sigma }(\omega {)}_j={\omega }_{P+j}\text{and}{T}_{\sigma }(\omega {)}_{P+j}={\omega }_j\text{for }1\le j\le k.$$

$$E=\Pi (A_{-L},...,A_0):=\{\omega \in \Omega :{\omega }_{-L}\in A_{-L},...,{\omega }_0\in A_0\},$$

$$\mu (T_{\sigma }^{-1}[E\cap F(A_1,...,A_k)\cap S^{-k}F(C_1,...,C_n)\cap S^{-P}F(B_1,...,B_k)\cap T])=$$

$$\mu (E\cap F(B_1,...,B_k)\cap S^{-k}F(C_1,...,C_n)\cap S^{-P}F(A_1,...,A_k)\cap T)$$

$${\int }_{E\cap F(A_1,...,A_k)\cap S^{-k}F(C_1,...,C_n)\cap T}{1}_{F(B_1,...,B_k)}\circ S^P{\varphi }_{\sigma }d\mu =$$

$${\int }_{E\cap F(B_1,...,B_k)\cap S^{-k}F(C_1,...,C_n)\cap T}{1}_{F(A_1,...,A_k)}\circ S^{P}d\mu .$$

$${\xi }_{B_1,...,B_k}^Q:=\frac{1}{Q}\sum _{P=R+k+1}^Q{1}_{F(B_1,...,B_k)}\circ S^P{\varphi }_{{\sigma }_{P,k}},$$

$$\text{and}{\psi }_{A_1,...,A_k}^Q:=\frac{1}{Q}\sum _{P=R+k+1}^Q{1}_{F(A_1,...,A_k)}\circ S^P.$$

$${\int }_{E\cap F(A_1,...,A_k)\cap S^{-k}F(C_1,...,C_n)\cap T}{\xi }_{B_1,...,B_k}^{Q}d\mu =$$

$${\int }_{E\cap F(B_1,...,B_k)\cap S^{-k}F(C_1,...,C_n)\cap T}{\psi }_{A_1,...,A_k}^{Q}d\mu ,$$

$${\int }_{E\cap S^{-k}F(C_1,...,C_n)\cap T}{1}_{F(A_1,...,A_k)}{\xi }_{B_1,...,B_k}^{Q}d\mu =$$

Full text

Corrigé

  

Doureid Hamdan

A family of two dimensionally determined ergodic processes

  

Doureid Hamdan

Ergodic quasi-exchangeable stationary processes are isomorphic to Bernoulli processes

  

Doureid Hamdan

A discrete time process, with law $\mu$, is quasi-exchangeable if for any finite permutation $\sigma$ of time indices, the law $\mu_{\sigma}$ of the resulting process is equivalent to $\mu$. For a quasi-exchangeable stationary process we prove mainly (1) that if the process is ergodic then it is isomorphic to a Bernoulli process and (2) that if the family of all Radon-Nikodym derivatives $\{{d\mu_{\sigma}\over d\mu}\}$ is uniformly integrable then the process is a mixture of Bernoulli processes, which generalizes De Finetti’s Theorem. We give application of (1) to some determinantal processes.

⁢⁢⁢Keywords: Ergodic processes, quasi exchangeable sequence, translation-invariant determinantal process, Bernoulli system, De Finetti’s Theorem.

2010 Mathematics Subject Classification: Primary 28D05, 37A05, 37A50, 60G09, 60G10. Secondary 60G55, 60J10.

1 Introduction, Notation

According to the generalization, by Hewitt and Savage, of De Finetti’s classical Theorem, an exchangeable sequence $(X_{n})_{n\geq 1}$ of random variables, with values in a presentable space, is a mixture of sequences of independent identically distributed random variables. ( The family of presentable spaces include the polish or locally compact spaces).

In the present paper we consider, more generally, the class of stationary quasi-exchangeable ( quasi symmetric, or quasi-invariant are also used) sequences of random variables ( Recall that a sequence $(X_{n})_{n\geq 1}$ of random variables is exchangeable [ respectively quasi-exchangeable] if its law $P$ is equal [ respectively equivalent ] to the law $P_{\sigma}$ of the sequence $(X_{\sigma(n)})_{n\geq 1}$, for any permutation $\sigma$ of finitely many coordinates.

For a stationary quasi-exchangeable sequence $(X_{n})_{n\geq 1},$ we prove mainly two facts:

(1) In general, we prove, Theorem 1, that if the dynamical system generated by $(X_{n})_{n\geq 1}$ is ergodic, then it is isomorphic to a Bernoulli system.

The result (1) implies, Theorem 3, that the discrete time stationary quasi-invariant determinantal processes are isomorphic to Bernoulli processes. These determinantal processes contain the discrete time translation-invariant determinantal processes in the family considered by Bufetov in [3] ( for example, the discrete sine process of Borodin Okounkov and Olshanski).

Also, we give( Corollary 2, Remark 3) a simple proof of De Finetti’s Theorem, when the state space $K$ is compact.

(2) Under the additional hypothesis that the family $\hskip 2.84544pt\{{dP_{\sigma}\over dP}:\sigma\textit{\hskip 2.84544ptis a finite permutation of }\hskip 2.84544pt\mathbb{N}\}\hskip 2.84544pt$ of Radon-Nikodym derivatives, be uniformly integrable, we prove( Theorem 2) that it is a mixture of i.i.d. sequences. This generalizes the De Finetti’s Theorem.

We establish, first, some definitions and notations.

**Definition 1

***A sequence $\hskip 2.84544pt(X_{n})_{n\geq 1}\hskip 2.84544pt$ of random variables is exchangeable if the law $P_{\sigma}$ of the process $(X_{\sigma(n)})_{n\geq 1}$ is equal the law $P$ of the process $(X_{n})_{n\geq 1}$, for every permutation $\sigma$ belonging to the group $H_{1}$ of all permutations of the set $\mathbb{N}$ of natural numbers, which leave fixed all but a finite number of integers.

Suppose that for any $n\geq 1$, $X_{n}$ takes values in the measurable space $(K,{\cal F})$. Then the sequence $(X_{n})_{n\geq 1}$ is exchangeable, if and only if

[TABLE]

holds for all $\hskip 2.84544ptn\geq 1$, $A_{1},...,A_{n}\in{\cal F}\hskip 2.84544pt$ and any permutation $\hskip 2.84544pt\tau\hskip 2.84544pt$ of $\hskip 2.84544pt\{1,...,n\}$, or equivalently, if and only if for any permutation $\hskip 2.84544pt\sigma\hskip 2.84544pt$ of $\hskip 2.84544pt\{1,...,n\}$ ( $\tau=\sigma^{-1}$),

[TABLE]

In the particular case where $\hskip 2.84544pt\tau\hskip 2.84544pt$ is defined by

[TABLE]

and when $\hskip 2.84544ptA_{n}=K,\hskip 2.84544pt$ equality (1) reads

[TABLE]

which proves that the law $P$ of an exchangeable sequence is invariant by the unilateral shift on $(K^{\mathbb{N}},{\cal F}^{\otimes\mathbb{N}})$.

