Permutation invariant strong law of large numbers for exchangeable sequences

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 Koml\'{o}s-Berkes theorem, the usual strong law of large numbers for exchangeable sequences and de Finetti's theorem.


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 [2], for nonnegative random variables in [3], for dependent, mixing random variables in [9,10], and for pairwise uncorrelated random variables in [4].
There is also a version of the SLLN for exchangeable sequence. More precisely, let (ξ n ) n∈N be an exchangeable sequence of random variables on a probability space (Ω, F , P), let E be its exchangeable σ-algebra, and let T be its tail σ-algebra. If the sequence (ξ n ) n∈N is integrable, then the SLLN for exchangeable sequences tells us that (ξ n ) n∈N is almost surely Cesàro convergent; more precisely, we have the following result.
1.1. Proposition. Let (ξ n ) n∈N ⊂ L 1 be an exchangeable sequence of integrable random variables. Then (ξ n ) n∈N is P-almost surely Cesàro convergent to the limit This result is well-known; see, for example [7,Example 12.15] or [6, page 185]. The goal of this note is to establish the following permutation invariant version of the SLLN for exchangeable sequences.
1.2. Theorem. Let (ξ n ) n∈N ⊂ L 1 be an exchangeable sequence of integrable random variables. We set ξ := E[ξ 1 |E ]. Then the following statements are true: (1) For every subsequence (n k ) k∈N and every permutation π : N → N the sequence (ξ n π(k) ) k∈N is P-almost surely Cesàro convergent to ξ. (2) For every permutation σ : N → N and every subsequence (m k ) k∈N the sequence (ξ σ(m k ) ) k∈N is P-almost surely Cesàro convergent to ξ.
Intuitively, the statement of Theorem 1.2 is plausible. Indeed, de Finetti's theorem, which is stated as Theorem 2.4 below, provides a connection between exchangeable sequences and conditional i.i.d. sequences, and in the present situation it implies that the sequence (ξ n ) n∈N is i.i.d. given E or given T .
Let us briefly indicate the main ideas for the proof of Theorem 1.2. Since exchangeability of the sequence is preserved under permutations, by Proposition 1.1 it follows that the sequences (ξ n π(k) ) k∈N and (ξ σ(m k ) ) k∈N are almost surely Cesàro convergent. However, it is not clear whether the limits of these two sequences coincide with ξ, because their exchangeable σ-algebras can be different from E , and accordingly their tail σ-algebras can be different from T . 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 [1]), which is stated as Theorem 2.3 below. This result is an extension of Komlós's theorem (see [8]); see also [5,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 [13]); see also [5,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 (n k ) k∈N such that for every permutation π : N → N the sequence (ξ n π(k) ) k∈N 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 ξ. For the identification of the limits we use results about conditional expectations which are provided in Appendix A.

Proof of the result
Let (Ω, F , P) be a probability space. We denote by L 1 = L 1 (Ω, F , P) the space of all equivalence classes of integrable random variables. Let (ξ n ) n∈N be a sequence of random variables. Furthermore, let E be the exchangeable σ-algebra of the sequence (ξ n ) n∈N , and let T be the tail σ-algebra of the sequence (ξ n ) n∈N . We assume that the sequence (ξ n ) n∈N is exchangeable; that is, for every finite permutation π : N → N we have or equivalently, for all k ∈ N, all pairwise different n 1 , . . . , n k ∈ N and all pairwise different m 1 , . . . , m k ∈ N we have P • (ξ n1 , . . . , ξ n k ) = P • (ξ m1 , . . . , ξ m k ).

2.1.
Remark. Note that for every subsequence (n k ) k∈N and every permutation π : N → N the sequence (ξ n π(k) ) k∈N is also exchangeable. Accordingly, for every permutation σ : N → N and every subsequence (m k ) k∈N the sequence (ξ σ(m k ) ) k∈N is also exchangeable.

2.2.
Lemma. The following statements are true: (1) For every subsequence (n k ) k∈N and every permutation σ : N → N there exist a permutation π : N → N and a subsequence (m k ) k∈N such that σ(m k ) = n π(k) for all k ∈ N.
(2) For every subsequence (n k ) k∈N and every permutation π : N → N there exists a permutation σ : N → N such that σ(n k ) = n π(k) for all k ∈ N. Proof.
Then π is a permutation. We define the subsequence (m k ) k∈N as m k := τ (π(k)) for each k ∈ N. Then we have σ(m k ) = n π(k) for each k ∈ N.
Then we have σ(n k ) = n π(k) for all k ∈ N.
For convenience of the reader, we state the Komlós-Berkes theorem and de Finetti's theorem, before we provide the proof of Theorem 1.2.

