2 Definitions

Let the alphabet A be a countable set, $\mathcal {X}=A^{\mathbb {Z}}$ , and $\mathcal {X}_{-} = A^{\mathbb {Z}_-}$ , where $\mathbb {Z}_-=\{0,-1,-2,\ldots \}$ . We endow $\mathcal {X}$ and $\mathcal {X}_-$ with the product topology and its corresponding Borel $\sigma $ -algebra. The topologies and $\sigma $ -algebras considered on subsets of $\mathcal {X}$ and $\mathcal {X}_-$ will always be the trace topologies and $\sigma $ -algebras. We denote by $x_i$ the ith coordinate of $x \in \mathcal {X}$ and, for $-\infty < i \leq j<\infty $ , we write $x^{-i}_{-j}:=(x_{-i},\ldots , x_{-j})$ , $x^{-i}_{-\infty }:=(x_{-i},x_{-i-1},\ldots )$ , and $x_i^{\infty }:=(\ldots ,x_{i+1},x_{i})$ . If $i < j$ , $x^i_j = \phi $ . For $x\in \mathcal {X}$ and $y \in \mathcal {X}_{-}$ , a concatenation $x^{0}_{-i}y$ is a new sequence $z\in \mathcal {X}_{-}$ with $z^{0}_{-i} = x^{-1}_{-i}$ and $z^{-i-1}_{-\infty } = y$ . We take $\phi $ to be the neutral element of the concatenation operation, that is, $\phi x=x$ for all $x\in \mathcal {X}_{-}$ . Note that we are using the convention, consistent with the concatenation operation, that when we scan an element $x\in \mathcal {X}$ from the left to the right we go further into the past.

Consider a measurable function $\phi : \mathcal {X}_{-}\to \mathbb {R}$ , which we call a potential. We say that $\phi $ is normalized if it satisfies

$$ \begin{align*} \sum_{a\in A}e^{\phi(ax)}=1 \end{align*} $$

for all $x\in \mathcal {X}_{-}$ . To a normalized potential $\phi $ we can associate a probability kernel g on the alphabet A by defining $g=e^{\phi }$ . The variation of order $k\geq 0$ of $\phi $ is defined by

$$ \begin{align*} \operatorname{{\mathrm{var}}}_{k}(\phi):=\frac{1}{2}\sup_{z \in \mathcal{X}}\sup_{x,y\in \mathcal{X}_-}\sum_{b\in A}|\phi(bz_{-k}^{-1}x)-\phi(bz_{-k}^{-1}y)|. \end{align*} $$

When A is finite, the variation is usually defined by taking the supremum over $b \in A$ instead of the sum. Nevertheless, our definition is more convenient when the alphabet is infinite and has appeared in the literature before [Reference Chazottes, Gallo and TakahashiCGT20]. The constant $1/2$ in the definition relates $\operatorname {{\mathrm {var}}}_k(e^{\phi })$ to total variation distance when $\phi $ is normalized.

We also define, for $k\geq 0$ , the $\chi ^2$ -variation of order k of $\phi $ as

$$ \begin{align*} \chi^2_{k}(\phi)=\sup_{z\in \mathcal{X}}\sup_{x,y\in \mathcal{X}_-}\sum_{b\in A}\frac{(e^{\phi(bz_{-k}^{-1}x)}-e^{\phi(bz_{-k}^{-1}y)})^2}{e^{\phi(bz_{-k}^{-1}y)}}. \end{align*} $$

The use of $\chi ^2$ -variation to measure the regularity of potentials seems to be new; therefore, it is interesting to compare it to variation, which is more standard. When $\phi $ is normalized, by using the Cauchy–Schwarz inequality, we have that $ \operatorname {{\mathrm {var}}}^2_{k}(e^\phi )\leq \tfrac {1}{4}\chi ^2_{k}(\phi )$ for any $k\geq 1$ . When the alphabet A is finite and $\phi $ is normalized, $\operatorname {{\mathrm {var}}}^2_k(\phi )$ and $\chi ^2_k(\phi )$ are comparable, that is, there exist positive constants $K_1$ and $K_2$ such that $K_1\operatorname {{\mathrm {var}}}^2_k(\phi )\leq \chi ^2_k(\phi )\leq K_2\operatorname {{\mathrm {var}}}^2_k(\phi )$ . The $\chi ^2$ -variation introduced in this work will be particularly useful to study asymptotic properties of positive probability kernels on infinite A (cf. §8.3).