Let

[TABLE]

Suppose that $(X_{n})_{n\geq 1}$, with law $P$, is exchangeable. Define the measure $\mu$ on $(\Omega,{\cal B})$, which extends $P$, by setting for all $l,k\geq 0$, and $\hskip 2.84544ptA_{-l},...,A_{k}\in{\cal F},\hskip 2.84544pt$

[TABLE]

Then, by (3), for all $p\geq 1$,

[TABLE]

so that $\mu$ extends to a probability measure on $\Omega$, which is also invariant by the shift transformation $\hskip 2.84544ptS:\hskip 2.84544pt(S\omega)_{n}=\omega_{n+1},\hskip 2.84544ptn\in\mathbb{Z},\hskip 2.84544pt\omega\in\Omega.$

Let $G$ be the group of all permutations of $\hskip 2.84544pt\mathbb{Z}\hskip 2.84544pt$ and $\hskip 2.84544ptH\subset G,$ be the subgroup of all permutations with finite support:

[TABLE]

For any $\tau\in G,\hskip 5.69046pt$ let the transformation $T_{\tau}:\Omega\rightarrow\Omega$, be defined for all $\hskip 5.69046pt\omega\in\Omega,\hskip 5.69046pt$ by

[TABLE]

Then $T_{\tau}$ is ${\cal B}$-measurable, and, when $K$ is a topological space, $T_{\tau}$ is continuous for the product topology on $\Omega$. Also for all $\sigma$ and $\tau$ in $H$,

[TABLE]

from which follows that

[TABLE]

Now for every $\sigma\in H$, any $N$ and $A_{1},...,A_{2N+1}\in{\cal F}$, one can see that the following equality holds

[TABLE]

and that it is also equivalent to

[TABLE]

In conclusion, the preceding shows that the exchangeability of the process $(X_{n})_{n\geq 1}$ is the same as the exchangeability of the process $(X^{\prime}_{n})_{n\in\mathbb{Z}}$ and it is also equivalent to the invariance of $\mu$, the law of $\hskip 2.84544pt(X_{n}^{\prime})_{n\in\mathbb{Z}},\hskip 2.84544pt$ by the transformation $T_{\tau}$, for all $\tau\in H$, and in particular, implies, as noted before, the invariance of $\hskip 2.84544pt\mu\hskip 2.84544pt$ by the shift $\hskip 2.84544ptS$. Henceforth, the process $(X_{n})$ we consider will be indexed by $\mathbb{Z}$, and furthermore $X_{n}$ will be the $n^{\textit{th}}$ coordinate function on $\hskip 5.69046pt\Omega:=K^{\mathbb{Z}}$.

A slight generalization of exchangeability is given by the following

**Definition 2

***We say that a sequence $(X_{n})_{n\in\mathbb{Z}}$ of random variables, with law $\mu$, is quasi exchangeable if $\mu\circ T_{\sigma}^{-1}$ is equivalent to $\mu$, for all permutation $\sigma\in H$.

In this case we denote the Radon-Nikodym derivative of $\hskip 2.84544pt\mu\circ T_{\sigma}^{-1}\hskip 2.84544pt$ with respect to $\hskip 5.69046pt\mu,\hskip 2.84544pt$ by $\hskip 2.84544pt\phi_{\sigma}$

[TABLE]

Since we are only interested exclusively in stationary sequences of random variables, the following remark may justify Definition 2.

**Remark 1

*** Naturally we shall say that a unilateral sequence $(X_{n})_{n\geq 1}$ of random variables is quasi-exchangeable if its law $P$ is equivalent to the law $P_{\sigma}$ of the sequence $(X_{\sigma(n)})_{n\geq 1},$ for all permutation $\sigma\in H_{1}$, where $H_{1}$ is the group of all finite permutations of the set $\mathbb{N}$ of natural numbers.

Then, if $(X_{n})_{n\geq 1}$, with law $P,$ is quasi-exchangeable and stationary, the sequence $(X^{\prime}_{n})_{n\in\mathbb{Z}}$, with law $\mu$ defined by $(4)$, will be shift invariant and quasi-exchangeable.

**Definition 3

***Let $(X_{n})_{n\in\mathbb{Z}}$ be a quasi exchangeable process, with law $\hskip 2.84544pt\mu.\hskip 2.84544pt$ If the family $\{{d\mu\circ T_{\sigma}^{-1}\over d\mu}:\sigma\in H\}$ of all Radon-Nikodym derivatives is uniformly integrable, we say that the process $X$ is quasi-exchangeable with uniformly integrable densities.

We shall also use the following notations.

If $L$ and $s$ are integers with $L\geq 0$ and $s\geq 1$, and $A_{-L},...,A_{s}$ are measurable subsets of $K$, we set

[TABLE]

and for all $I\subset\mathbb{Z}$,

[TABLE]

In the following particular cases:

[TABLE]

The same notation will be used for ${\cal B}_{I}$, in particular $\hskip 8.5359pt{\cal B}_{\mathbb{Z}}={\cal B}$.

Similarly, if $\mu$ is a measure on $\Omega$, then $\mu_{I}$ denotes the restriction of $\mu$ to the sigma-algebra ${\cal B}_{I}$. Also, $\hskip 5.69046ptM_{1}(Y)\hskip 5.69046pt$ will denote the set of all Radon probability measures on the topological

space $\hskip 5.69046ptY$, and if $(Z,{\cal G},m)$ is a probability space and ${\cal G}_{1}$ is a sub sigma-algebra of ${\cal G}$, the conditional expectation of a function $f\in L^{1}(Z,{\cal G},m)$ given ${\cal G}_{1}$ will be denoted by $E_{m}[f\mid{\cal G}_{1}]$, or, if there is no confusion on the measure $m$, simply by $E[f\mid{\cal G}_{1}]$.

Also, the smallest sigma-algebra which contains two sigma-algebras ${\cal F}_{1}$ and ${\cal F}_{2}$ is denoted by $\hskip 5.69046pt{\cal F}_{1}\vee{\cal F}_{2}$, and the smallest one containing a family $\{{\cal F}_{j}:j\in J\}$, of sigma-algebras is denoted $\bigvee_{j\in J}{\cal F}_{j}$. The complement of a subset $A$ is denoted $A^{c}$.

