Exchangeable Processes: de Finetti Theorem Revisited

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1 Exchangeable Processes: de Finetti Theorem Revisited Ehud Lehrer and Dimitry Shaiderman January 16, 2018 Abstract: A sequence of random variables is exchangeable if the joint distribution of any finite subsequence is invariant to permutations. de Finetti Theorem states that every exchangeable infinite sequence is a convex combination of i.i.d. processes. In this paper we explore the relationship between exchangeability and frequency-dependent posteriors. We show that any real-valued stationary process is exchangeable if and only if its posteriors depend only on the empirical frequency of past events. Classification numbers: 60A05, 60A10, 91A26 Keywords: Exchangeable process; de Finetti Theorem; Stationary process; Permutationsinvariant posteriors; Frequency-dependent posteriors; Urn Schemes. Tel Aviv University, Tel Aviv 69978, Israel and INSEAD, Bd. de Constance, Fontainebleau Cedex, France. lehrer@post.tau.ac.il. Lehrer acknowledges the support of the Israel Science Foundation, Grant 963/15. Tel Aviv University, Tel Aviv 69978, Israel. dimitrys@mail.tau.ac.il.

2 1 Introduction A sequence px n q 8 n1 of random variables is said to be exchangeable if the distribution of any finite cylinder is invariant to permutations, i.e., px 1,..., X m q d px πp1q,..., X πpmq q for every m P N and π P S m (the set of permutations over t1,..., mu). The notion of exchangeability turned out to be an important cornerstone in the justification of Bayesian statistics. This fact is due to the remarkable representation theorem of de Finetti [1]. This theorem states that if a process is exchangeable, then and only then, it is a convex combination of i.i.d. processes. That is, an exchangeable process is an i.i.d. process with unknown parameters selected according to a distribution over the set of parameters. Kreps [8] summarizes the importance of this result as follows:...de Finetti s theorem, which is, in my opinion, the fundamental theorem of statistical inference the theorem that from a subjectivist point of view makes sense out of most statistical procedures. Through the years there has been an effort to find other characterizations of exchangeability. A famous result is that of Kallenberg [5]. This result states that a stationary process is exchangeable if and only if its past samples affect the distribution of any future observation in the same manner, i.e., X n 1 X 1,..., X d n X n k X 1,..., X n for every n 0, k 1. Our interest in exchangeability arose in a recent work of Lehrer and Teper [7] who studied the relations between decision making theory and Bayesian updating. Consider a decision maker who updates her preferences. Suppose that after two histories that share the same empirical distributions she has the same preferences. The question arises as to whether a stationary process whose posteriors after any two histories that share the same frequency coincide is necessarily exchangeable. In section 3 we deal with discrete valued stationary sequences. We introduce two properties, called permutation invariant posteriors (PIP) and positivity, and show that any discrete valued stationary process is exchangeable if and only if it satisfies these properties. We then move on in section 4 to deal with real valued random processes. We give a natural generalization to the PIP property to the continuous case by defining the strong 1

3 PIP property. As it turns out, this property together with stationary are necessary and sufficient to the deduction of exchangeability. 2 Definitions and Existing Results Let px n q 8 n1 be a sequence of random variables defined over the probability space pω, F, Prq taking values in pr, BpRqq. Definition 1. The random sequence px n q 8 n1 is said to be exchangeable if the distribution of finite cylinders, is invariant under permutations of coordinates, i.e: px 1,..., X m q d px πp1q,..., X πpmq q for every m P N and π P S m. The seminal de Finetti Theorem [1] is the following: Theorem (de Finetti). A binary sequence px n q 8 n1 is exchangeable if and only if there exists a distribution function F on 0, 1 such that for all n P N: where x n 1 n measure F. ņ i1 PrpX 1 x 1,..., X n x n q θ nxn p1 θq n nxn df pθq» 1 0 x i is the sample mean. The sequence px n q 8 n1 is said to have de-finetti Hewitt and Savage [3] generalized de Finetti Theorem. Theorem (Hewitt and Savage). Let px n q 8 n1 be a sequence of exchangeable random variables taking values in some complete metric space X. Then there exists a unique probability measure ν on the set of probability measures P X on X, such that for any Borel sets ta i u in X :» Pr X 1 P A 1,..., X n P A n QpA 1 q QpA n q ν dq When a random sequence satisfies the conclusion of the theorem, it is said to have de Finetti measure ν. 2

