Global Central Limit Theorems for Stationary Markov Chains

Michael Lin

doi:10.2478/amsil-2025-0011

1. Introduction

Let P = P(x, A) be a Markov transition probability function on a general state space (S, Σ), with invariant probability measure m (i.e. m(·) = ∫_S P(x, ·)dm(x)). Let Ω := S^ℕ be the space of trajectories with σ-algebra 𝒜 := Σ^⊗^ℕ, and let ℙ_x be the probability measure on 𝒜 governing the chain with transition probability function P and initial distribution δ_x. The probability of the chain with initial distribution m is then ℙ_m = ∫_S ℙ_x dm(x). By invariance of m, ℙ_m is shift invariant on (Ω, 𝒜). Let X_n be the projection of Ω on the nth coordinate. Then (X_n) on (Ω, 𝒜, ℙ_m) is a stationary Markov chain with state space S.

For 1 ≤ p < ∞ we denote by L^p(m) the Banach space {f : S → ℝ : ∫_S | f |^p dm < ∞}, and put $L_{0}^{p} (m) = f \in L^{p} (m) : \int_{S} f dm = 0\}$ .

We assume m ergodic for P, which means (by one of the equivalent definitions) that if f ∈ L²(m) satisfies f(x) = ∫_S f(y)P(x, dy) m-a.e., then f is constant a.e. Then the chain is ergodic too, i.e. the shift θ on (Ω, 𝒜, ℙ_m), defined by θ(X_n)_n∈_ℕ = (X_n₊₁)_n∈_ℕ, is ergodic.

We say that a real centered $f \in L_{0}^{2} (m)$ satisfies the annealed CLT if in (Ω, ℙ_m) we have

\frac{1}{\sqrt{n}} \sum_{k = 1}^{n} f (X_{k}) \overset{𝒟}{\to} 𝒩 (0, σ^{2}), where 𝒩 (0, 0) : = δ_{0} .

We say that a real centered $f \in L_{0}^{2} (m)$ satisfies the L²-normalized CLT if

\frac{1}{σ_{n} (f)} \sum_{k = 1}^{n} f (X_{k}) \overset{𝒟}{\to} 𝒩 (0, 1),

provided

σ_{n} (f) : = {\sum_{k = 1}^{n} f (X_{k})‖}_{L^{2} (ℙ_{m})} > 0

for sufficiently large n ∈ ℕ.

We denote by P also the Markov operator defined as

P f (x) : = \int_{S} f (y) P (x, dy)

for every bounded measurable f and every x ∈ S. By invariance of m, P extends to all L¹(m) functions, and is a contraction of all L^p(m) spaces, 1 ≤ p ≤ ∞, meaning that it does not increase the norm of functions in these spaces. As previously mentioned, ergodicity implies that Pf = f ∈ L^p holds only for f constant. We denote by Pⁿ the n-fold composition of the operator P, and by Ef := ∫_S f dm the expectation (with respect to the probability measure m) of f ∈ L^p(m), p ≥ 1.

Following the early work of Doeblin, many efforts were made to identify conditions on an ergodic Markov operator P with invariant measure m which would ensure that every centered f ∈ L²(m) satisfies the annealed CLT – an L²-global annealed CLT for the chain.

2. History

Nagaev ([21]) used the following condition of Dobrushin: there exist k ∈ ℕ and δ < 1

sup_{x, y \in S} P^{k} (x, A) - P^{k} (y, A)| < δ, \forall A \in Σ .

This condition implies uniform geometric ergodicity: sup_x ‖Pⁿ(x, ·) − m‖_TV ≤ Mρⁿ for some M > 0 and 0 < ρ < 1. But the latter condition implies ‖Pⁿ − E‖_∞ → 0, which turns out to be equivalent to Doeblin’s condition; see [24, p. 213]. Ibragimov ([18]) used a strong mixing condition (φ-mixing), which also turns out to imply Doeblin’s condition. Davydov ([9], [10]) constructed a positive recurrent aperiodic chain with countable state space such that the CLT fails for some centered f ∈ L²(m).

Theorem 1 (M. Rosenblatt, [24]).

If ‖Pⁿ − E‖₂ → 0, then every centered f ∈ L²(m) satisfies the annealed CLT.

