Permutation Invariant Strong Law of Large Numbers for Exchangeable Sequences

Abstract

We provide a permutation invariant version of the strong law of large numbers for exchangeable sequences of random variables. The proof consists of a combination of the Komlós–Berkes theorem, the usual strong law of large numbers for exchangeable sequences, and de Finetti’s theorem.

1. Introduction

Kolmogorov’s strong law of large numbers (SLLN) for independent and identically distributed (i.i.d.) sequences of random variables has been generalized into several directions. It has, for example, been generalized for pairwise independent, identically distributed random variables in [1], for nonnegative random variables in [2], for dependent, mixing random variables in [3, 4], and for pairwise uncorrelated random variables in [5].

There is also a version of the SLLN for exchangeable sequences. More precisely, let be an exchangeable sequence of random variables on a probability space , let be its exchangeable -algebra, and let be its tail -algebra. If the sequence is integrable, then the SLLN for exchangeable sequences tells us that is almost surely Cesàro convergent; more precisely, we have the following result.

Proposition 1. Let be an exchangeable sequence of integrable random variables. Then, is -almost surely Cesàro convergent to the limit

This result is well known; see, for example, [6], Example 12.15, or [7], page 185. The goal of this note is to establish the following permutation invariant version of the SLLN for exchangeable sequences.

Theorem 1. Let be an exchangeable sequence of integrable random variables. We set . Then, the following statements are true:(1)For every subsequence and every permutation , the sequence is -almost surely Cesàro convergent to (2)For every permutation and every subsequence , the sequence is -almost surely Cesàro convergent to (3)We have -almost surely for each

Intuitively, the statement of Theorem 1 is plausible. Indeed, de Finetti’s theorem, which is stated as Theorem 3 in the following, provides a connection between exchangeable sequences and conditional i.i.d. sequences, and in the present situation, it implies that the sequence is i.i.d. given or given .

Let us briefly indicate the main ideas for the proof of Theorem 1. Since exchangeability of the sequence is preserved under permutations, by Proposition 1, it follows that the sequences and are almost surely Cesàro convergent. However, it is not clear whether the limits of these two sequences coincide with because their exchangeable -algebras can be different from , and accordingly, their tail -algebras can be different from . Nevertheless, note that, by exchangeability of the sequence, all these limits have the same distribution.

In order to overcome the problem regarding the identification of the limits, we use the Komlós–Berkes theorem (see [8]), which is stated as Theorem 2 in the following. This result is an extension of Komlós’s theorem (see [9]); see also [10], Thm. 5.2.1, for another extension of Komlós’s theorem. The Komlós–Berkes theorem was also used in order to prove the von Weizsäcker theorem (see [11]); see also [10], Thm. 5.2.3, for a similar result and [12] for a note on the von Weizsäcker theorem.

Coming back to the identification of the limits, the Komlós–Berkes theorem provides us with a subsequence such that, for every permutation , the sequence is almost surely Cesàro convergent to the same limit. Using this result, in three steps, we will show that, for every subsequence and every permutation, the corresponding sequence is almost surely Cesàro convergent to the same limit and that this limit is given by . For the identification of the limits, we use results about conditional expectations which are provided in the Appendix section.

2. Proof of the Result

Let be a probability space. We denote by the space of all equivalence classes of integrable random variables. Let be a sequence of random variables. Furthermore, let be the exchangeable -algebra of the sequence , and let be the tail -algebra of the sequence . We assume that the sequence is exchangeable; that is, for every finite permutation , we haveor equivalently, for all , all pairwise different , and all pairwise different , we have

Remark 1. Note that, for every subsequence and every permutation , the sequence is also exchangeable. Accordingly, for every permutation and every subsequence , the sequence is also exchangeable.

Lemma 1. The following statements are true:(1)For every subsequence and every permutation , there exist a permutation and a subsequence such that for all (2)For every subsequence and every permutation , there exists a permutation such that for all

Proof. (1)We define the one-to-one map as Then, there exists a permutation such that for all . Indeed, we define inductively as follows. Let be the unique index such that If are already defined for some , then let be the unique index such that Then, is a permutation. We define the subsequence as for each . Then, we have for each .(2)We define the permutation asThen, we have for all .

For convenience of the reader, we state the Komlós–Berkes theorem and de Finetti’s theorem before we provide the proof of Theorem 1.

Theorem 2. (Komlós–Berkes theorem). Let be a sequence of integrable random variables such that . Then, there exist a subsequence and an integrable random variable such that, for every permutation , the sequence is -almost surely Cesàro convergent to .

Proof. See [8].