Let $\eta =(\eta _n)_{\mathbb {Z}}$ be the canonical projections on $\mathcal {X}$ , that is, for all $x\in \mathcal {X}$ , $\eta _n(x)=x_n$ for all $n\in \mathbb {Z}$ . We say that a probability measure $\mu $ on $\mathcal {X}$ is compatible with a normalized potential $\phi $ if there exists a probability measure P on $\mathcal {X}_-$ such that

$$ \begin{align*} \mu[\eta_{-\infty}^{0}\in B]=P[B] \end{align*} $$

for all $B\subset \mathcal {X}_-$ measurable and if, for $n\geq 0$ ,

$$ \begin{align*} \mu[\eta_{n+1}=a\mid \eta_{-\infty}^{n}=x]=e^{\phi(ax)} \end{align*} $$

for every $a\in A$ and $\mu $ -a.e. x in $\mathcal {X}_{-}$ . Johansson, Öberg and Pollicott [Reference Johansson, Öberg and PollicottJÖP07] showed that if

$$ \begin{align*} \sum_{k = 0}^\infty \sup_{z \in \mathcal{X}}\sup_{x,y\in \mathcal{X}_-}\sum_{b\in A}(e^{\phi(bz_{-k}^{-1}x)/2}-e^{\phi(bz_{-k}^{-1}y)/2})^2 < \infty, \end{align*} $$

then there is at most one shift-invariant invariant compatible measure with $\phi $ . From [Reference ReissRei12, Lemma 3.3.9], for $k \geq 0$ , we have

$$ \begin{align*} \sum_{b\in A}(e^{\phi(bz_{-k}^{-1}x)/2}-e^{\phi(bz_{-k}^{-1}y)/2})^2 \leq \sum_{b\in A}\frac{(e^{\phi(bz_{-k}^{-1}x)}-e^{\phi(bz_{-k}^{-1}y)})^2}{e^{\phi(bz_{-k}^{-1}y)}} \end{align*} $$

and hence the summability of $\chi ^2_k(\phi )$ implies the existence of at most one shift-invariant invariant compatible measure.

When $\phi $ is not normalized, the definition of a compatible measure loses its meaning. Nevertheless, we can associate a set of shift-invariant measures called equilibrium states for not necessarily normalized $\phi $ [Reference WaltersWal75]. Equilibrium states are characterized via a variational principle and coincide with shift-invariant compatible measures when $\phi $ is normalized. An equilibrium state $\tilde {\mu }$ compatible with a normalized $\phi $ is also called a g-measure in ergodic theory [Reference KeaneKea72]. In probability literature, g-measures are known as chains of complete connections [Reference Doeblin and FortetDF37, Reference Iosifescu and GrigorescuIG90], chains of infinite order [Reference HarrisHar55, Reference KeaneKea72], random-step Markov processes [Reference KalikowKal90], and uniform martingales [Reference KalikowKal90]. Compatible measures that are not necessarily shift-invariant are called g-chains [Reference Johansson, Öberg and PollicottJOP12] or stochastic chains of unbounded memory [Reference Gallesco, Gallo and TakahashiGGT18]. When there is more than one shift-invariant measure compatible with $\phi $ , we say that there is a phase transition, otherwise we say that the shift-invariant compatible measure is unique.

4 Technical lemmas

Here we collect some results that we will use to prove Theorem 3.2. We first recall the definitions of the Kullback–Leibler and Pearson $\chi ^2$ divergences. Let P and Q be two probabilities on some discrete space $\mathcal {Y}$ . We define

$$ \begin{align*} D_{\text{KL}}(P||Q)=\sum_{y\in \mathcal{Y}}P(y)\ln\bigg(\frac{P(y)}{Q(y)}\bigg) \end{align*} $$

and

$$ \begin{align*} D_{\chi^2}(P||Q)=\sum_{y\in \mathcal{Y}}\frac{(P(y)-Q(y))^2}{Q(y)}. \end{align*} $$

It is well known that $D_{\text {KL}}(P||Q)\leq D_{\chi ^2}(P||Q)$ (cf. [Reference Sason and VerdúSV16, eq. 5]).

Lemma 4.1. Let $x, y\in \mathcal {X}_-$ and $\mu ,\nu \in \mathcal {P}(\phi )$ such that $\mu [\eta _{-\infty }^{0}\in \cdot \;]=\delta _x(\cdot )$ and $\nu [\eta _{-\infty }^{0}\in \cdot \;]=\delta _y(\cdot )$ . For all $n\geq 1$ , $0\leq k\leq n-1$ , and all $a, b, c\in \mathcal {X}$ , we have

(2) $$ \begin{align} D_{\text{KL}}&(\mu[\eta_{K_{n}}^{K_{n+1}-1}\in\; \cdot\; | \eta_1^{K_n-1}= a_{K_{n-k}}^{K_n-1} b_{1}^{K_{n-k}-1}]\nonumber\\ &\times || \nu [\eta_{K_{n}}^{K_{n+1}-1}\in \;\cdot \; | \eta_1^{K_n-1}=a_{K_{n-k}}^{K_n-1}c_{1}^{K_{n-k}-1}])\nonumber\\ &\leq \sum_{j=K_n}^{K_{n+1}-1}\chi^2_{j-K_{n-k}}(\phi). \end{align} $$

Proof Let us simply denote by D the left-hand term of inequality (2). We have by the chain rule property of the Kullback–Leibler divergence [Reference Cover and ThomasCT06, Theorem 2.5.3] that

$$ \begin{align*} D &=\sum_{i=K_n}^{K_{n+1}-1}D_{\text{KL}}( \mu[\eta_i\in \cdot \; | \eta_1^{i-1}=z_{K_n}^{i-1}a_{K_{n-k}}^{K_n-1} b_{1}^{K_{n-k}-1}] \nonumber\\ &\,\,\quad\phantom{\sum_{i=K_n}^{K_{n+1}-1}D_{\text{KL}}}\times|| \nu[\eta_i\in \cdot \; | \eta_1^{i-1}=z_{K_n}^{i-1}a_{KK_{n-k}}^{K_n-1}c_{1}^{K_{n-k}-1}] )\nonumber\\ &=:\sum_{i=K_n}^{K_{n+1}-1}D_i. \end{align*} $$