Rosenblatt proved that his condition is equivalent to ρ-mixing of the chain, and gave examples that it yields neither ‖Pⁿ − E‖_∞ → 0 nor ‖Pⁿ − E‖₁ → 0, although each of these conditions implies it; but ‖Pⁿ − E‖₂ → 0 if and only if ‖Pⁿ − E‖_p → 0 for some (every) 1 < p < ∞. Importantly, Rosenblatt’s condition does not necessarily imply Harris recurrence, see an example below.

Example (Random walks on the unit circle 𝕋).

Let μ be a probability measure on 𝕋, and define the convolution operator Pf = μ ∗ f, f ∈ L¹(𝕋, m), m the normalized Haar (Lebesgue) measure. It is shown in [11] that if ${lim}_{k| \to \infty} \hat{μ} (k) = 0$ , that is, the Fourier transform of μ vanishes at infinity (i.e. μ is Rajchman), then ‖Pⁿ − E‖₂ → 0. When μ is Rajchman with all its powers singular with respect to Lebesgue measure, P is not Harris recurrent.

A contraction T on a Banach space 𝒳 is called uniformly ergodic if $\frac{1}{n} \sum_{k = 1}^{n} T^{k}$ converges in the operator norm. The limit is a projection onto Fix(T) := {f ∈ 𝒳 : Tf = f} corresponding to the decomposition $X = Fix (T) \oplus \bar{(I - T) X}$ . A contraction T is uniformly ergodic if and only if (I−T)𝒳 is closed in 𝒳 ([20]).

When P is uniformly ergodic in L²(m), we have $L_{0}^{2} (m) = (I - P) L^{2} (m) = (I - P) L_{0}^{2} (m)$ . (Recall that $L_{0}^{2} (m) : = f \in L^{2} : Ef = 0\}$ ). If ‖Pⁿ − E‖₂ → 0, then P is uniformly ergodic on L²(m); moreover, the spectral radius $r (P_{| L_{0}^{2} (m)}) < 1$ , meaning P has a spectral gap in the complex $L_{0}^{2} (m)$ .

Theorem 2 (Gordin-Lifshits, [15]).

Let P be a Markov operator with invariant probability measure m, and assume that P is ergodic.

If f ∈ (I − P)L²(m), then f satisfies the annealed CLT, with

σ^{2} = σ_{f}^{2} : = lim_{n \to \infty} \frac{1}{n} {\sum_{k = 1}^{n} f (X_{k})‖}_{2}^{2} = {g‖}^{2} - {Pg‖}^{2},

where f = (I − P)g with $g \in L_{0}^{2} (m)$ .

When $σ_{f}^{2} > 0$ (which is the case when P^∗P is ergodic), f satisfies also the L²-normalized CLT, which follows from a theorem of Slutsky ([25]) (see [8, p. 254]).

By [7], f ∈ (I − P)L²(m) if and only if ${sup}_{n} {\sum_{k = 1}^{n} P^{k} f‖}_{2} < \infty$ .

Theorem 1 now follows from Corollary 3 below.

Corollary 3.

Let P be a Markov operator with invariant probability measure m, and assume that P is uniformly ergodic in L²(m) with limit equal to E. Then every $f \in L_{0}^{2} (m)$ satisfies the annealed CLT.

Note that uniform ergodicity does not necessarily imply Harris recurrence.

Problem 1.

Let P be a Markov operator with invariant probability measure m, and assume that P is ergodic. If every $f \in L_{0}^{2} (m)$ satisfies the annealed CLT, does it follow that P is uniformly ergodic in L²(m)?

3. Some ergodic properties

Theorem 4 (Derriennic-Lin, [11]).

Let P be a Markov operator with invariant probability measure m, and assume P is ergodic. Then the following conditions are equivalent:

(i) P is uniformly ergodic in L²(m).
(ii) For every $f \in L_{0}^{2} (m)$ we have ${sup}_{n \geq 1} {\frac{1}{n} \sum_{k = 1}^{n} f (X_{k})‖}_{L^{2} (ℙ_{m})}^{2} < \infty$ .
(iii) For every $f \in L_{0}^{2} (m)$ we have ${sup}_{n \geq 1} {\frac{1}{\sqrt{n}} \sum_{k = 1}^{n} P^{k} f‖}_{2} < \infty$ .
(iv) For every $f \in L_{0}^{2} (m)$ we have ${sup}_{n \geq 1} \sum_{k = 1}^{n} P^{k} f, f〉| < \infty$ .