4 Definition 2. The random sequence px n q 8 n1 is said to be stationary if its distribution is invariant under time shifts, that 0 and A P BpR 8 q we have: PrppX 1, X 2,...q P Aq PrppX k, X k 1,...q P Aq The next definition was introduced by Berti et. al. [4], following the results of Kallenberg [5]: Definition 3. The random sequence px n q 8 n1 is said to be conditionally identically distributed if: X n 1 X 1,..., X d n X n k X 1,..., X n for every n 0, k 1 That is, past events, effect the distribution of all the future steps in the sequence in the same manner. The following characterization is due to Kallenberg [5]. Theorem (Kallenberg). A sequence X 1, X 2,... of random variables is exchangeable iff it is stationary and conditionally identically distributed. Recently, Lehrer and Teper [7] dealt with Bayesian updating in a dynamic decision making setup. The question regarding the relation between stationarity and exchangeability rose again. They posed the question whether a discrete valued stationary stochastic process is necessarily exchangeable whenever any two posteriors that follow histories sharing the same frequency coincide. With the help of an additional property, we provide an affirmative answer. 3 The Main Result - Discrete Case Let px n q 8 n1 be a discrete valued random sequence defined on the probability space pω, G, Prq taking values in Γ 8 : Γ Γ..., where Γ R is a countable set. Define H n tph 1,..., h n q; h i P nu be the set of all finite histories of length n, and set H n H n. Let µ be the probability measure induced on pγ 8, σ phqq 1 by px n q 8 n1, that is: µ pph 1,..., h n qq Pr px 1 h 1,..., X n h n P N, ph 1,..., h n q P H n. We shall now define the Permutation Invariant Posteriors (PIP) property that captures the notion that any two posteriors that follow histories sharing the same frequency coincide. 1 σ phq is the σ algebra generated by H 3

5 3.1 The PIP property For each n P N define the set: ( H n : ph 1,..., h n q P H n ; µ ph πp1q,..., h πpnq q P S n The set H n contains all positive probability finite histories of length n, with the property that all finite histories with the same empirical frequency are feasible as well, i.e of positive probabilty. We next introduce the new property which plays a major role in our characterization. Definition 4. A stationary discrete valued random sequence px n q 8 n1 satisfies Permutation Invariant Posteriors (PIP) P ph 1,..., h m q P H m P S m : X m 1 ph 1,..., h m q d X m 1 ph πp1q,..., h πpmq q. The property PIP guarantees that the distribution of the random variable that follows a finite sequence of outcomes (i.e., the posterior) depends only on their empirical frequency, and not on their order. The main interest of our work lies in the relation between exchangeable processes and ones that satisfy the stationarity and PIP conditions. Notice that by de Finetti Theorem each exchangeable sequence must be stationary and satisfy the PIP condition. As for the inverse, we first show that stationary - PIP sequences need not be exchangeable. Example 1. Define the random sequence px n q 8 n1 inductively by: $ & X 1 U pt0, 1, 2uq % X n px n 1 1q pmod 3q. Notice that since H m 2, the PIP condition follows. Stationarity follows too as the outcome of the sequence is determined solely based on X 1, having uniform distribution. Nevertheless, Exchangeability does not hold as, PrpX 1 0, X 2 1, X 3 2q PrpX 1 1, X 2 0, X 3 2q. From now on we will focus on stationary and PIP processes where positive probability is retained under all permutations. The latter property is explored in the following subsection. 4