2 The main results

Recall that $\hskip 2.84544pt\Omega=K^{\mathbb{Z}},\hskip 2.84544pt$ and $\hskip 2.84544pt{\cal F}\hskip 2.84544pt$ is a sigma algebra of subsets of $\hskip 2.84544ptK.\hskip 2.84544pt$ In the following theorem we suppose that $\hskip 2.84544pt{\cal F}\hskip 2.84544pt$ is separable.

**Theorem 1

***Let $(X_{n})_{n\in\mathbb{Z}}$ be a stationary quasi exchangeable process, with law $\mu$, such that the dynamical system $(\Omega,S,\mu)$ is ergodic. Then the process is isomorphic to a Bernoulli process.

More precisely, if the state space $K$ is finite the system $(\Omega,S,\mu)$ is ”faiblement de Bernoulli”*.

Proof We consider first the case where the state space $K$ is finite, and then we prove that $(\Omega,S,\mu)$ is ”faiblement de Bernoulli”. For this, we shall use the ergodicity and the quasi-exchangeability under a particular infinite family of permutations, in order to find a sequence of measures converging to $\hskip 5.69046pt\mu_{\leq 0}\times\mu_{\geq 1}\hskip 5.69046pt$ on all cylinders and such that any measure in the sequence, coincides with $\mu$ on the two-sided tail sigma-algebra $\hskip 5.69046pt{\cal T}:=\bigcap_{n}({\cal B}_{\leq-n}\vee{\cal B}_{\geq n})$. The detail follows.

For all natural numbers $k$ and $P,\hskip 5.69046pt$ such that $\hskip 5.69046pt1\leq k<P,\hskip 5.69046pt$ let us consider the permutation ( involution) $\hskip 5.69046pt\sigma:=\sigma_{P,k}\in H\hskip 5.69046pt$ which translates the ”interval” $\hskip 2.84544ptI:=\mathbb{N}\cap[1,...,k]\hskip 2.84544pt$ by $\hskip 2.84544ptP$, translates the "interval" $\hskip 2.84544ptP+I=\mathbb{N}\cap[P+1,...,P+k]\hskip 2.84544pt$ by $\hskip 2.84544pt-P,\hskip 2.84544pt$ and leaves fixed all $\hskip 2.84544ptn\in\mathbb{Z},\hskip 2.84544pt$ which are not in the disjoint union $\hskip 2.84544ptI\cup(P+I)\hskip 2.84544pt$, that is, which is defined by

[TABLE]

so that

[TABLE]

Recall that $\hskip 2.84544pt\phi_{\sigma}:={d\mu\circ T_{\sigma}^{-1}\over d\mu},\hskip 2.84544pt$ and for all natural numbers $\hskip 5.69046ptL\geq 0$ and $n\geq 1$, consider any measurable subsets $\hskip 5.69046ptA_{-L},...,A_{k},\hskip 5.69046pt$ $B_{1},...,B_{k}$ and $\hskip 5.69046ptC_{1},...,C_{n}\hskip 5.69046pt$ of $\hskip 2.84544ptK,\hskip 2.84544pt$ and any $\hskip 8.5359ptT\in{\cal T}.\hskip 8.5359pt$ Using the notation as in (10) and (11), and setting

[TABLE]

then, for $\hskip 2.84544ptn+k<P,\hskip 2.84544pt$ the equality

[TABLE]

which holds by the definitions of $\sigma$ and $T_{\sigma}$, reads also, in view of quasi-invariance, as

[TABLE]

For every integer $\hskip 2.84544ptQ\geq R+k+1,\hskip 2.84544pt$ where $R$ is a fixed integer, $\hskip 2.84544ptR\geq n,\hskip 2.84544pt$ consider the following two functions $\xi_{B_{1},...,B_{k}}^{Q}\in L^{1}(\mu)$ and $\psi_{A_{1},...,A_{k}}^{Q}\in L^{\infty}(\mu)$, defined by

[TABLE]

Then, by $(13)$, we obtain

[TABLE]

which we write as

[TABLE]

This last equality, holding true for all $L\geq 1,\hskip 2.84544pt$ $1\leq n\leq R$, all $E$ as in (12), all $C_{1},...,C_{n}$, and all $T\in{\cal T}$, means that

[TABLE]

where ${\cal M}_{R}$ is the sigma-algebra

[TABLE]

Now, $\hskip 2.84544ptk\hskip 2.84544pt$ being fixed, by ergodicity, the sequence $\hskip 2.84544pt(\psi_{A_{1},...,A_{k}}^{Q})_{Q\geq R+k+1}\hskip 2.84544pt$ converges in $L^{2}(\mu)$ norm to the constant $\mu(F(A_{1},...,A_{k}))$. Then, by Cauchy Schwartz for example, the sequence $\hskip 2.84544pt(1_{F(B_{1},...,B_{k})}\psi_{A_{1},...,A_{k}}^{Q})_{Q\geq R+k+1}\hskip 2.84544pt$ converges in $\hskip 2.84544ptL^{2}(\mu)\hskip 2.84544pt$ norm also to $\mu(F(A_{1},...,A_{k}))1_{F(B_{1},...,B_{k})}$. It follows that the sequence in the left side of $(14)$ converges in norm $\hskip 2.84544ptL^{2}(\mu),\hskip 2.84544pt$

to $\hskip 2.84544pt\mu(F(A_{1},...,A_{k}))E[1_{F(B_{1},...,B_{k})}\mid{\cal M}_{R}]$.

In particular,

[TABLE]

The sequence $\hskip 5.69046pt(\xi_{(B_{1},...,B_{k})}^{Q})_{Q\geq R+k+1},\hskip 5.69046pt$ is bounded in $\hskip 5.69046ptL^{1}(\mu):\hskip 5.69046pt$

[TABLE]

Then it is bounded in the bidual $\hskip 5.69046ptL^{\infty*}(\mu)\hskip 5.69046pt$ of $\hskip 5.69046ptL^{1}(\mu).\hskip 5.69046pt$ Hence, by Alaoglu-Bourbaki Theorem( [6], Theorem 2, p.424), this sequence has at least one weak-star cluster point. Let $\hskip 2.84544pt\eta_{(B_{1},...,B_{k})}^{k}\in L^{\infty*}(\mu)\hskip 2.84544pt$ be such a cluster point. Then $\hskip 5.69046pt\eta_{B_{1},...,B_{k}}^{k}\hskip 5.69046pt$ is positive, because for each $Q$, $\hskip 5.69046pt\xi_{(B_{1},...,B_{k})}^{Q}\hskip 5.69046pt$ is positive. Also, for any $x^{*}\in L^{\infty}(\mu)$, there exists a subsequence of natural numbers $(Q_{j})_{j\geq 1}=(Q_{j}(x^{*}))_{j\geq 1}$, which may depend on $x^{*}$, converging to infinity such that

[TABLE]

Taking, in particular, $\hskip 2.84544ptx^{*}=x_{0}^{*},\hskip 2.84544pt$ where $\hskip 2.84544ptx_{0}^{*}=1_{M\cap F(A_{1},...,A_{k})}$, we obtain

[TABLE]

Then, in view of $(15)$, the limit in $(16)$ is independent of the sequence $(Q_{j}(x_{0}^{*}))$, and the following equality holds

[TABLE]

for all $M\in{\cal M}_{R}$.