Note that P is a contraction also of each complex L^p(m) space, 1 ≤ p ≤ ∞, and it is uniformly ergodic in the complex L^p(m) iff it is uniformly ergodic in the real L^p(m). A similar statement holds also for norm convergence of Pⁿ.

Theorem 5.

Let P be a Markov operator with invariant probability measure m. If P is uniformly ergodic on L^p(m), 1 ≤ p < ∞, and is weakly mixing on the complex L^p(m) (the only unimodular eigenvalue of P is 1), then ‖Pⁿ − E‖_p → 0.

The proof primarily relies on positivity and ergodicity.

Lemma 6.

If P^∗P is ergodic, then for every $f \in L_{0}^{2} (m)$ we have Pⁿ f → 0 weakly in L²(m); thus the shift θ on (Ω, 𝒜, ℙ_m) is weakly mixing, hence totally ergodic (all powers θ^k are ergodic). Moreover, ‖(P^∗P)ⁿ f ‖₂ → 0 for every $f \in L_{0}^{2} (m)$ if and only if P^∗P is ergodic.

Proof

We assume that P^∗P is ergodic. Let 𝒦 be the unitary space of P:

K : = g \in L^{2} (m) : {P^{n} g‖}_{2} = {P^{* n} g‖}_{2} = {g‖}_{2} for every n \geq 1\} .

Clearly

{Pg‖}_{2}^{2} = {g‖}_{2}^{2}

if and only if

P^{*} Pg, g〉 = {g‖}_{2}^{2}

. Hence, by the Cauchy-Schwarz inequality, g ∈ 𝒦 implies P^∗Pg = g, and the ergodicity of P^∗P implies that 𝒦 contains only the constant functions. Any f centered is therefore orthogonal to 𝒦, and by [13] both Pⁿ f → 0 and P^∗n f → 0 weakly in L²(m). Thus P is weakly mixing.

The weak mixing of P implies that the shift θ is weakly mixing; see [1, Section 2].

The operator P^∗P is symmetric positive semi-definite in the complex L²(m), so its spectrum is a subset of [0, 1]. If P^∗P is ergodic, then for centered f ∈ L²(m) we have ‖(P^∗P)ⁿ f ‖₂ → 0 by the spectral theorem.

Conversely, if ‖(P^∗P)ⁿ f ‖₂ → 0 for every centered f ∈ L²(m), then obviously P^∗P is ergodic.

Lemma 7.

Let the shift θ be totally ergodic on (Ω, 𝒜, ℙ_m), which is the case when P^∗P is ergodic. If f ≠ 0 belongs to $L_{0}^{2} (m)$ , then σ_n(f) > 0 for every n ≥ 1.

Proof

By stationarity of the chain (X_n), σ_n(f) = 0 implies

{\sum_{k = 0}^{n - 1} f (X_{k})‖}_{L^{2} (ℙ_{m})} = 0,

\begin{array}{l} f (X_{0}) \circ θ^{n} - f (X_{0}) & = f (X_{n}) + \sum_{k = 0}^{n - 1} f (X_{k})] - f (X_{0}) \\ = \sum_{k = 1}^{n} f (X_{k}) = \sum_{k = 0}^{n - 1} f (X_{k})] \circ θ = 0 . \end{array}

By ergodicity of θⁿ, f(X₀) is a constant, which is zero since f is centered.

4. Global central limit theorems

Theorem 8.

Let P be a Markov operator with invariant probability measure m. If P^∗P is ergodic and P is uniformly ergodic, then ‖Pⁿ − E‖₂ → 0, and every centered 0 ≠ f ∈ L²(m) satisfies a non-degenerate annealed CLT and the L²-normalized CLT.