Then we use the well-known bound

$$ \begin{align*} D_i&\leq D_{\chi^2}(\mu[\eta_i\in \cdot \; | \eta_1^{i-1}=z_{K_n}^{i-1}a_{K_{n-k}}^{K_n-1} b_{1}^{K_{n-k}-1}]\nonumber\\ &\phantom{\leq D_{\chi^2}}\times || \nu[\eta_i\in \cdot \; | \eta_1^{i-1}=z_{K_n}^{i-1}a_{K_{n-k}}^{K_n-1}c_{1}^{K_{n-k}-1}])\nonumber\\ &\leq \chi^2_{i-K_n}(\phi) \end{align*} $$

to conclude the proof.

Lemma 4.2. For $\alpha> 1$ and $0<a<b$ , we have

(3) $$ \begin{align} \frac{(b+1)^{\alpha}-a^{\alpha}}{b^{\alpha}-a^{\alpha}}\geq \frac{(b+1)^{\alpha-1}-a^{\alpha-1}}{b^{\alpha-1}-a^{\alpha-1}}. \end{align} $$

Proof By algebraic computations, we obtain that (3) is equivalent to

$$ \begin{align*} \bigg(\frac{b}{a}\bigg)^{\alpha-1}\geq 1+(b-a)\bigg(1-\bigg(\frac{b}{b+1}\bigg)^{\alpha-1}\bigg). \end{align*} $$

This last inequality is obtained from the Bernoulli inequality $(1+x)^{r}\geq 1+rx$ , for $r>0$ and $x>-1$ , observing that

$$ \begin{align*} \bigg(\frac{b}{a}\bigg)^{\alpha-1}=\bigg(1+\frac{b-a}{a}\bigg)^{\alpha-1}\geq 1+(\alpha-1)\frac{b-a}{a} \end{align*} $$

and

$$ \begin{align*} \bigg(\frac{b}{b+1}\bigg)^{\alpha-1}=\bigg(1-\frac{1}{b+1}\bigg)^{\alpha-1}\geq 1-(\alpha-1)\frac{1}{b+1}.\\[-40pt] \end{align*} $$

Define, for all $\delta>0$ , $\beta \geq 1$ , $k\geq 3$ , and $n\geq k+1$ ,

$$ \begin{align*} \Delta^n_k:=(n^{\beta}-(n-k)^{\beta}-2)^{-\delta}-((n+1)^{\beta}-(n-k)^{\beta})^{-\delta}. \end{align*} $$

Proof The statement of the lemma is trivial for $\beta =1$ . For $\beta>1$ , consider the function $f:[4,\infty )\to \mathbb {R}^+$ defined by

$$ \begin{align*} f(x)=(x^{\beta}-(x-k)^{\beta}-2)^{-\delta}-((x+1)^{\beta}-(x-k)^{\beta})^{-\delta}. \end{align*} $$

In order to prove the result, it is enough to show that the derivative of f is negative. Since

$$ \begin{align*} f'(x)&=-\delta \beta[(x^{\beta}-(x-k)^{\beta}-2)^{-\delta-1}(x^{\beta-1}-(x-k)^{\beta-1})\nonumber\\ &\phantom{**}-((x+1)^{\beta}-(x-k)^{\beta})^{-\delta-1}((x+1)^{\beta-1}-(x-k)^{\beta-1})], \end{align*} $$

it is enough to show that

$$ \begin{align*} \frac{(x+1)^{\beta}-(x-k)^{\beta}}{x^{\beta}-(x-k)^{\beta}-2}\geq \frac{(x+1)^{\beta-1}-(x-k)^{\beta-1}}{x^{\beta-1}-(x-k)^{\beta-1}}. \end{align*} $$

But this last inequality follows from Lemma 4.2.

Lemma 4.4. For all $\delta>0$ , $\beta \geq 1$ , and $k\geq 3$ , we have

$$ \begin{align*} \Delta_k^{k+1}\leq 4\frac{2^{\beta} 4^{\delta}\beta}{k^{\delta\beta+1}}. \end{align*} $$

Proof Observe that for $k\geq 3$ ,

(4) $$ \begin{align} \Delta_k^{k+1}&=\int_{(k+1)^{\beta}-3}^{(k+2)^{\beta}-1}\frac{1}{x^{1+\delta}}dx \nonumber\\[5pt] & \leq \frac{(k+2)^{\beta}-(k+1)^{\beta}+2}{((k+1)^{\beta}-3)^{1+\delta}} \leq \frac{\beta (k+2)^{\beta-1}+2}{((k+1)^{\beta}-3)^{1+\delta}} \leq 4\frac{2^{\beta} 4^{\delta}\beta}{k^{\delta\beta+1}}, \end{align} $$

where, to obtain the second inequality in (4), we used the inequality

$$ \begin{align*} (a+b)^{\alpha}\leq a^{\alpha}+\alpha b (a+b)^{\alpha-1} \end{align*} $$

for $\alpha \geq 1$ and $a, b\geq 0$ . This inequality can be obtained using the fundamental theorem of calculus applied to the function $f(x)=x^{\alpha }$ .