6 3.2 The Positivity property Definition 5. We say that a discrete valued random sequence px n q 8 n1 has the positivity property P N and for each ph 1,..., h m q P H m : µ pph 1,..., h m qq 0 ùñ µ ph πp1q,..., h πpmq q P S m. I.e., the class of positive-probability finite length histories is invariant under all permutation of the indices. A discrete valued random process is said to be positive if it satisfies the positivity property. The following example shows that the PIP and positivity properties alone are not sufficient for exchangeability. Example 2. Our example can be found in [2]. Assume we have an urn with one red ball and two black balls. At each stage we draw a ball and replace it by two balls with the same color as the ball just drawn. The probability drawing a red ball in each stage is the proportion of black balls in the urn at that time. Let X n be the color of the ball drawn at stage n. By the construction the random sequence px n q 8 n1 satisfies both PIP and positivity properties. A simple calculation shows that PrpX 1 red, X 2 blackq which differs from 3 PrpX 1 black, X 2 redq Hence, this random sequence is not exchangeable. 4 To conclude the discussion regarding the insufficiency of a partial set of the discussed properties for exchangeability, let us now demonstrate that stationarity and positivity do not imply exchangeability. Example 3. Consider a Markov Chain on a two - states space t0, 1u with the following transition matrix: T and initial state distribution PrpX 1 0q 1 PrpX 1 1q 4. As the distribution 19 v p 4 q satisfies the equation v vt, we get the v is a stationary distribution vector of, the chain, and thus the resulting random sequence px n q 8 n1 is stationary. By construction

7 it is also positive. However we have: PrpX 1 1, X 2 0, X 3 0q The sequence px n q 8 n1 is therefore not exchangeable PrpX 1 0, X 2 1, X 3 0q. From now on we will deal only with stationary processes that satisfy both positivity and PIP assumptions. Such processes will be referred to as stationary and positive PIP. 3.3 Discrete Case - The main theorem It is clear that any exchangeable process is stationary and positive PIP. Our main theorem establishes the inverse direction. Theorem 1. A discrete valued random sequence is exchangeable iff it stationary and positive PIP. 3.4 The Proof The first step in the proof is to show that a stationary and positive PIP sequence can be extended backwards in time. We then use Levy s upwards theorem to understand the asymptotic behaviour of such processes. It turns out that the result of the discrete case is important in its own right, as it allows to give a straightforward and implicit criteria for exchangeability in the urn schemes introduced by Hill et. al. [2] (see Subsection 3.5 below). Section 4 extends our results to the continuous case. Theorem (Extension Theorem). Assume px n q 8 n1 is a stationary and positive PIP random sequence. There exist random variables X 0, X 1, X 2,... such P N Y t0u: px n, X n 1,...q d px 1, X 2,...q. In P N Y t0u the random sequence px n, X n 1,...q is stationary and positive PIP as well. 6