Moreover, if 0 ≠ f ∈ L³(m) is centered, then

(1)

sup_{t \in ℝ} ℙ_{m} \frac{\sum_{k = 1}^{n} f (X_{k})}{σ_{f} \sqrt{n}} \leq t\} - \frac{1}{\sqrt{2 π}} \int_{- \infty}^{t} e^{- x^{2} / 2} dx| = O (\frac{1}{\sqrt{n}}) .

Proof

Ergodicity of P^∗P implies ergodicity of P, by Lemma 6. The assumption of uniform ergodicity implies that every $f \in L_{0}^{2} (m)$ is of the form f = (I − P)g with g ∈ L²(m) centered.

Fix 0 ≠ f = (I − P)g with g ∈ L²(m) centered. By the Gordin-Lifshits CLT, the annealed CLT holds for f, with variance of the limit expressed as

σ^{2} = σ_{f}^{2} = lim_{n \to \infty} {\frac{1}{\sqrt{n}} \sum_{k = 1}^{n} f (X_{k})‖}_{L^{2} (ℙ_{m})}^{2} = {g‖}_{2}^{2} - {Pg‖}_{2}^{2}, g \in L_{0}^{2} (m) .

Hence σ_f = 0 if and only if P^∗Pg = g. If σ_f = 0, then g is constant by the ergodicity of P^∗P. Since g is centered, σ_f = 0 implies g = 0, so f = 0.

By Lemma 6 the shift is totally ergodic, so Lemma 7 yields σ_n(f) > 0 for n ≥ 1. Thus, for centered f ≠ 0 we have n^−1/2σ_n(f) → σ_f > 0, so the annealed CLT implies the L²-normalized CLT, by Slutsky’s theorem [25].

Ergodicity of P^∗P implies weak mixing of P (Lemma 6), so uniform ergodicity yields ‖Pⁿ − E‖₂ → 0 (by Theorem 5). For 0 ≠ f ∈ L³(m) centered σ_f > 0 as shown above, and (1) holds by [17].

Corollary 9.

Let P be a Markov operator with invariant probability measure m, and assume that P is ergodic and uniformly ergodic. Every centered 0 ≠ f ∈ L²(m) satisfies a non-degenerate annealed CLT if and only if P^∗P is ergodic.

Proof

When P^∗P is ergodic Theorem 8 applies. For the converse, if P^∗Pg = g for non-constant g ∈ L²(m), then P^∗P(g − Eg) = g − Eg, and f = (I − P)(g − Eg) ≠ 0 satisfies the CLT with σ_f = 0.

Proposition 10.

Let P be a Markov operator with invariant probability measure m, and assume that P is normal in L²(m), i.e. P^∗P = PP^∗. If ‖Pⁿ − E‖₂ → 0, then P^∗P is ergodic, and Theorem 8 applies.

Proof

Let P^∗Pg = g ∈ L²(m). Since P^∗E = E, normality yields ‖g − Eg‖₂ = ‖(P^∗P)ⁿ g − Eg‖₂ = ‖P^∗nPⁿg − P^∗nEg‖₂ ≤ ‖Pⁿg − Eg‖₂ → 0.

Example

In general, ‖Pⁿ − E‖₂ → 0 does not imply that P^∗P is ergodic.

Let us define P on S := {1, 2, 3} by the matrix $\frac{1}{2} \frac{1}{2} 0 ‖ 0 0 1 ‖ \frac{1}{2} \frac{1}{2} 0]$ . The invariant probability vector is $(\frac{1}{3}, \frac{1}{3}, \frac{1}{3})$ , and P^∗ is given by the adjoint matrix. P has no non-trivial invariant sets, its only unimodular eigenvalue is 1, but P^∗P is not ergodic.

Problem 2.

If a Markov operator P is ergodic, and every centered nonzero f ∈ L²(m) satisfies a non-degenerate annealed CLT, does ‖Pⁿ − E‖₂ → 0?

Note that P^∗P is ergodic (proof of Corollary 9), so P is weakly mixing.

Below we present a sufficient “moment improving” condition for uniform ergodicity (called hyperboundedness); this condition is sometimes easy to check.

Theorem 11 (Glück, [14]).