To obtain the last inequality in (4), we used that for $k\geq 3$ ,

$$ \begin{align*} \beta (k+2)^{\beta-1}+2\leq 2\beta(k+2)^{\beta-1}\leq 2^{\beta}\beta k^{\beta-1} \end{align*} $$

and

$$ \begin{align*} (k+1)^{\beta}-3=(k+1)^{\beta}\bigg(1-\frac{3}{(k+1)^{\beta}}\bigg)\geq \frac{(k+1)^{\beta}}{4}\geq \frac{k^{\beta}}{4}. \\[-40pt]\end{align*} $$

Finally, we recall the following lemma in [Reference Bressaud, Fernández and GalvesBFG99] (see also Lemma A.4 in [Reference GiacominGia07]) that gives an estimate for the renewal sequence that will appear in the proof of Theorem 3.2. We state the lemma using a notation that is adapted to our purpose.

Lemma 4.5. (Proposition 2, item (iv) in [Reference Bressaud, Fernández and GalvesBFG99])

Let $(f_k)_{k \geq 1}$ be a sequence of positive real numbers such that $\sum _{k=1}^\infty f_k < 1$ . Suppose that $(u_k)_{k \geq 1}$ is a sequence with $u_0 = 1$ and satisfies the renewal equation

$$ \begin{align*} u_n = \sum_{k=1}^n f_k u_{n-k}. \end{align*} $$

If $f_n \leq c_1/n^{1+\alpha }$ for some $\alpha>0$ and a positive constant $c_1$ , then $u_n \leq c_2/n^{1+\alpha }$ , where $c_2$ is a constant that depends on $(f_k)_{k \geq 1}$ .

5 Proof of Theorem 3.2

Let $x, y\in \mathcal {X}_-$ and $\mu , \nu $ compatible measures such that $\mu [\eta _{-\infty }^{0}\in \cdot \;]=\delta _x(\cdot )$ and $\nu [\eta _{-\infty }^{0}\in \cdot \;]=\delta _y(\cdot )$ . We now construct the coupling of $\mu $ and $\nu $ , that we call $\mathbb {P}^{x,y}$ , as follows. We start by defining

$$ \begin{align*}\mathbb{P}^{x,y}[\hat{\eta}_{-\infty}^{0}\in\cdot\;, \hat{\omega}_{-\infty}^{0}\in\cdot\;]=\delta_x \otimes\delta_y.\end{align*} $$

Then, for all $n\geq 1$ , given the pasts $\hat {\eta }_{-\infty }^{K_n-1}$ and $\hat {\omega }_{-\infty }^{K_n-1}$ , we maximally couple $\hat {\eta }_{K_n}^{K_{n+1}-1}$ and $\hat {\omega }_{K_n}^{K_{n+1}-1}$ to complete the construction of $\mathbb {P}^{x,y}$ .

Next, we show that $\mathbb {P}^{x,y}$ satisfies the inequality in Theorem 3.2. For all $n\geq 1$ and $0\leq k\leq n-1$ , define

$$ \begin{align*} q^n_k=\sup_{x,y,a,b \in\mathcal{X}}\mathbb{P}^{x,y}[X_{n}=1\mid X_{n-k}^{n-1}=0, \hat{\eta}^{K_{n-k}-1}_{1} = a_{1}^{K_{n-k}-1}, \hat{\omega}^{K_{n-k}-1}_{1} = b_{1}^{K_{n-k}-1}] \end{align*} $$

with the convention that if $k>l$ , then elements of the form $a_k^l$ are dropped from the conditional part. The shorthand notation $X_{n-k}^{n-1}=0$ means that $X_{n-k} = 0, \ldots , X_{n-1} = 0$ . Observe that for all $n\geq 1$ and $0\leq k\leq n-2$ , we have $q^n_{k} \geq q^n_{k+1}$ .

We start by proving the following lemma.

Lemma 5.1. Suppose that $(\chi ^2_n(\phi ))_{j\geq 0}\in \ell ^1$ . Then there exists $\varepsilon>0$ such that for all ${k\geq 0}$ and all $n\geq k+1$ ,

(5) $$ \begin{align} q_k^n\leq\sqrt{1-\exp\bigg(-\sum_{j=0}^{\infty}\chi_j^2(\phi)\bigg)}\leq 1-\varepsilon. \end{align} $$

For $k\geq 1$ and $n\geq k+1$ , we also have

(6) $$ \begin{align} q_k^n\leq \sqrt{\frac{1}{2}\sum_{j=K_n}^{K_{n+1}-1}\chi^2_{j-K_{n-k}}(\phi)}. \end{align} $$