In the particular case, where $\hskip 5.69046ptB_{1}=...=B_{k}=K,\hskip 5.69046pt$ let us denote $\hskip 5.69046pt\eta_{B_{1},...,B_{k}}^{k}\hskip 5.69046pt$ simply by $\hskip 5.69046pt\eta^{k}.\hskip 5.69046pt$ Then, by $(17)$,

[TABLE]

which means that, under $\eta^{k}$, the algebra ${\cal A}_{\{1,...,k\}}$ and the sigma algebra ${\cal M}_{R}$ are independent( Note that as element of $L^{\infty*}(\mu)$, $\eta^{k}$ may have a non null purely finitely additive part). Recall that

[TABLE]

and set

[TABLE]

Then $\hskip 2.84544pt{\cal M}\subset{\cal M}_{R},\hskip 2.84544pt$ and, by $(18)$, we have in particular

[TABLE]

Also, if we denote by $\hskip 2.84544pt{\cal D}_{k},\hskip 2.84544pt$ the smallest algebra containing $\hskip 2.84544pt{\cal M}\cup{\cal A}_{\{1,...,k\}},\hskip 2.84544pt$ then $\hskip 2.84544pt{\cal D}_{k}\subset{\cal D}_{k+1},\hskip 2.84544pt$ and $(19)$ implies that $\hskip 2.84544pt\eta^{k+1}\hskip 2.84544pt$ extends $\hskip 2.84544pt\eta^{k}\hskip 2.84544pt$ from $\hskip 2.84544pt{\cal D}_{k}\hskip 2.84544pt$ to $\hskip 2.84544pt{\cal D}_{k+1}$. By induction we then have, for every $n\geq 0$,

[TABLE]

Now, the sequence $\hskip 5.69046pt(\eta^{k})_{k\geq 1}\hskip 5.69046pt$ is norm bounded in $\hskip 5.69046ptL^{\infty*}(\mu),\hskip 5.69046pt$ since by positivity and by $(18)$ we have

[TABLE]

Then $\hskip 2.84544pt(\eta^{k})_{k\geq 1}\hskip 2.84544pt$ has at least one weak-star cluster point. By $(20)$, every such cluster point, say $\hskip 5.69046pt\eta,\hskip 5.69046pt$ verifies

[TABLE]

which means

[TABLE]

In particular

[TABLE]

Notice that $(22)$ implies that $\hskip 5.69046pt\eta\hskip 5.69046pt$ and $\hskip 5.69046pt\mu_{\leq 0}\times\mu_{\geq 1}\hskip 5.69046pt$ coincide on all cylinders. We recall that, for any subset $\hskip 5.69046ptI\subset\mathbb{Z}$, $\hskip 5.69046pt\mu_{I}$ denotes the restriction of $\mu$ to the sigma algebra ${\cal B}_{I}$. Also $(21)$ implies that $\hskip 5.69046pt\eta\hskip 5.69046pt$ and $\hskip 5.69046pt\mu\hskip 5.69046pt$ coincide on $\hskip 5.69046pt{\cal T}.$

It follows that if $\hskip 2.84544pt{\cal L}\hskip 2.84544pt$ is the algebra generated by $\hskip 2.84544pt{\cal M}\cup{\cal A}_{\geq 1}$, then clearly $\hskip 2.84544pt{\cal L}\hskip 2.84544pt$ contains $\hskip 2.84544pt{\cal A}\hskip 2.84544pt$ and, by $(21)$, that $\hskip 2.84544pt\eta\hskip 2.84544pt$ and $\hskip 2.84544pt\mu_{{}_{\leq 0}}\times\mu_{{}_{\geq 1}}\hskip 2.84544pt$ coincide also on $\hskip 2.84544pt{\cal L},\hskip 2.84544pt$ so that $\hskip 2.84544pt\eta\hskip 2.84544pt$ is countably additive on $\hskip 2.84544pt{\cal L}.\hskip 2.84544pt$ We also note that $\hskip 2.84544pt{\cal B}\hskip 2.84544pt$ is the smallest sigma-algebra containing $\hskip 2.84544pt{\cal L}.\hskip 2.84544pt$ Then the unique countably additive measure $\hskip 2.84544pt\tilde{\eta}\hskip 2.84544pt$ extending $\hskip 2.84544pt\eta\hskip 2.84544pt$ to the sigma algebra $\hskip 2.84544pt{\cal B},\hskip 2.84544pt$ is the measure $\hskip 2.84544pt\mu_{\leq 0}\times\mu_{\geq 1}.\hskip 2.84544pt$

Then, in particular, $\hskip 5.69046pt\tilde{\eta}\hskip 5.69046pt$ verifies

[TABLE]

Since ${\cal T}\subset{\cal M}$, we also have, by $(21)$, as noted before,

[TABLE]

In conclusion, the countably additive measure $\hskip 2.84544pt\tilde{\eta}\hskip 2.84544pt$ on $\hskip 2.84544pt{\cal B}\hskip 2.84544pt$ satisfies the equalities $(23)$ and $(24)$, which means, when the state space $K$ is finite, that the system $(\Omega,S,\mu)$ is ”faiblement de Bernoulli” ( [11], Définition 3). Since in this case, ”faiblement de Bernoulli” is equivalent to weak Bernoulli ( [11], Proposition 2) and also, a system which is weak Bernoulli is isomorphic to a Bernoulli system [7], the proof is complete in this finite case.

To end the proof in the general case, we establish first the following Lemma, and we appeal to a result of [17] (Theorem 2), saying that if $T$ is a one to one invertible measure preserving transformation on a measure algebra ${\cal L}$, if ${\cal L}$ is the increasing union of invariant sub-sigma algebras ${\cal L}_{i}$ such that $T$ restricted to each ${\cal L}_{i}$ is a Bernoulli shift, then $T$ is a generalized Bernoulli shift.

**Lemma 1

*** Let $\Omega=K^{\mathbb{Z}}$, $S$ the shift on $\Omega$ and $\mu$ be an $S$-invariant probability measure on $\Omega$. Let $X_{n}$ be the $n^{\textit{th}}$ coordinate function on $\Omega$ and ${\cal P}=\{A_{0},...,A_{k-1}\}$ be a finite measurable partition of $K$.