Let be a sub--algebra. A sequence of random variables is called independent and identically distributed (i.i.d.) given if for every finite subset and all Borel sets , , we have -almost surelyand for all and every Borel set , we have -almost surely

Theorem 3 (de Finetti’s theorem). Let be a sequence of random variables. Then, the following statements are equivalent:(i)The sequence is exchangeable(ii)There exists a sub--algebra such that is i.i.d. given If the previous conditions are fulfilled, then we can choose or .

Proof. See, for example, Thm. 12.24 of [6].

Now, we are ready to provide the proof of Theorem 1.

Proof. of Theorem 1. By the Komlós–Berkes theorem (see Theorem 2), there exist a subsequence and an integrable random variable such that, for every permutation , the sequence is -almost surely Cesàro convergent to . Now, we proceed with the following three steps: Step 1: first, we show that, for every permutation , the sequence is -almost surely Cesàro convergent to . Indeed, by Lemma 1, there exist a permutation and a subsequence such that for each . By Remark 1 and Proposition 1, we have where denotes the exchangeable -algebra of the sequence and denotes the tail -algebra of the sequence . Furthermore, by Remark 1 and Proposition 1, the sequence is -almost surely Cesàro convergent to the random variable where denotes the exchangeable -algebra of the sequence and denotes the tail -algebra of the sequence . By de Finetti’s theorem (see Theorem 3), we have Since and , by (10), we obtain By exchangeability of the sequence , we have and hence, by Proposition A.1, we obtain -almost surely . In particular, if , then by (11) and de Finetti’s theorem (see Theorem 3), we obtain -almost surely Step 2: now, let be an arbitrary permutation, and let be an arbitrary subsequence. Then, the sequence is -almost surely Cesàro convergent to . Indeed, by Step 1 and de Finetti’s theorem (see Theorem 3), the sequence is -almost surely Cesàro convergent to Furthermore, by Remark 1 and Proposition 1, the sequence is -almost surely Cesàro convergent to the random variable Since , by (16), we obtain By exchangeability of the sequence , we have and hence, by Proposition A.1, we obtain -almost surely . Consequently, the sequence is -almost surely Cesàro convergent to . Step 3: now, let be an arbitrary subsequence, and let be an arbitrary permutation. By Lemma 1, there exists a permutation such that for all . Therefore, by Step 2, the sequence is -almost surely Cesàro convergent to , which concludes the proof.

We can extend the statement of Theorem 1 as follows.

Proposition 2. Let be an exchangeable sequence of integrable random variables. We set . Then, for every subsequence and all permutations , the sequence is -almost surely Cesàro convergent to . Furthermore, we have -almost surely for each .

Proof. By Lemma 1, there exists a permutation such that for all . The mapping given by is also a permutation, and we have for all . Therefore, applying Theorem 1 concludes the proof.

We conclude this section with the following consequence regarding Komlós’s theorem for exchangeable sequences, namely, let be an exchangeable sequence of random variables. Then, Theorem 1 shows that both extensions of Komlós’s theorem (the Komlós–Berkes theorem from [8], which we have stated as Theorem 2, and Thm. 5.2.1 of [10]) are true with the original sequence ; that is, we do not have to pass to a subsequence .

Appendix

Results about Conditional Expectations

We require the following results about conditional expectations. Since these results were not immediately available in the literature, we provide the proofs. For the following, let be a sub--algebra.

Lemma A.1. Let be a square-integrable random variable such that . Then, we have -almost surely .

Proof. Setting , we have , and hence,completing the proof.

Lemma A.2. Let be a nonnegative random variable, and let be a concave function such that -almost surelyThen, we have -almost surely

Proof. By Jensen’s inequality for concave functions and conditional expectations, we have -almost surelySuppose that (A.3) does not hold true. Then, we have -almost surelywhere denotes the convex cone of all equivalence classes of nonnegative random variables. Hence, we obtain -almost surelywhich contradicts (A.2).

Lemma A.3. Let be an integrable random variable such thatThen, we have -almost surely .

Proof. First, we assume that is nonnegative. Let be arbitrary. By (A.7) and Lemma A.2, we have -almost surelyTherefore, by taking into account (A.7), we haveSince , by Lemma A.1 and (A.8), we deduce that -almost surelySince was arbitrary, it follows that -almost surely .
Now, let be arbitrary. Sinceby (A.7), we haveBy the first part of the proof, we deduce that -almost surely and , and hence, .

Proposition A.1. Let and be sequences of random variables, and let be an integrable random variable. We assume that for each and that and as . Then, we have -almost surely .

Proof. Noting that , this is a consequence of Lemma A.3.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

The author gratefully acknowledges the financial support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) (Project no. 444121509).