Now, on some probability space $(\Omega , \mathcal {F}, P)$ , consider the random process $Y=(Y_n)_{n\geq 0}$ with values in $\{0,1\}$ such that $Y_0=1$ and, for $n\geq 1$ and $0\leq k\leq n-1$ ,

$$ \begin{align*} P[Y_n=1\mid Y^{n-1}_{n-k}=0, Y_{n-k-1}=1, Y_1^{n-k-2}]=q^n_k. \end{align*} $$

For all $m\geq 1$ and $a,b \in \{0,1\}^{m}$ , we say that $a\geq b$ if $a_i \geq b_i$ for $i \in \{1,\ldots , m\}$ . By construction, for all $n\geq 2$ , $a,b \in \{0,1\}^{n-1}$ , and $a\geq b$ , we have $P[Y_n = 1 | Y^{n-1}_1 = a] \geq \mathbb {P}^{x,y}[X_n = 1 | X^{n-1}_1 = b]$ . Therefore, by applying Strassen’s theorem on stochastic domination [Reference LindvallLin99] inductively on n, we can construct a coupling measure Q such that, for $a,b \in \{0,1\}^{n-1}$ and $a\geq b$ , we have $Q[Y_n \geq X_n| Y^{n-1}_1 = a, X^{n-1}_1 = b] = 1$ . Therefore, for all $n\geq 1$ , we have $Q[Y_n \geq X_n] = 1$ , which implies that

(7) $$ \begin{align} P[Y_n = 1] \geq \mathbb{P}^{x,y}[X_n = 1] \end{align} $$

for all $n \geq 1$ .

Now, consider the process $Z=(Z_n)_{n\geq 0}$ with values in $\{0,1\}$ such that $Z_0=1$ and, for $n\geq 1$ and $0\leq k\leq n-1$ ,

(8) $$ \begin{align} P[Z_n=1\mid Z^{n-1}_{n-k}=0, Z_{n-k-1}=1, Z_1^{n-k-2}]=b_k:=\sup_{n\geq k+1} q_k^n. \end{align} $$

Observe that, for all $k \geq 0$ , we have $b_k \geq b_{k+1}$ . Using the same argument used to show (7), we have that $P[Z_n = 1] \geq P[Y_n = 1]$ for all $n\geq 1$ . Also, by (6) and Lemmas 4.3 and 4.4, we have that for $k\geq 3$ and $n\geq k+1$ ,

$$ \begin{align*} 2(q_k^n)^2&\leq \sum_{j=K_n}^{K_{n+1}-1}\chi^2_{j-K_{n-k}}(\phi)\\ &\leq C\sum_{j=\lfloor n^{\beta}\rfloor}^{\lfloor(n+1)^{\beta}\rfloor-1}\frac{1}{(j-\lfloor (n-k)^{\beta}\rfloor)^{1+\delta}}=C\sum_{j=\lfloor n^{\beta}\rfloor-\lfloor (n-k)^{\beta}\rfloor}^{\lfloor(n+1)^{\beta}\rfloor-\lfloor (n-k)^{\beta}\rfloor-1}\frac{1}{j^{1+\delta}}\\ &\leq C\int_{n^{\beta}-(n-k)^{\beta}-2}^{(n+1)^{\beta}-(n-k)^{\beta}}\frac{1}{x^{1+\delta}}dx=C\frac{\Delta_k^n}{\delta}\leq C\delta^{-1}\Delta_k^{k+1}\leq 4C\frac{2^{\beta} 4^{\delta}\beta\delta^{-1}}{k^{\delta\beta+1}}. \end{align*} $$

Using (5), we obtain that $b_k\leq (2C({2^{\beta } 4^{\delta }\beta \delta ^{-1}})/({k^{\delta \beta +1}}))^{1/2}\wedge (1-\varepsilon )$ for all $k\geq 3$ and $b_k\leq 1-\varepsilon $ for $0\leq k\leq 2$ .

Next, let $f_i:= b_{i-1}\prod _{k=0}^{i-2}(1-b_k)$ for $i\geq 1$ (with the convention that $\prod _{j=0}^{-1}=1$ ) and $u_i:= P[Z_i = 1]$ for $i \geq 0$ . We have that

$$ \begin{align*}P[Z_n = 1] = \sum_{k= 1}^nP[Z_n = 1, Z^{n-1}_{n-k+1} = 0| Z_{n-k} = 1]P[Z_{n-k} = 1]\end{align*} $$

and hence the following renewal equation holds:

$$ \begin{align*}u_n = \sum_{k = 1}^nf_{k}u_{n-k}.\end{align*} $$

By definition, we have that $\sum _{k= 1}^\infty f_k = 1-\prod _{k = 1}^\infty (1-b_k)$ . If $\beta>\delta ^{-1}$ , we have $\sum _{k = 0}^\infty b_k < \infty $ . Hence, $\sum _{k= 1}^\infty f_k < 1$ . Moreover, when $\beta>\delta ^{-1}$ , we have that $b_k \leq c_1k^{-(\delta \beta +1)/2}$ for some positive constant $c_1$ that depends on $C, \delta $ , and $\beta $ . From Lemma 4.5, we have that, for all $n\geq 1$ ,

$$ \begin{align*} u_n\leq \frac{C_1}{n^{({\delta\beta+1})/{2}}}, \end{align*} $$

where $C_1$ is a positive constant that depends on $C, \delta $ , and $\beta $ . Because $P[Z_n = 1] \geq \mathbb {P}^{x,y}[X_n = 1]$ , we obtain that for all $n\geq 1$ ,

$$ \begin{align*} \mathbb{P}^{x,y}[X_n=1]\leq \frac{C_1}{n^{({\delta\beta+1})/{2}}} \end{align*} $$

for $\beta>\delta ^{-1}$ . Because the bound is uniform on $x,y \in \mathcal {X}_-$ , we obtain the desired result.