Let $\Omega_{1}:=\{0,...,k-1\}^{\mathbb{Z}},$ $S_{1}$ the shift on $\Omega_{1}$, $Y_{n}$ the $n^{\textit{th}}$ coordinate function on $\Omega_{1}$ and $\theta:\Omega\rightarrow\Omega_{1}$ the factor map defined by:*

[TABLE]

for any $x=(x_{n})_{n\in\mathbb{Z}}\in\Omega$, that is

[TABLE]

Let, for all $n$,

[TABLE]

*Then

(1) the quasi-exchangeability[ respectively exchangeability] of the process $X=(X_{n})$ implies the same property for the process $Z=(Z_{n})$.

(2) the ergodicity of the process $(X_{n})$ implies the ergodicity of $(Z_{n})$.

Proof Let $\nu:=\mu\circ\theta^{-1}$ and, for any permutation $\tau$, $R_{\tau}:\Omega_{1}\rightarrow\Omega_{1},$ the transformation defined by

[TABLE]

Then the quasi-exchangeability[ respectively exchangeability] of $(Z_{n})$ follows from the commutation relationship

[TABLE]

because if this relationship holds we will get

[TABLE]

and then

[TABLE]

[Respectively

[TABLE]

The commutation relation is a consequence of the following

[TABLE]

where $x\in\Omega,n\in\mathbb{Z}$, and $j=0,1,...,k-1$.

**Corollary 1

***Under the same conditions of the above Lemma, if the process $X$ with law $\mu$ is quasi-exchangeable and the system $(\Omega,S,\mu)$ is ergodic, then the system $(\Omega_{1},S_{1},\mu\circ\theta^{-1})$ is isomorphic to a Bernoulli system.

From this Corollary, it follows that if ${\cal P}$ is a finite measurable partition of $K$ and if $\hskip 2.84544pt{\cal P}_{S}:=\bigvee_{n\in\mathbb{Z}}S^{n}{\cal P},\hskip 2.84544pt$ is the smallest sigma-algebra containing $\hskip 2.84544pt\cup_{n\in\mathbb{Z}}S^{n}{\cal P},\hskip 2.84544pt$ then the system $(\Omega,S,{\cal P}_{S})$ is isomorphic to a Bernoulli system and is a factor of $(\Omega,S,{\cal B}(\Omega))$.

Now, if ${\cal Q}$ is a finite partition of $K$, which refines ${\cal P}$, then ${\cal P}_{S}\subset{\cal Q}_{S}$, that is $(\Omega,{\cal P}_{S},\mu)$ is a factor of $(\Omega,{\cal Q}_{S},\mu)$. Then, because ${\cal B}$ is generated by $\cup\{{\cal P}_{S}:{\cal P}\textit{ is a finite partition of}\hskip 2.84544ptK\}$, and since this last union can also be written as an increasing union $\hskip 2.84544pt\cup_{i}{\cal P}_{S}^{i},\hskip 2.84544pt$ where for each $\hskip 2.84544pti,\hskip 2.84544pt$ $(\Omega,S,{\cal P}^{i}_{S})\hskip 2.84544pt$ is isomorphic to a Bernoulli shift, we conclude, by the above mentioned Theorem of [17], that the system $(\Omega,S,{\cal B},\mu)$ is a generalized Bernoulli system, and the proof of Theorem 1 is achieved.$\square$

Recall that the exchangeability means the equalities $\mu\circ T_{\sigma}^{-1}=\mu$, for all $\sigma\in H$, so that the following result generalizes De Finetti’s Theorem.

**Theorem 2

***Let $X=(X_{n})_{n\in\mathbb{Z}}$ be a stationary process. Then the following properties are equivalent

 (1) $X$ is exchangeable.

(2) $X$ is quasi-exchangeable with uniformly integrable densities.

(3) $X$ is a mixture of Bernoulli processes.*

Proof The equivalence $(1)\iff(3)$ is the generalization by Hewitt-Savage of De Finetti’s Theorem to presentable spaces. The implication $(1)\Rightarrow(2)$ is trivial. We show that $(2)$ implies $(3)$. For this, let $\mu$ denotes the law of the process, and let $H_{N}:=\{\sigma\in H:\sigma(n)=n,\forall n,\mid n\mid>N\}$. Then $H_{N}$ is a subgroup of $H$ and $H=\cup_{N}H_{N}$. For every $N\geq 1$, consider the measure

[TABLE]

where $\textit{card}(H_{N})=(2N+1)!$ denotes the cardinality of $H_{N}$.

Scheme of the proof: We prove first that any cluster point $\nu$ of the sequence $(\nu_{N})_{N\geq 1}$ is invariant by $T_{\sigma}$, for any permutation $\sigma\in H$ (and thus the Hewitt-Savage generalization of De Finetti’s Theorem applies to that cluster point $\nu$). The hypothesis implies that $\nu$ is absolutely continuous with respect to $\mu$, and using the definition of $\nu$, we prove that also $\mu$ is absolutely continuous with respect to $\nu$ and we conclude using the Hewitt-Savage mentioned Theorem( or also by Corollary 2 and Remark 3 below).

The details are as follows.

Notice that the uniform integrability of the family $\{\phi_{\sigma}:\sigma\in H\}$ implies the same property for the family $\{{1\over\textit{card}(H_{N})}\sum_{\sigma\in H_{N}}\phi_{\sigma}:N\geq 1\}$, so that, if $\hskip 5.69046ptf_{N}:={1\over\textit{card}(H_{N})}\sum_{\sigma\in H_{N}}\phi_{\sigma}$, then, according to Dunford-Pettis Theorem, $\hskip 5.69046pt(f_{N})$ admits a subsequence $\hskip 5.69046pt(f_{N_{k}}),\hskip 5.69046pt$ which converges weakly in $\hskip 5.69046ptL^{1}(\mu),\hskip 5.69046pt$ to some $f\in L^{1}(\mu).\hskip 5.69046pt$ It follows that the sequence of probability measures $\nu_{N_{k}}$ converges to $\nu:=f\mu$, in the sense that

[TABLE]

Now, because, for every $\hskip 2.84544pt\tau\in H,\hskip 2.84544pt$ there is $k_{0}(\tau)$ such that $\tau\in H_{N_{k}}$, for any $k\geq k_{0}(\tau)$, the measure $\nu$ is invariant by $T_{\tau}$, for all $\tau\in H$. In fact let $A$ be a measurable set, and $\tau\in H$. Then, with $M:=N_{k}$, and using (5), the following equalities hold

[TABLE]

But, as noted before, for $k$ big enough, $\hskip 2.84544pt\tau\in H_{N_{k}}=H_{M},\hskip 2.84544pt$ so that $\hskip 2.84544ptH_{M}=(H_{M})\tau,\hskip 2.84544pt$ since $\hskip 2.84544ptH_{M}\hskip 2.84544pt$ is a group, and then

[TABLE]

and consequently $\hskip 2.84544pt\nu(T_{\tau}^{-1}(A))=\nu(A)$.