Let P be a Markov operator with invariant probability measure m, assumed to be ergodic. Assume that for some 1 ≤ s < r < ∞ we have P L^s(m) ⊂ L^r(m). Then P is uniformly ergodic in all L^p(m) spaces, 1 < p < ∞ (i.e. ${\frac{1}{n} \sum_{k = 1}^{n} P^{k} - E‖}_{p} \to 0$ ); hence (by Corollary 3) every centered f ∈ L²(m) satisfies the annealed CLT.

Example (A hyperbounded Markov operator).

Let (S, m) be the unit circle with normalized Lebesgue measure. Let 0 ≤ g ∈ L²(m) with ∫ g dm = 1, and define P by P f = g ∗ f. Then m is invariant, P is ergodic and normal in L²(m). Since ‖P f ‖₂ = ‖g ∗ f ‖₂ ≤ ‖g‖₂‖f ‖₁ for f ∈ L¹(m), P maps L¹(m) into L²(m).

Proposition 12 (Becker, [2])).

A power-bounded operator T (i.e. sup_n≥₀ ‖Tⁿ‖ < ∞) on a Banach space 𝒳 is uniformly ergodic if and only if for every $f \in \bar{(I - T) X}$ the series ∑_n_≥₁ n⁻¹Tⁿf converges in 𝒳.

Proposition 13.

Let P be a Markov operator with invariant probability measure m, assumed to be ergodic. Then the following conditions are equivalent:

(i) The Markov chain is ρ-mixing1.
(ii) ‖Pⁿ − E‖₂ → 0.
(iii) For every $f \in L_{0}^{2} (m)$ the series $\sum_{k = 1}^{\infty} P^{k} f, f〉$ converges.
(iv) For every $f \in L_{0}^{2} (m)$ we have $\sum_{n = 1}^{\infty} {P^{n} f‖}_{2}^{2} < \infty$ .
(v) There exists 1 ≤ p < ∞ such that for every $f \in L_{0}^{p} (m)$ there exists r > 1 with $\sum_{n = 1}^{\infty} {P^{n} f‖}_{p}^{r} < \infty$ .

If either of the above conditions holds, then the annealed CLT holds for every $f \in L_{0}^{2} (m)$ . The variance of the limiting normal distribution is

σ_{f}^{2} = {f‖}_{2}^{2} + 2 \sum_{k = 1}^{\infty} P^{k} f, f〉 .

Proof

The equivalence of (i) and (ii) is by [24, p. 207].

By [11, Proposition 3.1], condition (ii) is equivalent to the existence of ρ < 1 and M > 0 such that ‖Pⁿ − E‖₂ ≤ M ρⁿ for n ≥ 1. This yields (iii) and (iv).

(iii) implies uniform ergodicity, by Theorem 4. By [13, Lemma 2.1], (iii) implies Pⁿ f → 0 weakly in L²(m) for every $f \in L_{0}^{2} (m)$ ; hence P is weakly mixing. Now (ii) holds by Theorem 5.

Obviously (iv) implies (v) with p = 2.

If (v) holds, then for every centered f ∈ L^p(m), Hölder’s inequality, applied with s = r/(r − 1), yields

\sum_{n = 1}^{\infty} \frac{{P^{n} f‖}_{p}}{n} \leq {(\sum_{n = 1}^{\infty} \frac{1}{n^{s}})}^{\frac{1}{s}} {(\sum_{n = 1}^{\infty} {P^{n} f‖}_{p}^{r})}^{\frac{1}{r}} < \infty .

Hence the series

\sum_{n = 1}^{\infty} \frac{P^{n} f}{n}

is convergent in L^p-norm when f ∈ L^p(m) is centered. By Becker’s Proposition 12, P is then uniformly ergodic in L^p(m). Since condition (v) implies that P has no unimodular eigenvalues, we have ‖Pⁿ − E‖_p → 0 (by Theorem 5), and by [24, Theorem VII.4.1] (ii) holds.

Finally, (ii) implies the CLT statement by Theorem 1. By Theorem 2 the variance of the limit is lim_n→∞ σ_n(f)²/n.

Proposition 14.

Let P be a Markov operator with invariant probability measure m. If every $0 \neq f \in L_{0}^{2} (m)$ satisfies the L²-normalized CLT, then P^∗P is ergodic. Consequently (Lemma 6 and Theorem 5), if P is uniformly ergodic, then ‖Pⁿ − E‖₂ → 0.