6 Proofs of Corollaries 3.3 and 3.4

6.1 Proof of Corollary 3.3

For $k\in [K_n, K_{n+1})$ and all $y,z \in \mathcal {X}_-$ , using the coupling inequality for total variation distance (cf. [Reference ThorissonTho00]), we have

$$ \begin{align*} d_{\text{TV}}(\mu^y[\eta_k \in \cdot\;],\mu^z[\eta_k \in \cdot\;])\leq \mathbb{P}[\hat{\eta}_k\neq \hat{\omega}_k]\leq \mathbb{P}[X_n=1]. \end{align*} $$

Then, by Theorem 3.2, we obtain

$$ \begin{align*} \mathbb{P}[\hat{\eta}_k\neq \hat{\omega}_k]\leq \frac{C_1}{n^{({\beta \delta+1})/{2}}} \end{align*} $$

for all $\beta \geq 1$ and $\beta>\delta ^{-1}$ . If $\delta>1$ , just take $\beta =1$ . In this case $k=n$ and thus we obtain Corollary 3.3 with a constant $C_2$ that depends on C and $\delta $ . If $\delta \in (0,1)$ , since $k\leq (n+1)^{\beta }$ , we have $n\geq k^{1/\beta }-1$ . This leads to

$$ \begin{align*} \mathbb{P}[\hat{\eta}_k\neq \hat{\omega}_k]\leq \frac{C_3}{k^{({\beta \delta+1})/{2\beta}}} \end{align*} $$

for all $k\geq 1$ , where $C_3$ is a positive constant that depends on $C, \delta $ , and $\beta $ . Now, observe that for any $0< \delta '<\delta $ , we can choose $\beta $ such that $\beta \geq 1$ , $\beta>\delta ^{-1}$ , and $({\beta \delta +1})/ {2\beta }\geq \delta '$ .

6.2 Proof of Corollary 3.4

Consider $k\in [K_n, K_{n+1})$ . Let

$$ \begin{align*}\theta=\inf\{n\geq 1: \hat{\eta}_k=\hat{\omega}_k\; \text{for all}\; k\geq n\}\end{align*} $$

with the convention that $\inf \emptyset =\infty $ . We start by observing that

$$ \begin{align*} \mathbb{P}[\theta>k]\leq \mathbb{P}\bigg[\bigcup_{j\geq n} \{ X_j=1\}\bigg]\leq \sum_{j\geq n} \mathbb{P}[X_j=1]. \end{align*} $$

By Theorem 3.2, we obtain

$$ \begin{align*} \mathbb{P}[\theta>k]\leq C_1\sum_{j\geq n} \frac{1}{n^{({\beta \delta+1})/{2}}}\leq \frac{C_1'}{n^{({\beta\delta-1})/{2}}} \end{align*} $$

for $\beta \geq 1$ , $\beta>\delta ^{-1}$ , and $C_1'$ a positive constant that depends on $C, \delta $ , and $\beta $ . Since $n\geq k^{1/\beta }-1$ , we obtain

$$ \begin{align*} \mathbb{P}[\theta>k]\leq \frac{C_4}{k^{({\beta\delta-1})/{2\beta}}} \end{align*} $$

for all $k\geq 1$ and $C_4$ a positive constant that depends on $C, \delta $ , and $\beta $ . Finally, notice that for all $\delta '<\delta $ , we can choose $\beta $ large enough such that $({\beta \delta -1})/{2\beta }\geq \delta '$ . Using the coupling inequality (cf. [Reference ThorissonTho00]), we conclude that

$$ \begin{align*} \qquad\ \ \quad M(n) := \sup_{y,z} d_{\text{TV}}(\mu^y[(\eta_j)_{j\geq n} \in \cdot\;],\mu^z[(\eta_j)_{j\geq n} \in \cdot\;]) \leq \mathbb{P}[\theta>k].\end{align*} $$