It follows that $\nu$ is also invariant by the shift $S$, (hence it will be equal $\mu$, if $\mu$ was ergodic, and hus $\mu$ will be invariant by $H$, hence $\mu$ will be Bernoulli, by the Hewitt-Savage generalization of De Finetti’s Theorem, or by Proposition 1 below).

Suppose now that the system is not necessarily ergodic. By a slight adaptation of the Hewitt-Savage generalization of De Finetti’s Theorem [8] or also, in the case where $K$ is compact, by Corollary 2 and Remark 3 below, being invariant by all $T_{\tau}$, the measure $\nu$ is given by an average of independent measures:

[TABLE]

for all $A\in{\cal B}_{a}(K^{\mathbb{Z}})$, the Baire sigma algebra of $\hskip 5.69046ptK^{\mathbb{Z}},\hskip 5.69046pt$ where $\tilde{K}=M_{1}(K^{\mathbb{Z}})$ and where, for any probability measure $\pi$ on $K$, the probability measure $\hskip 2.84544pt\tilde{\pi}\hskip 2.84544pt$ denotes the corresponding product measure on $\hskip 5.69046ptK^{\mathbb{Z}}$: $\hskip 5.69046pt\tilde{\pi}:=\pi^{\otimes\mathbb{Z}}$.

By repeated use of Lebesgue dominated convergence theorem and monotone convergence theorem, we deduce from $(25)$ that, for any ${\cal B}_{a}(K^{\mathbb{Z}})$ measurable and $\nu$-integrable $h$,

[TABLE]

Also, $\hskip 2.84544pt\nu=f\mu\hskip 2.84544pt$ is absolutely continuous with repect to $\hskip 2.84544pt\mu,\hskip 2.84544pt$ and thus, $f$ is $S$-invariant $\mu$ almost everywhere.

Now the following implications, which hold for all $\sigma\in H$,

[TABLE]

(the last implications use the equivalence of $\hskip 5.69046pt\mu,\hskip 5.69046pt$ $\hskip 5.69046pt\mu\circ T_{\sigma}^{-1}\hskip 5.69046pt$ and $\hskip 5.69046pt\mu\circ T_{\sigma}$) implie that the set $\hskip 2.84544ptA_{0}:=\{f=0\}\hskip 2.84544pt$ is mod($\mu$) invariant by $\hskip 2.84544ptT_{\sigma}$, for all $\hskip 5.69046pt\sigma$. Then, in particular

[TABLE]

so that

[TABLE]

Then $\hskip 2.84544pt\nu\hskip 2.84544pt$ is equivalent to $\hskip 2.84544pt\mu$ so that, for some $S$-invariant $g\in L^{1}(\nu)$, $\mu=g\nu$.

Now, according to Lemma 2 below, for some $E$ with $\nu(E)=1$, $\hskip 2.84544ptg=g_{1}$ on $E$ and $g_{1}$ is ${\cal B}_{a}(K^{\mathbb{Z}})$ measurable. Then, setting $\hskip 2.84544ptF:=\cap_{n\in\mathbb{Z}}S^{n}E,\hskip 2.84544pt$ we obtain $\hskip 2.84544ptg=g\circ S^{n}=g_{1}\circ S^{n}=g_{1}\hskip 2.84544pt$ on $\hskip 2.84544ptF.\hskip 2.84544pt$ We note that $\hskip 2.84544ptg_{1}\circ S^{n}\hskip 2.84544pt$ is $\hskip 2.84544pt{\cal B}_{a}(K^{\mathbb{Z}})\hskip 2.84544pt$ measurable for every $\hskip 2.84544ptn.\hskip 2.84544pt$ Then, for any $\hskip 2.84544ptA\in{\cal B}_{a}(K^{\mathbb{Z}}),\hskip 2.84544pt$ and any $n$,

[TABLE]

and then, since this last equality holds for every $n$, we have

[TABLE]

Then, using Lebesgue and the mixing property of the Bernoulli system $\hskip 2.84544pt(K^{\mathbb{Z}},S,\tilde{\pi}),\hskip 2.84544pt$ we obtain

[TABLE]

because

[TABLE]

and

[TABLE]

Now setting $\hskip 2.84544ptv(\tilde{\pi}):=\int_{K^{\mathbb{Z}}}g_{1}d\tilde{\pi},\hskip 2.84544pt$ and $\hskip 2.84544pt\beta_{1}=v\beta,\hskip 2.84544pt$ the equality $(27)$ reads

[TABLE]

that is

[TABLE]

and this ends the proof, because

[TABLE]

The following Lemma 2 was used in the proof of Theorem 2. We leave its proof to the reader.

Lemma 2( [9], ex. (12,63), p. 186)

*Let $(X,{\cal B},\nu)$ be a measure space and $f$ be a $\nu$-measurable function. Then

there exists a ${\cal B}$-measurable function $g$, which is equal to $f$, $\nu$ almost everywhere. *

In particular

• If $\mu$ is a Borel (respectively Baire) measure on a topological space and if $f\in L^{1}(\mu)$ then there exists a Borel( respectively Baire) function $g$ such that $f=g\hskip 5.69046pt$ $\mu$ almost everywhere.

The simplicity of the proof together with the absence of topological hypothesis on the state space $K$, justifie the following particular case of De Finetti’s theorem.

**Proposition 1

***If $\mu$ is invariant by $T_{\sigma}$ for all $\sigma\in H$, and ergodic for the shift $S$, then $\mu$ is the product measure.

Proof Let $N\geq 1$, and for any $\hskip 2.84544ptP>N+1,\hskip 2.84544pt$ let $\hskip 2.84544pt\tau:=\tau_{N,P}\in H,\hskip 2.84544pt$ be the transposition defined by

[TABLE]

Then, because $\hskip 2.84544pt\mu\circ T^{-1}_{\tau_{N,P}}=\mu,\hskip 2.84544pt$ for all $\hskip 2.84544ptP\hskip 2.84544pt$ with $\hskip 2.84544ptN+1<P\leq Q,\hskip 2.84544pt$ the following equalities

[TABLE]

hold for all $\hskip 2.84544ptQ>N+1,\hskip 2.84544pt$ and show then, by ergodicity, that

[TABLE]

The proof is achieved.$\square$

Recall that, if $\mu$ is a probability measure on $\Omega$, then a measurable set $A$ is $\mu$-almost everywhere shift-invariant if $1_{A}=1_{A}\circ S$ in $L^{1}(\mu)$, and also that it is $\mu$-almost everywhere exchangeable if $1_{A}=1_{A}\circ T_{\sigma}$ in $L^{1}(\mu)$, for any $\sigma\in H$.