5. α-mixing

Rosenblatt in [24] introduced a certain “strong mixing” condition, now called α-mixing, and proved that for the stationary chain generated by P with invariant probability measure m, α-mixing is equivalent to

4 α (n) : = sup_{\int f dm = 0} \frac{{P^{n} f‖}_{1}}{{f‖}_{\infty}} \to 0 as n \to \infty .

The above supremum is bounded by ‖Pⁿ − E‖₂, so ρ-mixing implies α-mixing. Clearly α-mixing implies ‖Pⁿ g − Eg‖₂ → 0 for every g ∈ L²(m), hence total ergodicity of the shift θ.

A stationary Markov chain which is Harris recurrent and aperiodic is α-mixing; see [4, Section 3.2].

Theorem 15.

Let P be a Markov operator with invariant probability measure m, and assume that the chain is α-mixing. If every $0 \neq f \in L_{0}^{2} (m)$ satisfies the L²-normalized CLT, then P^∗P is ergodic, every $0 \neq f \in L_{0}^{2} (m)$ satisfies a non-degenerate annealed CLT, and ‖Pⁿ − E‖₂ → 0.

Proof

By Proposition 14 P^∗P is ergodic, so the shift is totally ergodic. Hence for $0 \neq f \in L_{0}^{2} (m)$ , σ_n(f) > 0 for every n ≥ 1, by Lemma 7.

Let γ ∈ (0, 1) be fixed. Fix $0 \neq f \in L_{0}^{2} (m)$ , and put σ_n = σ_n(f). Since the chain is α-mixing, the stationary sequence {f(X_j)} is also α-mixing. By a result in [19], the L²-normalized CLT implies that there exists a function L(t), t > 0, slowly varying at ∞, such that $σ_{n}^{2} = nL (n)$ . By a property of slowly varying functions, we obtain $n^{- (γ + 1)} σ_{n}^{2} = n^{- γ} L (n) \to 0$ . Then

\frac{1}{n^{(γ + 1) / 2}} {\sum_{k = 1}^{n} P^{k} f‖}_{2} \leq \frac{1}{n^{(γ + 1) / 2}} {\sum_{k = 1}^{n} f (X_{k})‖}_{L^{2} (ℙ_{m})} = n^{- (γ + 1) / 2} σ_{n} \to 0 .

The above convergence holds for every

f \in L_{0}^{2} (m)

. Denoting ϵ = (1 − γ)/2, we apply it to f = g − Eg, g ∈ L²(m), to obtain

n^{ε} {\frac{1}{n} \sum_{k = 1}^{n} P^{k} g - Eg‖}_{2} = \frac{1}{n^{(γ + 1) / 2}} {\sum_{k = 1}^{n} P^{k} (g - Eg)‖}_{2} \leq C_{g} \forall n (g \in L^{2} (m)) .

By the Banach-Steinhaus theorem, the norms

n^{ε} {\frac{1}{n} \sum_{k = 1}^{n} P^{k} - E‖}_{2}\}

are bounded, so

{\frac{1}{n} \sum_{k = 1}^{n} P^{k} - E‖}_{2} \leq \frac{K}{n^{ε}} \to 0

. Thus P is uniformly ergodic. Theorem 8 yields ‖Pⁿ − E‖₂ → 0 and the non-degenerate annealed CLT for every

0 \neq f \in L_{0}^{2} (m)

Theorem 16.

Let P be an ergodic Markov operator with invariant probability measure m. Then the following conditions are equivalent:

(i) ‖Pⁿ − E‖₂ → 0 and P^∗P is ergodic.
(ii) The chain is α-mixing and every $0 \neq f \in L_{0}^{2} (m)$ satisfies the L²-normalized CLT.
(iii) Every $0 \neq f \in L_{0}^{2} (m)$ satisfies a non-degenerate annealed CLT and the L²-normalized CLT.

Proof

(i) implies (ii) follows from Theorem 8 and the fact that ρ-mixing implies α-mixing (combined with Proposition 13).

(ii) implies (i): Indeed, P^∗P is ergodic by Proposition 14, and ‖Pⁿ − E‖₂ → 0 by Theorem 15.