8 Proof. This result follows by Lemma 9.12 in [6], which states that a stationary sequence can be extended backwards to a two sided stationary sequence px k q kpz, with the property that px n, X n 1,...q d px 1, X P N Y t0u. From this point up to the end of this section, let px k q kpz be the extension, given by the extension theorem, of the stationary and positive PIP discrete valued sequence px n q 8 n1. We need a few notations. Let Γ 8 : 8 n1 i1 8 tx n h i h i P Γu. Γ 8 can be thought of as the set of infinite histories occurring in the past. Denote by h ph 1, h 2,...q P Γ 8 the event tx 1 h 1, X 2 h 2,...u. We will refer to each h ph 1, h 2,...q P Γ 8 as a history. For any h P Γ 8 and n P N let h n tx n h n,..., X 2 h 2, X 1 h 1 u. Finally, F n : σpx n,..., X 1 q is the σ algebra generated by X n,..., X 1 and F 8 : σ F n be n 1 the σ-algebra generated by F n. n 1 Levy upwards theorem implies that for Pr almost every h P Γ 8 the sequence Pr px 0 γ h n q converges as n Ñ 8. Denote the limit by Pr px 0 γ hq. The next lemma is a cornerstone of our proof. It supports the intuition that Pr px 0 γ hq is the asymptotic " rate of occurrence of the * letter γ in the history h. For each γ P Γ define the set A γ h P Γ 8 ; Pr px 0 γ hq 0. Lemma 1. Pr almost every history h P A γ contains the letter γ infinitely many times. Proof. For each k P Z define C k tx k γ, X l ku. The sets pc k q kpz are disjoint. By stationarity all pc k q kpz have the same probability. Thus, Pr C k 0. (1) kpz By eq. (1), Pr almost every h P Γ 8 either contains no γ letters, or contains an infinite number of them. Define, B 1 tx n P Nu. We obtain that Pr almost every h P Γ 8 zb 1 contains infinitely many γ letters. In case Pr pb 1 q 0 the proof is complete. Otherwise, Pr pb 1 q 0. It is sufficient to prove that for Pr almost every h P B 1, Pr px 0 γ hq 0. (2) 7

9 Assume the contrary. It implies that there exists D B 1 with PrpDq 0 such that for Pr almost every h P D, Pr px 0 γ hq 0. Thus, Dn and D n D with PrpD n q 0 such that Pr px 0 γ hq 1 n for Pr almost every h P D n. By Pr-a.s convergence of! Pr px 0 γ h n q, there exists M 0 such that the set D n,m : h P D n ; Pr px 0 γ h n q ) M has positive probability. Using stationarity, we get: 2n Pr px 0 γ D n,mq PrpD n,mq lim Pr px 0 γ h n q Pr ph n q nñ8 th n ; hpd n,m u 1 2n lim nñ8 th n ; hpd n,m u 1 2n PrpD n,mq, implying Pr px 0 γ D n,mq 1 2n. We thus obtain, Pr px 0 γ B 1 q Pr px 0 γ D n,mq Pr pd n,mq Pr pb 1 q Pr ph n q 1 Pr pd n,mq 2n Pr pb 1 q 0. (3) eq. (3) stands in contradiction with 0 PrpC 0 q Pr px 0 γ B 1 q PrpB 1 q, which implies that Pr px 0 γ B 1 q 0. Proposition 1. For every γ 1, γ 2 P Γ we have, 2 E 1pX 0 γ 1, X 1 γ 2 q F 8 E 1pX 0 γ 2, X 1 γ 1 q F 8 (4) Proof. By Levy s upwards theorem, eq. (4) is equivalent to E 1pX 0 γ 1, X 1 γ 2 q F n lim nñ8 E 1pX 0 γ 2, X 1 γ 1 q F n 0 which is true if and only if for Pr almost every history h P Γ 8 we have Pr X 0 γ 1, X 1 γ 2 h n Pr X 0 γ 2, X 1 γ 1 h n ÝÝÝÑ nñ8 0, By the stationarity assumption, the latter holds if and only if for Pr almost every history h P Γ 8 we have Pr X 0 γ 1 h n Pr X 0 γ 2 ph n, γ 1 q Pr X 0 γ 2 h n Pr X 0 γ 1 ph n, γ 2 q ÝÝÝÑ nñ pF q stands for the indicator function of the set F. 8