Theorem (Komlós-Berkes theorem)
. Let (ξ n ) n∈N ⊂ L 1 be a sequence of integrable random variables such that sup n∈N E[|ξ n |] < ∞. Then there exist a subsequence (n k ) k∈N and an integrable random variable ξ ∈ L 1 such that for every permutation π : N → N the sequence (ξ n π(k) ) k∈N is P-almost surely Cesàro convergent to ξ.
Let G ⊂ F be a sub σ-algebra. A sequence (ξ n ) n∈N of random variables is called independent and identically distributed (i.i.d.) given G if for every finite subset I ⊂ N and all Borel sets B i ∈ B(R), i ∈ I we have P-almost surely and for all n, m ∈ N and every Borel set B ∈ B(R) we have P-almost surely P(ξ n ∈ B|G ) = P(ξ m ∈ B|G ). (identical distributions given G ) 2.4. Theorem (De Finetti's theorem). Let (ξ n ) n∈N be a sequence of random variables. Then the following statements are equivalent: (i) The sequence (ξ n ) n∈N is exchangeable. (ii) There exists a sub σ-algebra G ⊂ F such that (ξ n ) n∈N is i.i.d. given G .
If the previous conditions are fulfilled, then we can choose G = E or G = T .
Proof. See, for example [7,Thm. 12.24]. Now, we are ready to provide the proof of Theorem 1.2.
Proof of Theorem 1.2. By the Komlós-Berkes theorem (see Theorem 2.3) there exist a subsequence (n k ) k∈N and an integrable random variable ξ ∈ L 1 such that for every permutation π : N → N the sequence (ξ n π(k) ) k∈N is P-almost surely Cesàro convergent to ξ. Now, we proceed with the following three steps: Step 1: First, we show that for every permutation σ : N → N the sequence (ξ σ(n) ) n∈N is P-almost surely Cesàro convergent to ξ. Indeed, by Lemma 2.2 there exist a permutation π : N → N and a subsequence (m k ) k∈N such that σ(m k ) = n π(k) for each k ∈ N. By Remark 2.1 and Proposition 1.1 we have ξ = E[ξ n π(1) | E (n π(k) ) k∈N ] = E[ξ n π(1) | T (n π(k) ) k∈N ], (2.1) where E (n π(k) ) k∈N denotes the exchangeable σ-algebra of the sequence (ξ n π(k) ) k∈N , and T (n π(k) ) k∈N denotes the tail σ-algebra of the sequence (ξ n π(k) ) k∈N . Furthermore, by Remark 2.1 and Proposition 1.1 the sequence (ξ σ(n) ) n∈N is P-almost surely Cesàro convergent to the random variable where E (σ(n)) n∈N denotes the exchangeable σ-algebra of the sequence (ξ σ(n) ) n∈N , and T (σ(n)) n∈N denotes the tail σ-algebra of the sequence (ξ σ(n) ) n∈N . By de Finetti's theorem (see Theorem 2.4) we have Since E (n π(k) ) k∈N ⊂ E (σ(n)) n∈N and T (n π(k) ) k∈N ⊂ T (σ(n)) n∈N , by (2.1) we obtain By exchangeability of the sequence (ξ n ) n∈N we have and hence, by Proposition A.4 we obtain P-almost surely ξ = η. In particular, if σ = Id, then by (2.2) and de Finetti's theorem (see Theorem 2.4) we obtain P-almost surely Step 2: Now, let σ : N → N be an arbitrary permutation, and let (m k ) k∈N be an arbitrary subsequence. Then the sequence (ξ σ(m k ) ) k∈N is P-almost surely Cesàro convergent to ξ. Indeed, by Step 1 and de Finetti's theorem (see Theorem 2.4) the sequence (ξ σ(n) ) n∈N is P-almost surely Cesàro convergent to Furthermore, by Remark 2.1 and Proposition 1.1 the sequence (ξ σ(m k ) ) k∈N is Palmost surely Cesàro convergent to the random variable Since E (σ(m k )) k∈N ⊂ E (σ(n)) n∈N , by (2.3) we obtain By exchangeability of the sequence (ξ n ) n∈N we have and hence, by Proposition A.4 we obtain P-almost surely ξ = ζ. Consequently, the sequence (ξ σ(m k ) ) k∈N is P-almost surely Cesàro convergent to ξ.
Step 3: Now, let (n k ) k∈N be an arbitrary subsequence, and let π : N → N be an arbitrary permutation. By Lemma 2.2 there exists a permutation σ : N → N such that σ(n k ) = n π(k) for all k ∈ N. Therefore, by Step 2 the sequence (ξ n π(k) ) k∈N is P-almost surely Cesàro convergent to ξ, which concludes the proof.
We can extend the statement of Theorem 1.2 as follows.
Proof. By Lemma 2.2 there exists a permutation τ : N → N such that τ (n k ) = n π(k) for all k ∈ N. The mapping ρ : N → N given by ρ := σ • τ is also a permutation, and we have σ(n π(k) ) = ρ(n k ) for all k ∈ N. Therefore, applying Theorem 1.2 concludes the proof.
We conclude this section with the following consequence regarding Komlós's theorem for exchangeable sequences. Namely, let (ξ n ) n∈N ⊂ L 1 be an exchangeable sequence of random variables. Then Theorem 1.2 shows that both extensions of Komlós's theorem (the Komlós-Berkes theorem from [1], which we have stated as Theorem 2.3, and [5, Thm. 5.2.1]) are true with the original sequence (ξ n ) n∈N ; that is, we do not have to pass to a subsequence (ξ n k ) k∈N .

Appendix A. 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 what follows, let G ⊂ F be a sub σ-algebra.
A.1. Lemma. Let X ∈ L 2 be a square-integrable random variable such that P• X = P • E[X|G ]. Then we have P-almost surely X = E[X|G ].

Proof. Setting
completing the proof.
A.2. Lemma. Let X ∈ L 1 be a nonnegative random variable, and let ϕ : R + → R + be a concave function such that P-almost surely Proof. First, we assume that X ∈ L 1 is nonnegative. Let n ∈ N be arbitrary. By