7 Proof of Theorem 3.6

Consider $K_m=\lfloor m^{\beta } \rfloor $ for $m\ge 1$ . For each $x, y\in \mathcal {X}_-$ , we consider a probability space $(\Omega , \mathcal {F}, \mathbb {P}^{x,y})$ that supports the random elements $\tilde {\eta }$ , $\tilde {\omega }$ , and $\tilde {Z}$ defined as follows. Let $\tilde {\eta }_*\mathbb {P}^{x,y}$ , $\tilde {\omega }_*\mathbb {P}^{x,y}$ be compatible with $\phi $ and $\tilde {\eta }_{-\infty }^{0}=x,\, \tilde {\omega }_{-\infty }^{0}=y$ . Also, for all $m\geq 1$ given the pasts $\tilde {\eta }_{-\infty }^{K_m-1}$ and $\tilde {\omega }_{-\infty }^{K_m-1}$ , the blocks $\tilde {\eta }_{K_m}^{K_{m+1}-1}$ and $\tilde {\omega }_{K_m}^{K_{m+1}-1}$ are maximally coupled. Under $\mathbb {P}^{x,y}$ , the process $\tilde {Z}$ has the same law as the process Z defined in §5 and verifies for all $m\geq 1$ (this is indeed possible since Z stochastically dominates X; see §5). We denote by $\mathbb {E}^{x,y}$ the expectation with respect to $\mathbb {P}^{x,y}$ .

Fix some $n\in \mathbb {N}$ and let k be such that $n\in [K_{k-1}, K_{k})$ . We will show that

(9) $$ \begin{align} \bigg| \int f\circ T^n \;{\hat f}\,d\tilde{\mu}-\int f\,d\tilde{\mu}\int {\hat f}\,d\tilde{\mu} \bigg|\leq c_1 P[Z_k=1] \end{align} $$

for some positive constant $c_1$ that depends only on C, $\delta $ , and $\beta $ . From this point, Theorem 3.6 is easily obtained following the proof of Corollary 3.3. To obtain (9), we follow the argument developed in [Reference Bressaud, Fernández and GalvesBFG99, §5]. Using (3.7) in [Reference Bressaud, Fernández and GalvesBFG99], we first observe that

$$ \begin{align*} \bigg| \int f\circ T^n \;{\hat f}\,d\tilde{\mu}-\int f\,d\tilde{\mu}\int {\hat f}\,d\tilde{\mu}\bigg | \leq \|f\|_1\sup_{x,y} \mathbb{E}^{x,y}[|\hat{f}(\tilde{\eta}_{-\infty}^n)-\hat{f}(\tilde{\omega}_{-\infty}^{n}) |]. \end{align*} $$

For $k\geq 1$ , let

$$ \begin{align*} \theta_k=\inf\{0\leq m\leq k:\tilde{Z}_{k-m}=1\}. \end{align*} $$

We have

(10)

Observe that for all $0\leq j\leq k$ ,

$$ \begin{align*} \mathbb{P}^{x,y}[\theta_k=j]&=P[Z_k=0,\ldots,Z_{k-j+1}=0,Z_{k-j}=1]\nonumber\\ &=\prod_{l=1}^j(1-b_{k-j+l})P[Z_{k-j}=1], \end{align*} $$

where the $b_{k-j+l}$ are from (8). Now, observe that for all $i\geq 1$ , we have

$$ \begin{align*} P[Z_i=1]&=\sum_{k=1}^ib_{i-k}P[Z_{i-1}=0\mid Z_k^{i-1}=0, Z_{k-1}=1]\nonumber\\ &=\sum_{k=1}^i b_{i-k}\prod_{l=1}^{i-k}(1-b_{i-k-l})P[Z_{k-1}=1]. \end{align*} $$

Thus, we have that for all $i\geq 1$ ,

$$ \begin{align*} P[Z_i=1]&=\sum_{k=1}^if_kP[Z_{i-k}=1] \end{align*} $$

with $f_k:=b_{k-1}\prod _{l=0}^{k-2}(1-b_l)$ , $k\geq 1$ . From this, we obtain

$$ \begin{align*} \mathbb{E}^{x,y}[|\hat{f}(\tilde{\eta}_{-\infty}^n)-\hat{f}(\tilde{\omega}_{-\infty}^{n}) |]\,{\leq}\, &\|\hat{f}\|_{\phi}\bigg( \operatorname{{\mathrm{var}}}_{n-K_{k}+1}(e^\phi)\sum_{l=1}^k f_lP[Z_{k-l}=1]\nonumber\\ &\kern1.5pt{+}\, \sum_{l=1}^k \operatorname{{\mathrm{var}}}_{n-K_{k-l}+1}(e^\phi)\prod_{m=1}^{l}(1-b_{k-l+m})P[Z_{k-l}=1] \bigg). \end{align*} $$

We deduce that

$$ \begin{align*} \sup_{x,y}\mathbb{E}^{x,y}[|\hat{f}(\tilde{\eta}_{-\infty}^n)-\hat{f}(\tilde{\omega}_{-\infty}^{n}) |]&\leq \kappa \sum_{l=1}^k f_lP[Z_{k-l}=1]=\kappa P[Z_{k}=1] \end{align*} $$

with

$$ \begin{align*}\kappa:=\operatorname{{\mathrm{var}}}_{n-K_{k}+1}(e^\phi)+\sup_{1\leq l\leq k}\frac{\operatorname{{\mathrm{var}}}_{n-K_{k-l}+1}(e^\phi)}{f_l}. \end{align*} $$

Finally, since $({\operatorname {{\mathrm {var}}}_{n-K_{k-l}+1}(e^\phi )})/({b_{l-1}})\leq 2$ and $\prod _{j=0}^{\infty }(1-b_j)>0$ (using that $b_0 < 1$ and $\sum _{j=0}^\infty b_j<\infty $ ), we observe that

$$ \begin{align*} \kappa\leq 1+\frac{\operatorname{{\mathrm{var}}}_{n-K_{k-l}+1}(e^\phi)}{b_{l-1}\prod_{j=0}^{\infty}(1-b_j)}\leq c_2 \end{align*} $$

for some positive constant $c_2$ depending on C, $\delta $ , and $\beta $ .