10 Using the positivity and PIP and properties we get that eq. (4) holds if and only if for Pr almost every history h P Γ 8 : Pr X 0 γ 1 h n Pr X 0 γ 2 pγ 1, h n q Pr X 0 γ 2 h n Pr X 0 γ 1 pγ 2, h n q ÝÝÝÑ nñ8 0. Consider now the sets A γ1 and A γ2 (recall the notation introduced just before Lemma 1). The obove criteria obviously holds on pa γ1 Y A γ2 q c. For A γ1 X A γ2, Lemma 1 implies that Pr almost every h P A γ1 X A γ2 (5) contains each of the γ i letters (i 1, 2) infinitely many times. Let tn k u k 1 and tn l u l 1 be two subsequences such that h nk γ 1 and h nl γ 2. Thus, and PrpX 0 γ 2 pγ 1, h n k 1 qq PrpX 0 γ 2 h n k q ÝÝÝÑ kñ8 PrpX 0 γ 1 pγ 2, h n l 1 qq PrpX 0 γ 1 h n l q ÝÝÑ lñ8 PrpX 0 γ 2 hq, PrpX 0 γ 1 hq. Since Levy s upward theorem ensures that the limit is Pr-a.s unique, we get: lim PrpX 0 γ 1 h n qprpx 0 γ 2 pγ 1, h n qq lim PrpX 0 γ 1 h nk 1 qprpx 0 γ 2 h n k q nñ8 kñ8 Prpγ 1 hqprpγ 2 hq. Analogously, lim nñ8 PrpX 0 γ 2 h n qprpx 0 γ 1 pγ 2, h n qq Prpγ 2 hqprpγ 1 hq, establishing eq. (5) for Pr almost every h P A γ1 X A γ2. Finally let us show that the criteria holds for Pr almost every h P A γ1 X pa γ2 q c (the case A γ2 X pa γ1 q c follows in a similar fashion). By Lemma 1 we have that lim PrpX 0 γ 2 h n qprpx 0 γ 1 ph n, γ 2 qq 0 nñ8 Lemma 1 also implies the existence of a subsequence tn k u k 1 such that h nk γ 1. Thus by the Pr-a.s uniqueness of the limit argument we get: lim PrpX 0 γ 1 h n qprpx 0 γ 2 pγ 1, h n qq lim PrpX 0 γ 1 h nk 1 qprpx 0 γ 2 h n k q nñ8 kñ8 which concludes the proof of the proposition. Prpγ 1 hqprpγ 2 hq 0, The next corollary provides a property that guarantees that a discrete stationary and positive PIP sequence is exchangeable. 9

11 Corollary 1. Let w ph 1,..., h n q P H n with µpwq 0. Then, µpx n 1 γ 1, X n 2 γ 2 w µ X n 1 γ 2, X n 2 γ 1 w (6) for every γ 1, γ 2 P Γ. Proof. By stationarity tx n h 1,...X 1 h n u P F n F 8 is of positive probability. Thus, by eq. (4), µppw, γ 1, γ 2 qq E 1pX n h 1,..., X 1 h n, X 0 γ 1, X 1 γ 2 q E E 1pX 0 γ 1, X 1 γ 2 q F 8 1pX n h 1,..., X 1 h n q E E 1pX 0 γ 2, X 1 γ 1 q F 8 1pX n h 1,..., X 1 h n q E 1pX n h 1,..., X 1 h n, X 0 γ 2, X 1 γ 1 q µppw, γ 2, γ 1 qq. This implies µppγ 1, γ 2 q w µ pγ 2, γ 1 q w, as desired. We are now in a position to prove the main result of this section, Theorem 1. Proof of Theorem 1 A discrete valued exchangeable random sequence is clearly stationary and positive PIP. For the other direction let us first define for each word h ph 1,..., h n q P H n and permutation π P S n, π phq : ph πp1q,..., h πpnq q. The proof will follow by induction on the length of the words. Let n 2. By taking the expected value on eq. (4), we get µppγ 1, γ 2 qq µppγ 2, γ 1 qq for any γ 1, γ 2 P Γ. To prove the induction step assume µpph n, γ 1 qq 0 for some word h n P H n. For π P S n, we have by the induction hypothesis, together with positivity and PIP, µppπph n q, γ 1 qq µpγ 1 πph n qqµpπph n qq µpγ 1 h n qµph n q µpph n, γ 1 qq. (7) Next, fix π P S n 1 and assume πph n, γ 1 q ps n, γ 2 q for some γ 2 γ 1 and some s n P H n. There exists w n 1 P H n 1 and ρ P S n such that ρppw n 1, γ 1 qq s n. By the induction step we have µps n q µppw n 1, γ 1 qq. Combining the positivity and PIP properties together with 10