Let us use the following notation

[TABLE]

We have seen that $\cal{S}$ is a subset of $M_{1}(\Omega,S)$, the space of all shift invariant probability measures on $\Omega$. It is clear, when $K$ is a compact space, that $\cal S$ is closed in $M_{1}(\Omega,S)$ for the weak star topology $\sigma(C(K)^{*},C(K))$. Then ${\cal S}$ is convex compact for this topology. We need the following Lemma [15], which is easy to prove.

**Lemma 3

**For all $\mu\in{\cal S}$, the sigma-algebra ${\cal I}_{\mu}$ of $\mu$-invariant sets is equal to the sigma-algebra ${\cal E}_{xch}$ of $\mu$-exchangeable sets:

[TABLE]

**Lemma 4

*** The set of measures which are extreme points of ${\cal S}$ is the set of all Bernoulli measures.

Proof Let $\mu\in{\cal S}$ be an extreme point. By Proposition 1, it suffices to show that $\mu$ is ergodic for the shift $S$. Suppose that $\mu$ is not ergodic. Then there exists an invariant set $A$, with $\mu(A)\in]0,1[$. Then $A\in{\cal E}_{xch}$ also, because ${\cal I}_{\mu}={\cal E}_{xch}$, by Lemma 2. Set then $\mu_{1}={1\over\mu(A)}1_{A}\mu$ and $\mu_{2}={1\over\mu(A^{c})}1_{A^{c}}\mu$. Then $\mu_{1},\mu_{2}\in{\cal S}$, $\mu_{1}\neq\mu_{2}$ and $\mu=\mu(A)\mu_{1}+\mu(A^{c})\mu_{2}$, so that $\mu$ is not extremal in ${\cal S}$, contradicting the hypothesis, and we conclude that $\mu$ is ergodic.$\square$

**Remark 2

***It follows from Lemma 4 that the set of exreme points of ${\cal S}$ is closed.

**Corollary 2

*** Let $(X_{n})$ be an exchangeable sequence with values in a compact metrizable space $K$, with law $\mu$. Then there exists a probability measure $\eta$ supported on the Bernoulli measures such that for every Borel subset $A$ of $K^{\mathbb{Z}}$,*

[TABLE]

where $Ber$ stands for the set of all Bernoulli probability measures on $\Omega=K^{\mathbb{Z}}$. In particular for any $k\geq 1$, all Borel sets $A_{1},...,A_{k}$ in $K$,

[TABLE]

**Proof

**Since $\Omega$ is compact metrizable $M_{1}(\Omega,S)$ is compact metrizable. Then ${\cal S}$ is compact metrizable also, and so, by Choquet Theorem ([18], p.14), there is a probability measure $\eta$ on ${\cal S}$, which represents $\mu$ and is supported by the extreme points of ${\cal S}$ and this ends the proof by Lemma 4. $\square$

**Remark 3

***Under the conditions as in the Corollary above, but assuming only the state space $K$ to be compact Haussdorff, the same conclusion holds, with Borel replaced by Baire.

**Corollary 3

*** If the state space $K$ is compact metrizable, the set ${\cal S}$ is a simplex.

Proof For any $\hskip 2.84544pt\mu\in{\cal S},\hskip 2.84544pt$ the decomposition $\hskip 5.69046pt\mu=\int_{{\cal E}}\xi d\eta(\xi),\hskip 5.69046pt$ on the ergodic measures, which is unique since $\hskip 5.69046ptM_{1}(\Omega,S)\hskip 5.69046pt$ is a simplex, is, by Corollary 2, given( represented) by a measure $\eta$ concentrated on the subset $\hskip 5.69046ptBer.\hskip 5.69046pt$ Then, because, by Lemma 3, $Ber$ is the set of extreme points of ${\cal S}$, we conclude that every $\mu\in{\cal S}$ is the barycenter of a unique measure which is supported by the extreme points of $\cal{S}$.

Since ${\cal S}$, being a closed subset of the compact metrizable space $M_{1}(\Omega,S)$, is metrizable, it follows then, by Choquet Theorem( [ 18], p. 60), that ${\cal S}$ is a simplex.$\square$

3 Application:

Bernoullicity of some Determinantal processes

We recall some properties of simple point processes and determinantal processes. For more details we refer to [5, 19, 20, 22, 4, 10]. Let $E$ be a locally compact Polish space. A locally finite subset of $E$ is called a configuration in $E$. Let $\textit{Conf}\hskip 2.84544pt(E)$ be the space of configurations in $E$. We identify a configuration $\xi$, defined as a set, with the atomic integer valued measure $\sum_{x\in\xi}\delta_{x}$, on $E$ and then the space $\textit{Conf}\hskip 2.84544pt(E)$ is endowed with the vague topology of measures on $E$, and the corresponding Borel sigma-algebra ${\cal B}(\textit{Conf}\hskip 2.84544pt(E))$.

A simple random point process with phase space $E$ is a probability measure $P$ on the measurable space $(\textit{Conf}\hskip 2.84544pt(E),{\cal B}(\textit{Conf}\hskip 2.84544pt(E))$. If $B$ is a bounded Borel subset of $E$, let $\eta_{B}$ be the function defined on $\textit{Conf}\hskip 2.84544pt(E),$ by

[TABLE]

The joint distribution of the random variables $\eta_{B}$ determines the measure $P$. Let $\lambda$ be a measure on $E$. A locally integrable function $\rho^{(n)}$ on the cartesian product $E^{n}$, is called a $n$-point correlation function of $P$ if

[TABLE]

for all disjoint bounded Borel subsets $B(1),...,B(m)$ of $E$, and all $k_{1},...,k_{m}\in\mathbb{N}$, with $k_{1}+...+k_{m}=n$( [22] Def. 2 , [4] Def 2, [10])).

Another natural way to investigate a point process is to consider expectations of functions $\tilde{\phi}$ of the form

[TABLE]

where $\phi$ is a measurable function on $E$, with support in some bounded Borel set. Note that if $\mid\phi(x)\mid<1$, for all $x\in E$, then we can write

[TABLE]

Taking, in particular $\phi=-1+\textit{exp}(-\psi)$, with $\psi\geq 0$ having bounded support, we obtain

[TABLE]

leading to Laplace transform of $P$, which proves then that $P$ can be characterized by the expectations of $\tilde{\phi}$’s as in $(d_{1})$.