(i) implies (iii) by Theorem 8.

(iii) implies (i): First of all, P^∗P is ergodic by Proposition 14. Further, fix $0 \neq f \in L_{0}^{2} (m)$ . We shall prove that $σ_{n} (f) / \sqrt{n}\}$ is bounded. For the sake of contradiction, suppose it is not bounded. Then there is an increasing sequence {n_k}_k∈_ℕ such that $\sqrt{n_{k}} / σ_{n_{k}} (f)$ converges to zero, whence

(2)

\frac{1}{σ_{n_{k}} (f)} \sum_{j = 1}^{n_{k}} f (X_{j}) = \frac{\sqrt{n_{k}}}{σ_{n_{k}} (f)} \cdot \frac{1}{\sqrt{n_{k}}} \sum_{j = 1}^{n_{k}} f (X_{j}) .

The left-hand side of (2) converges in distribution to 𝒩(0, 1) by the assumption of the L²-normalized CLT for f; the right-hand side converges to 𝒩(0, 0), by the assumed annealed CLT for f and Slutsky’s theorem, leading to a contradiction. Hence

σ_{n} (f) / \sqrt{n}\}

is bounded for every

f \in L_{0}^{2} (m)

. By Theorem 4, P is uniformly ergodic. By Proposition 14, P^∗P is ergodic, so P is weakly mixing by Lemma 6, and then ‖Pⁿ − E‖₂ → 0 by Theorem 5.

Problem 3.

Assume that P is a Makov operator with invariant probability measure m such that

lim_{n \to \infty} {P^{n} g - Eg‖}_{2} = lim_{n} {P^{* n} g - Eg‖}_{2} = 0 for every g \in L^{2} (m),

and assume that every non-zero

f \in L_{0}^{2} (m)

satisfies the L²-normalized CLT. Does it follow that P is uniformly ergodic in L²(m)?

If yes, then ‖Pⁿ − E‖₂ → 0 by Theorem 5, since P is weakly mixing by the strong convergence of Pⁿ. Note that the assumption implies that P^∗P is ergodic, by Proposition 14.

By Theorem 15, the answer is yes for P which is Harris recurrent and aperiodic.

Example (P not uniformly ergodic with (P^∗P) ergodic).

Let Q be ergodic with invariant probability measure m which is not uniformly ergodic. For ε ∈ (0, 1) define P = P_ε := εI +(1 − ε)Q. We shall prove that P^∗P is ergodic. Clearly m is invariant also for P and for P^∗P. For A ∈ Σ we have

P^{*} P 1_{A} = ε^{2} 1_{A} + ε (1 - ε) (Q^{*} 1_{A} + Q 1_{A}) + {(1 - ε)}^{2} Q^{*} Q 1_{A} .

If P^∗P1_A = 1_A a.e., then for almost every x ∉ A the above summands are zero, so in particular Q1_A ≤ 1_A a.e. Since m is invariant, Q1_A = 1_A, and A is trivial by the ergodicity of Q. By definition (I − P)L²(m) = (I − Q)L²(m), so when Q is not uniformly ergodic (I − P)L²(m) is not closed; hence P is not uniformly ergodic.

6. Geometric ergodicity

Definition.

A Markov operator P with invariant probability measure m is called geometrically ergodic if, for some ρ < 1,

M_{x} : = sup_{n} ρ^{- n} {P^{n} (x, \cdot) - m‖}_{TV} < \infty a . e .

Geometric ergodicity implies aperiodic Harris recurrence and α-mixing, with the α-mixing coefficients α(n) converging to 0 exponentially fast; see [4, Section 3.2].

Theorem 17 (Doukhan-Massart-Rio, [12]).

Let Σ be countably generated and let P be a geometrically ergodic Markov operator. Then any centered f with ∫ | f |² log⁺ | f | dm < ∞ satisfies the annealed CLT.

Theorem 18 (Roberts-Tweedie, [23]).

Let Σ be countably generated, and let P be a Harris positive recurrent Markov chain. If ‖Pⁿ − E‖₂ → 0, then P is geometrically ergodic.