12 Corollary 1 we get, µpπph n, γ 1 qq µpps n, γ 2 qq µpγ 2 s n qµps n q µpγ 2 pw n 1, γ 1 qqµppw n 1, γ 1 qq µppγ 1, γ 2 q w n 1 qµpw n 1 q µppγ 2, γ 1 q w n 1 qµpw n 1 q µppw n 1, γ 2, γ 1 qq µpph n, γ 1 qq, where the last equality is due to eq. (7). For events pa 1... A m q Γ m of positive probability and π P S m we have: µ ppa 1... A m qq... µ pps 1,..., s m qq s 1 PA 1 s mpa m... µ ps πp1q,..., s πpmq q s πp1q PA πp1q s πpmq PA πpmq µ pa πp1q... A πpmq q, as desired. 3.5 Application to Urn Schemes Bruce et. al. [2] tried to give a characterization to exchangeable urn schemes in terms of their de Finetti measures. They defined the notion of generalized t-color urn schemes as follows: Definition 6. Suppose Y ty 1, Y 2,...u is a sequence of random variables with possible values in t1,..., tu. The sequence Y is a generalized t - color urn scheme (each ball drawn from the urn is replaced by two balls of the same color) if for n 1, 2,... and 1 j t the conditional probability Pr Y n 1 j Y 1 c 1,..., Y n c n that the next ball drawn is of color j depends only on the proportion of balls of each color at stage n. That is, Pr Y n 1 j Y 1 c 1,..., Y n c n fj r1 n 1 m n, r 2 n 2 m n,..., r t n t m n, (8) where r i is the number of balls of color i in the urn initially, n i the number of c j s equal to i, and m t r i the total number of balls in the urn initially. i1 11

13 In their work Bruce et. al. [2] found a characterization for exchangeable generalized t-color urn schemes with t 2 but were unable to generalize it for t 3. eq. (8) clearly ensures that generalized t-color urn schemes satisfy the PIP property. A straightforward implementation of Theorem 1 thus yields the following observation. Proposition 2. A generalized stationary t-color urn scheme which satisfies the positivity property is exchangeable. 4 The Continuous Case In this section we will deal with continuous real valued random sequences px n q 8 n1 defined on a probability space pω, F, Prq. Let µ be the probability measure induced on pr 8, BpR 8 qq by px n q 8 n1. We will define in subsection 4.1 the strong PIP property for continuous processes. As it turns out the strong PIP property together with stationarity ensures exchangeability, see Theorem 2. Subsection 4.3 is devoted to the proof of the new criteria given in Theorem The strong PIP property We wish to define a continuous variant of the PIP property in order to get a similar type characterization of exchangeable processes to that in Theorem 1. A natural way to translate the PIP property to the continuous case is the following: Define first the sets: ( O m : pa 1... A m q P BpR m q; µpa πp1q... A πpmq q P S m. The set O m contains all the events of the form pa 1... A m q P BpR m q that keep having positive probability under any permutation of the coordinates. Definition 7. A stationary random sequence px n q 8 n1 satisfies the Strong Permutation Invariant Posteriors property (strong PIP) P pa 1... A m q P O m P S m : µ px m 1 P A A 1... A m q µ X m 1 P A A πp1q... A P BpRq. Notice that in the case of discrete-valued processes the strong PIP property implies the PIP property. As for the other direction notice that the stationary PIP process in Example 12

