On the CLT of additive functionals of Markov chains

In this paper we study the additive functionals of Markov chains via conditioning with respect to both past and future of the chain. We shall point out new sufficient projective conditions, which assure that the variance of partial sums of n consecutive random variables of a stationary Markov chain is linear in n. The paper also addresses the central limit theorem problem and is listing several open questions.


Introduction
Throughout the paper assume that (ξ n ) n∈Z is a stationary Markov chain defined on a probability space (Ω, F , P) with values in a measurable space (S, A). We suppose that there is a regular conditional distribution for ξ 1 given ξ 0 denoted by Q(x, A) = P(ξ 1 ∈ A| ξ 0 = x). In addition Q denotes the Markov operator acting via (Qf )(x) = S f (s)Q(x, ds). Denote by F n = σ(ξ k , k ≤ n) and F n = σ(ξ k , k ≥ n). The invariant distribution is denoted by π(A) = P(ξ 0 ∈ A) and Q * denotes the adjoint of Q. Next, let L 2 0 (π) be the set of measurable functions on S such that f 2 dπ < ∞ and f dπ = 0. For a function f ∈L 2 0 (π) let Note that for every k ∈ Z, Q k f (ξ 0 ) = E(X k |ξ 0 ), while (Q * ) k f (ξ 0 ) = E(X −k |ξ 0 ). We denote by ||X|| the norm in L 2 (P) and by ||f || π the norm in L 2 0 (π). Sometimes, we shall also use the notations V n = I + Q + ... + Q n and V * n = I + Q * + ... + (Q * ) n .
In some statements, we assume that the stationary Markov chain is ergodic, i.e. the only invariant functions Qf = f are the constant functions. Concerning the central limit theorem for additive functionals of a stationary and ergodic Markov chain, many of the results in the literature are given under sufficient conditions either in terms of Q k f or V n f. Among them we mention the pioneering works by Gordin (1969), Heyde (1974), McLeish (1975) and Volný (1993) among others. For a survey see Peligrad (2010) and the book by Merlevède et al. (2019). Maxwell and Woodroofe (2000) introduced a more general condition than in the papers mentioned above, namely In the same paper, they showed that (1) implies that and where ⇒ denotes the convergence in distribution and N (0, σ 2 ) is a normally distributed random variable. Later on, Peligrad and Utev (2005) established the functional form of the CLT under (1).
There are examples of Markov chains pointing out that, in general, condition (1) is as sharp as possible is some sense. Peligrad and Utev (2005) constructed an example showing that for any sequence of positive constants (a n ), a n → 0, there exists a stationary Markov chain such that n≥1 a n ||E(S n |ξ 0 )|| n 3/2 < ∞ but S n / √ n is not stochastically bounded. This example and other counterexamples provided by Volný (2010), Dedecker (2015) and Cuny and Lin (2016), show that, in general, condition does not assure that (2) holds and also does not assure (3). However, by using Proposition 2.2 in Cuny (2011), which connects (4) with a spectral condition, we know that (4) is sufficient for the CLT given in (3), in case when the Markov chain is normal (QQ * = Q * Q), as shown by Gordin and Lifshitz (1981) and independently in Derriennic and Lin (1996). Gordin and Lifshitz (1981). Assume that the Markov chain is normal, stationary and ergodic and satisfies (4). Then (2) and (3) hold.
A natural question is to ask what will be a natural generalization of Theorem 1 to Markov processes, which are not necessarily normal. In other words what will be a natural minimal condition to be added to (4), which will insure (2) and (3).
A possibility is to impose besides (4) a similar condition, but conditioning this time with respect to the future of the process n≥1 ||E(S n |ξ n )|| 2 n 2 < ∞.
It is interesting to point out that for normal Markov chains conditions (4) and (5) coincide.
In the operator notation, conditions (4) and (5) could be written in the following alternative form: This paper has double scope. First, in Section 2, we shall raise some open questions concerning the CLT for the additive functionals of a Markov chain under conditions related to (6). In the following section we support these conjectures by proving some partial results. We shall show, for instance, that (6) implies (2) and we shall comment that the CLT holds up to a random centering. The proofs are given in Sections 4 and 5.

Open problems
We shall list here several natural open problems.

Problem 2 For a stationary and ergodic Markov chain is it true (or not) that condition (6) implies that the CLT in (3) holds?
In terms of the individual random variables, let us note that, by the triangle inequality, stationarity and Lemma 14 in the last section, applied with a k = ||E(X k |ξ 0 )|| (and also with a k = ||E(X −k |ξ 0 )|| 2 ), we obtain Then, clearly (4) is implied by and (5) is implied by These two last conditions can be reformulated as: As a matter of fact there are summability conditions that interpolates between (6) and (9), which are known under the name of square root conditions. Following Derriennic and Lin (2001) the operator √ I − Q is defined by with δ n > 0, n ≥ 1 and n≥1 δ n = 1. By the equivalent definitions in Corollary 2.12 in Derriennic and Lin (2001) (4) and also f ∈ √ 1 − Q * L 2 (π) implies (5). On the other hand, by Proposition 9.2 in Cuny and Lin (2016), condition . Furthermore, according to Corollary 4.7 in Cohen et al. (2017), for normal contractions f ∈ √ 1 − QL 2 (π) is equivalent to (4). We mention that Volný (2010) constructed an example of a (non-normal) Markov operator Q and f ∈ √ 1 − QL 2 (π) for which the asymptotic variance of ||S n || 2 /n does not exist. Note (2) holds (see Proposition 1 in Derriennic and Lin, 2001 and the remarks following this proposition).
This considerations suggest that the following conjecture deserves to be studied, of course, in case the answer to Problem 2 is negative.