The point process $P$, all of whose correlation functions $\rho^{(n)}$ exist, is called determinantal if there exists a function $\hskip 2.84544ptk:E\times E\rightarrow\mathbb{R}$, such that for all $n$ and $x_{1},...,x_{n}\in E$,

[TABLE]

The function $k$ above is called the correlation kernel of the process $P$. It is not unique. For example, if $f(x)\neq 0,\forall x\in E$ and $k^{\prime}(x,y)={f(x)k(x,y)\over f(y)}$, then

[TABLE]

In [19, 20] (see also [22] Theorem 3 p. 934) it is proved that if $K$ is a bounded symmetric integral operator on $L^{2}(E,\lambda)$ with kernel $k$, which is also of locally trace class and with spectrum contained in $[0,1]$, then there exists a unique probability measure $P$ on $\textit{Conf}(E)$, such that for every nonnegative continuous function $\psi$ with compact support,

[TABLE]

where the determinant is the Fredholm determinant ( see [21]), and where the

operator $\hskip 11.38092pt\sqrt{(1-e^{-\psi})}K\sqrt{(1-e^{-\psi})}\hskip 11.38092pt$ denotes the integral operator with kernel

[TABLE]

and moreover the correlation functions of $P$ are given by

[TABLE]

In the particular case, where $E$ is countable( in fact $E=\mathbb{Z}$) and $\lambda$ is the counting measure, which is relevant to our purpose, by identifying a subset $A$ of $E$ with its indicator function $1_{A}$, we can take the configuration space to be equal $\{0,1\}^{E}$ and then $(d_{2})$ is equivalent to $(d_{3})$ below ([20] Theorem 1.1, see also [12] and [14] p. 319)

[TABLE]

for any finite subset $A$ of $E$.

We have the following result

**Theorem 3

***Any stationary discrete time quasi-invariant determinantal process $X=(X_{n})_{n\in\mathbb{Z}}$, with phase space $\mathbb{Z}$ is isomorphic to a Bernoulli system.

Proof Let $\mu$ be the law of the process $X$. Since $\Omega:=\{0,1\}^{\mathbb{Z}}$ is the configuration space of the process $X$, $\hskip 5.69046pt\mu$ is a shift invariant probability measure on $\Omega$. It follows from Theorem 7 in [22], that a translation invariant determinantal random point field, with one-particle space $E=\mathbb{Z}$, is mixing of any order and then in particular it is ergodic, meaning in our setting that the system $(\Omega,S,\mu)$ is ergodic. Hence the process $X$ satisfies the hypothesis in Theorem 1, and therefore it is isomorphic to a Bernoulli process. $\square$

In [3] Bufetov considered a class of determinantal processes with phase space $F$, where $F=\mathbb{R}$ (the continuous case), or $F=$ a countable subset of $\mathbb{R}$, without accumulation points( the discrete case), corresponding to projection operators with integrable kernels. In the discrete case, he proved( [3], Theorem 1.6) that they are quasi-invariant, which means quasi-exchangeable. It follows then from the preceding Theorem 3, that the processes in this class, corresponding to the phase space $F=\mathbb{Z}$, and which are translation invariant, are isomorphic to Bernoulli systems.

This applies, in particular, to the discrete sine process. Recall that the discrete sine kernel which is translation invariant kernel on the lattice $\mathbb{Z}$ is defined by

[TABLE]

where $a$ is a real number ($-2\leq a\leq 2$)( [1] p. 486).

Remark 4:

• There exists processes generating dynamical systems isomorphic to Bernoulli and which are not quasi-exhangeable*.

Before giving some examples, let us, when $K$ is a finite set $K:=\{0,...,k-1\}$, denote the cylinder

[TABLE]

Then, for any permutation $\hskip 2.84544pt\sigma\in H,\hskip 2.84544pt$ with $\hskip 2.84544pt\tau:=\sigma^{-1}$, and such that the support of $\sigma$ is included in $\{0,1,...,n\}$,

[TABLE]

so that, for a stationary Markov chain defined by a stochastic matrix $\Pi=(\Pi_{i,j})_{i;j=0,...,k-1}$ and an invariant probability vector $p=(p(0),p(1),...,p(k-1))$,

[TABLE]

(In general,

[TABLE]

As a simple example, consider the stationary Markov chain with state space $K:=\{0,1\}$, defined by the matrix $\Pi$ given by

[TABLE]

and the invariant row probability vector $\hskip 2.84544ptp\hskip 2.84544pt$( meaning $\hskip 2.84544ptp\Pi=p$)

[TABLE]

Now the matrix $\hskip 2.84544pt\Pi^{2}\hskip 2.84544pt$ is equal

[TABLE]

so that for all $\hskip 2.84544ptt\hskip 2.84544pt$ with $\hskip 2.84544pt0<t<1,\hskip 2.84544pt$ $\hskip 2.84544pt\Pi\hskip 2.84544pt$ is irreducible and aperiodic, and then this Markov chain is isomorphic to a Bernoulli shift.

For $n=2$, for example, let $\tau=\sigma^{-1}$, be the transposition defined by

[TABLE]

and take $C:=[[0,1,1]]$ so that if $D:=T_{\sigma}^{-1}C$, then $T_{\sigma}^{-1}D=C$, and hence the following lolds

[TABLE]

and proves that $\mu$ [resp. $\mu\circ T_{\sigma}^{-1}$] is not absolutely continuous with respect to $\mu\circ T_{\sigma}^{-1}$ [resp. $\mu$]

More generally, we have the following

**Proposition 2

*** Any stationary Markov chain with finite state space $K$, with irreducible and aperiodic transition matrix $\Pi$ having at least one zero entry, is isomorphic to Bernoulli but it is not quasi-exhangeable.

Proof Observe first, due to the irreducibility and aperiodicity of $\Pi$, that if $p$ is the row invariant probability vector, then all the coordinates of $p$ are $>0$. Let $i_{0},j_{0}\in K$, such that $\Pi_{i_{0},j_{0}}=0$. Then the following holds

[TABLE]

Let, with notation as in (28),

[TABLE]

and $\hskip 8.5359pt\sigma\in H,\hskip 8.5359pt$ be the transposition defined by

[TABLE]

Then

[TABLE]

But

[TABLE]

so that, $\mu\circ T_{\sigma}^{-1}$[ respectively $\mu$] is not absolutely continuous with respect to $\mu$[ respectively $\mu\circ T_{\sigma}^{-1}$], because

[TABLE]

The proof is achieved because any mixing Markov chain is isomorphic to a Bernoulli System. $\square$

Acknowledgments. Jean Paul Thouvenot posed to me the question whether or not quasi-invariance implies complete positive entropy. He suggests many improvements and complements. I am greatly indebted to him. I am deeply grateful to him for many useful discussions as well as for valuable comments which improve the presentation of the paper, and also for bringing to my attention some appropriate references.

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Sorbonne Université, UMR 8001, Laboratoire de Probabilités, Statistique et Modélisation, Boîte courrier 158, 4 Place Jussieu, F-75752 Paris Cedex 05, France.

E-mail: doureid.hamdan@upmc.fr