14 1 does not satisfy the strong PIP property. The reason is that the set t0, 1, 2u m 1 t2u is a member of O m, but µ X m 1 0 t0, 1, 2u m 1 t2u 1 0 µ X m 1 0 t0, 1, 2u m 2 t2u t0, 1, 2u. Hence, the strong PIP property was given its name for a reason. 4.2 The Continuous Case - The main theorem Theorem 2. A real valued random sequence is exchangeable iff it is stationary and satisfies the strong PIP property. 4.3 The Proof We start the proof of Theorem 2 with the following technical lemma. Lemma 2. Assume px n q 8 n1 is a discrete-valued stationary sequence. Let w ph 1,..., h n q P H n with µpwq 0. P t1,..., nu there exists k P N and a word s ph 1,..., h n,..., h i q P H n k with µpsq 0 (i.e. µpx 1 h 1, X 2 h 2,..., X n h n, X n k h i q 0). Proof. For each m P N define C m tx m h i, X l h mu. The sets pc m q mpn are disjoint. By stationarity all pc m q mpn have the same probability. Thus, Pr C m 0. (9) mpn Eq. (9) implies that with probability 1 each word contains either infinite number of h i s or none. Define, 3 N 1 w X tx n 1 h i u N k w X tx n 1 h i u X... X tx n k 1 h i u X tx n k h i u, k 2 Since w contains the letter h i and the sets tn k u kpn are disjoint we have: 8 0 µpwq µ N k µ pn k q. 3 We abuse notation: w stands also for the event tx 1 h 1,..., X n h n u. kpn k1 13

15 Hence there exists k such that µ pn k q 0. The latter implies the existence of s P H n k of the form s ph 1,..., h n,..., h i q with µpsq 0. The next lemma gives an insight into the special structure of stationary and strong PIP sequences. Lemma 3. Assume that px n q 8 n1 is a discrete-valued stationary strong PIP sequence. Let w pw 1,..., w n q P H n with µpw πp1q,..., w πpnq q P S n. Then w Ω m P O n m for all m P N. Proof. The proof is by induction on n. Let n 1. If µpw 1 q 0, stationarity implies w 1 Ω m P O 1 m for all m P N. The induction step: Assume by contradiction that there exists a permutation ti 1,..., i n u of t1,..., nu and non-negative integers j 1,..., j n 1 such that, µ Ω j 1 w i1 Ω j 2... Ω jn w in Ω j n 1 0, (10) so that if j k 0 for some k n, then w ik 1 Ω j k w ik w ik 1 w ik. Stationarity together with eq. (10) implies: µ w i1 Ω j 2... Ω jn w in 0. (11) Since µpw πp1q,..., w πpnq q P S n, stationarity implies µpw ρpi1 q,..., w ρpin 1 qq P S n 1. Thus by the induction step w i1 Ω j 2... w in 1 Ω jn P O n 1 j2... j n. By eq. (11), the strong PIP property and stationarity we get: 0 µ w in w i1 Ω j 2... w in 1 Ω jn µ w in Ω j 2... j n w i1... w in 1 µ w i 1,..., w in 1, w in, µ w i1,..., w in 1 implying µ w i1,..., w in 1, w in 0, thus contradicting the assumption of the lemma. Lemma 4. Assume that px n q 8 n1 is a discrete-valued stationary strong PIP sequence. Let h ph 1,..., h n q P H n with µphq 0. Then, for every permutation ti 1,..., i n u we have: µ ph i1,..., h in q 0. (12) 14