that the CLT in (3) holds?
Finally, if the answer to Problem 3 is negative we could ask the following question:

Results
We give here a few results in support of the open problems which have been raised in the previous section. Point (a) of the next theorem deals with the variance of partial sums, which plays a very important role in the CLT.
Theorem 5 Assume that conditions (4) and (5) hold. Then: (a) The limit in (2) holds, namely (b) The following limit exists and, if the chain is ergodic As an immediate consequence, by the discussion in the previous section, we also have the following corollaries:

Corollary 6 The conclusion of Theorem 5 also holds for
Corollary 7 The conclusion of Theorem 5 also holds under the couple of conditions (7) and (8).

Remark 8 Let us mention that the conclusion of Corollary 6 does not hold
if we assume only that f ∈ √ 1 − QL 2 (π) as shown in Volný (2010). Also, the conclusion of Corollary 7 does not hold under just (7). Dedecker (2015) constructed a relevant example, which has been reformulated in Proposition 9.5. in Cuny and Lin (2016). This example shows that there exists a Markov operator Q on some L 2 (π) and a function f ∈ L 2 0 (π) satisfying k≥1 (log k)||Q k f || 2 π < ∞ and such that ||S n || 2 /n → ∞ as n → ∞.
An interesting question asked in Problem 2 is whether the random centering in the point (c) of Theorem (5) can be avoided altogether. We can prove this fact under the condition n≥1 ||E(S n |ξ 0 , ξ n )|| 2 n 2 < ∞, namely:

Proofs
Let us comment first about conditions (4) and (5). We are going to establish two lemmas (Lemma 12 and Lemma 13) showing that condition (4) implies that while condition (5) implies that As a matter of fact, we can replace conditions (4) and (5) in Theorem 5 by conditions (14) and (15).

Proof of point (a) of Theorem 5
The proof of point (a) of Theorem 5 is related to the proof of Proposition 2.1 in Peligrad and Utev (2005), but it takes advantage of the Markov property. It includes several steps.

Upper bound on a subsequence
We shall establish first the following recurrence formula which has interest in itself. Denote Then, for 2 r−1 ≤ n < 2 r , we have the following bound: To establish it, denote byS n = 2n k=n+1 ξ k . So, by stationarity Note that, by the properties of conditional expectation and by the Markov property, We see that, by recurrence we have Now, by Hölder's inequality and stationarity, |E[E(S n |ξ n )E(S n |ξ n )]| ≤ ||E(S n |ξ n )|| · ||E(S n |ξ 0 )||, and (16) follows.

Limit on a subsequence
Note that, if sup r ∆ 2 r < ∞, then converges as r → ∞, say to L. Then, by (17), we have that where σ 2 = E(X 2 0 ) + L.

Limiting variance for S n / √ n
We show here that if conditions (4) and (5)  Then, we apply the following representation Clearly, for a j = 0, U 2 j = 0. Then we use the representation Now, (18) implies the convergence It remains to prove that |J n |/n → 0. Let 0 ≤ i < j < r. Then, by the properties of Markov chains and Hölder's inequality Hence, We can easily see that E|J n |/n → 0 because of (14) and (15).
Proof of Corollary 12.
By the properties of the conditional expectation we see that condition (12) implies that (4) and (5) are satisfied and therefore, by point (b) of Theorem 5, the limit in (10) exists. If this limit is not 0, note that (12) cannot be satisfied. Therefore (12) implies that the limit in (10) is 0. We can apply now Theorem 3.1 in Billingsley (1999) to conclude that in (11) the random centering is not needed if we assume (12).

Auxiliary results
The following lemma holds for any subadditive sequence (V m ) m≥1 of positive numbers. Its proof is inspired by Lemma 2.8. in Peligrad and Utev (2005). Because of the subtle differences we shall give it here. The main difference is that the sequence V 2 m is not subadditive.

Lemma 12
For any positive subadditive sequence (V m ) m≥1 of positive numbers we have Proof. We recall first a property on the page 806 in Peligrad and Utev (2005). Consider two positive integers M and N and the set Thus, |A c | ≤ |A| and so N − M = |D| = |A| + |A c | ≤ 2|A| and the property is proved. Now, in order to continue the proof, we add the variables in blocks in the following way: We are going to apply the above property with M = 4 r and N = 4 r+1 . So Then, by the subadditivity property, we have V 2 2r+1 ≤ 2V 2 2r , so that and, as a consequence and the proof is complete.
Next lemma contains examples of subadditive sequences which are relevant for the proofs.  Proof. Note that By changing the order of summation, taking into account that for i ≥ 1 we have k≥i k −2 ≤ 2i −1 and applying the Cauchy-Schwartz inequality, we obtain which completes the proof of the lemma.