Note that ‖Pⁿ − E‖₂ → 0 does not necessarily imply Harris recurrence; therefore Harris recurrrence must be assumed.

Note.

The converse may fail – in [3] and [16] are examples of P geometrically ergodic with some centered f ∈ L²(m) which does not satisfy the annealed CLT, so lim_n→∞ ‖Pⁿ − E‖₂ > 0.

Theorem 19.

Let P be a Markov operator with invariant probability measure m, and assume that P is normal in L²(m). Then ‖Pⁿ − E‖₂ → 0 if (and only if) the α-mixing coefficients converge to zero (at least) exponentially fast.

Bradley ([5]) proved the theorem when P is symmetric.

In general, if P is geometrically ergodic, then P is Harris aperiodic and the α-mixing coefficients converge to zero exponentially fast. We do not know if a Harris aperiodic P whose α–mixing coefficients converge to zero exponentially fast is geometrically ergodic.

Corollary 20.

Let Σ be countably generated. If a Markov operator P is geometrically ergodic, and is additionally normal in L²(m), then

{P^{n} - E‖}_{2} \to 0 .

The symmetric case is in [22]. For S countable Corollary 20 is established in [26].

Remarks.

P in Theorem 19 need not be Harris recurrent.
When Σ is countably generated and P is Harris recurrent and normal in L²(m), Theorems 18 and 19 yield that exponential decay to 0 of α(n), geometric ergodicity and ρ-mixing are equivalent.

In Bradley’s and Häggström’s examples P is geometrically ergodic, and every centered f ∈ L^p(m), p > 2, satisfies the CLT, by Theorem 17; however, P does not have a spectral gap in L^p(m), i.e. lim_n→∞ ‖Pⁿ − E‖_p > 0, since otherwise it would imply lim_n→∞ ‖Pⁿ − E‖₂ = 0 ([24]), and so the CLT for every centered f ∈ L²(m). By Corollary 20, P in such examples cannot be normal in L²(m).

The examples of Bradley and Häggström show that without normality Theorem 19 fails, although we have geometric ergodicity.

Problem 4.

Let P be a Harris aperiodic Markov chain, and suppose that every centered f such that ∫ | f |² log⁺ | f | dm < ∞ satisfies the annealed CLT. Does this imply that P is geometrically ergodic? (Is a converse of Theorem 17 true?).

Dedecker informed the author that an example of Bradley ([6]) exhibits P Harris recurrent which is not geometrically ergodic, such that every $f \in L_{0}^{p} (m)$ , p > 2, satisfies the annealed CLT. In Problem 4 we (necessarily) assume more, i.e. that the annealed CLT is satisfied by a strictly larger subset of $L_{0}^{2} (m)$ .

Acknowledgements

The author is grateful to the referee for his careful reading of the manuscript and for his detailed comments which improved the presentation.

The author thanks the Department of Mathematics of the University of Silesia in Katowice for its warm hospitality.

Global Central Limit Theorems for Stationary Markov Chains

Full Article

1. Introduction

2. History

Theorem 1 (M. Rosenblatt, [24]).

Example (Random walks on the unit circle 𝕋).

Theorem 2 (Gordin-Lifshits, [15]).

Corollary 3.

Problem 1.

3. Some ergodic properties

Theorem 4 (Derriennic-Lin, [11]).

Theorem 5.

Lemma 6.

Proof

Lemma 7.

Proof

4. Global central limit theorems

Theorem 8.

(1)

Proof

Corollary 9.

Proof

Proposition 10.

Proof

Example

Problem 2.

Theorem 11 (Glück, [14]).

Example (A hyperbounded Markov operator).

Proposition 12 (Becker, [2])).

Proposition 13.

Proof

Proposition 14.

5. α-mixing

Theorem 15.

Proof

Theorem 16.

Proof

(2)

Problem 3.

Example (P not uniformly ergodic with (P∗P) ergodic).

6. Geometric ergodicity

Definition.

Theorem 17 (Doukhan-Massart-Rio, [12]).

Theorem 18 (Roberts-Tweedie, [23]).

Note.

Theorem 19.

Corollary 20.

Remarks.

Problem 4.

Acknowledgements

Example (P not uniformly ergodic with (P^∗P) ergodic).