16 Proof. The proof is by induction on n. The case where n 1 follows trivially. The induction step: let ti 1,..., i n u be a permutation of t1,..., nu. An iterative use of Lemma 2 guaratntees that there exists m P N and a word s P H m of the form: s ph 1,..., h n,..., h i1,..., h i2,..., h in 1,..., h in q, such that µpsq 0. Hence there exists w P H m n with w ph i1,..., h i2,..., h in 1,..., h in q and µpwq 0. This implies the existence of numbers tj p up1 n 1 n 1 0 with j p m 2n such that µ h i1 Ω j 1... Ω j n 1 h in 0. (13) By the induction step we have that µph ρpi1 q,..., h ρpin 1 qq P S n 1, which by Lemma 3 implies h i1 Ω j 1... h in 1 Ω j n 1 P O m n 1. Thus, by eq. (13), the strong PIP property and stationarity we get: 0 µ h in h i1 Ω j 1... h in 1 Ω j n 1 µ h in Ω m 2n h i1... h in 1 µ h i 1,..., h in 1, h in, µ h i1,..., h in 1 which implies µ h i1,..., h in 1, h in 0. This completes the proof. Proposition 3. A discrete-valued stationary process px n q npn is exchangeable iff it satisfies the strong PIP property. Proof. Assume px n q npn satisfies the strong PIP property. By Lemma 4 it satisfies the positivity property. Thus, Theorem 1 implies that it is exchangeable. The inverse direction follows easily from de Finneti Theorem. We are now in a position to prove Theorem 2. p1 Proof of Theorem 2 As in the discrete case it follows from the generalized de Finetti representation theorem that real - valued exchangeable random sequences are stationary and strong PIP. For the other direction define Zi n l " l 1 2 on the event X n i P 2, l * n 2 n for any l P Z and n P N. The sequence pzi nq8 i1 is a discrete random sequence. Also as px i q 8 i1 is a stationary and strong PIP sequence, it follows that pz n i q8 i1 is one as well for 15

17 every n P N. Thus, by Corollary 3 we get that pz n i q8 i1 is exchangeable for all n P N. The latter yields that the sequence px i q 8 i1 is exchangeable on dyadic cubes 4. Indeed for every m, n P N and any permutation π P S m we get: l1 1 µ 2, l 1 lm 1,...,, l m Pr Z n n 2 n 2 n 2 n 1 l 1 2,..., n Zn m l m 2 n Pr Z1 n l πp1q 2,..., n Zn m l πpmq (14) 2 n lπp1q 1 µ, l πp1q lπpmq 1,...,, l πpmq. 2 n 2 n 2 n 2 n As the restriction of µ to R m, BpR m q is a regular measure, and as any open set in R m can be decomposed to a disjoint union of dyadic cubes, we have the following identity: µpbq inf # 8 k0 µpq k q; Q k R m are disjoint dyadic cubes and B k Q k +. (15) For each π P S m define the map πpx 1,..., x m q : px πp1q,..., x πpmq q. By eq. (14) we have that for each dyadic cube Q R m : µpqq µpπpqqq, (16) where πpqq tπpx 1,..., x m q; px 1,..., x m q P Qu. Combining eqs. (15) and (16), we get that for every π P S m, µ ppa 1... A m qq µ pπpa 1... A m qq µ pa πp1q... A πpmq q, proving that real-valued stationary and strong - PIP sequences are exchangeable. completes the proof of Theorem 2. This References [1] Finetti, B. de (1931), Fuzione caratteristica di un fenomeno aleatorio, JAtti Acc. Naz. Lincei 4, [2] Bruce M. Hill, David Lane and William Sudderth (1987), Exchangeable Urn Processes, The Annals of Probability, Vol. 15, No. 4: A dyadic cube Q R m is a set of the form z 2 n 16 1 m 0, for some n P N, z P Z m 2 n.

18 [3] Hewitt, E. and Savage, L.J. (1955), Symmetric measures on Cartesian products, Trans. Amer. Math. Soc. 80, [4] P. Berti, L. Pratelli, and P. Rigo (2004), Limit theorems for a class of identically distributed random variables, The Annals of Probability, 32: [5] Kallenberg, O. (1988), Spreading and predictable sampling in exchangeable sequences and processes, Annals of Probability, 16: [6] Kallenberg, O., Foundations of Modern Probability, Springer, [7] Lehrer, E. and Teper, R. (2015), The dynamics of preferences, predictive probabilities, and learning, mimeo. [8] Kreps, D. M. (1988), Notes on the Theory of Choice, Westview Press. 17