7.1 Proof of Corollary 3.9

The potential $\phi $ is summable; therefore, there exists a normalized potential $\psi $ with the same unique equilibrium state as $\phi $ [Reference WaltersWal75, Theorem 3.2]. If $\operatorname {{\mathrm {var}}}_{k}(\phi )\leq {C}/{k^{({3+ \delta })/{2}}}$ , then $\operatorname {{\mathrm {var}}}_{k}(\psi )\leq {C'}/{k^{({1+ \delta })/{2}}}$ for some constant $C'> 0$ that depends only on C (see, for example, [Reference PollicottPol00, Proposition 1]). Closely following the proof of Theorem 3.6 using the measures compatible with $\psi $ , we obtain the desired results. The only difference is that in (10) we use the bound

$$ \begin{align*} \mathbb{E}^{x,y}[|\hat{f}(\tilde{\eta}_{-\infty}^n)-\hat{f}(\tilde{\omega}_{-\infty}^{n}) |] \leq \|\hat{f}\|_{\phi} \sum_{j=0}^k \operatorname{{\mathrm{var}}}_{n-K_{k-j}+1}(e^{\phi})\mathbb{P}^{x,y}[\theta_k=j] \end{align*} $$

instead of

$$ \begin{align*} \mathbb{E}^{x,y}[|\hat{f}(\tilde{\eta}_{-\infty}^n)-\hat{f}(\tilde{\omega}_{-\infty}^{n}) |] \leq \|\hat{f}\|_{\psi} \sum_{j=0}^k \operatorname{{\mathrm{var}}}_{n-K_{k-j}+1}(e^{\psi})\mathbb{P}^{x,y}[\theta_k=j]. \\[-24pt] \end{align*} $$

8 Applications

8.1 Functional central limit theorem (FCLT) for potentials with non-summable variations

Let $\sigma> 0$ . A function $h:A\to \mathbb {R}$ satisfies the functional central limit theorem (FCLT), also called the weak invariance principle, when the process $\{\zeta _n(t), t \in [0,1], n\geq 1\}$ defined by

$$ \begin{align*} \zeta_n(t) = \frac{1}{\sigma \sqrt{n}} \sum_{i = 0}^{\lfloor nt \rfloor} h\circ \eta_i \end{align*} $$

converges weakly to a standard Brownian motion on $D[0,1]$ . Tyran-Kamińska [Reference Tyran-KamińskaTK05, Section 4.3] showed that the FCLT holds when the potential $\phi $ has summable variations. A straightforward application of Theorem 3.2 in [Reference Tyran-KamińskaTK05] and our Corollary 3.3 is the following FCLT for potentials with non-summable variations.

Proposition 8.1. Assume that the alphabet A is finite and $\phi $ satisfies Assumption ( $\mathcal {A}$ ) with $\delta \in (1/2,1]$ . Let $\mu $ be shift-invariant and compatible with $\phi $ . Also, let $h:A \to \mathbb {R}$ be a function such that $\int h\circ \eta _0\; d\mu = 0$ . If $\sigma ^2 :=\int (h\circ \eta _0)^2\; d\mu> 0$ , then h satisfies the FCLT.

Proof Because the alphabet is finite, we have that $\sigma ^2 \leq \|h\|^2_\infty < \infty $ , as required by Theorem 3.2 in [Reference Tyran-KamińskaTK05]. It remains to verify the condition on the mixing rate. Using the processes $\{\tilde {\eta }_i, i \in \mathbb {Z}\}$ and $\{\tilde {\omega }_i, i \in \mathbb {Z}\}$ introduced in the proof of Theorem 3.6, it is sufficient to check that there exists $\gamma> 1/2$ such that

(11) $$ \begin{align} \limsup_{n \rightarrow \infty} n^\gamma \sup_{x,y}\mathbb{E}^{x,y}[|h(\tilde{\eta}_n)-h(\tilde{\omega}_n)|] < \infty. \end{align} $$

We have from Corollary 3.3 that, for all $x,y \in \mathcal {X}_-$ ,

$$ \begin{align*} \mathbb{E}^{x,y}[|h(\tilde{\eta}_n)-h(\tilde{\omega}_n)|] &\leq 2\|h\|_{\infty} \mathbb{P}^{x,y}[\tilde{\eta}_n\neq \tilde{\omega}_n]\\ &\leq \frac{c_1}{n^{\delta'}}, \end{align*} $$

where $\delta ' < \delta $ and $c_1 $ is a positive constant that depends on $C, \delta , \delta '$ , and h. Taking $\delta '$ and $\gamma $ such that $1/2 < \gamma \leq \delta ' <\delta $ , we obtain (11).

8.2 Hoeffding-type inequality for potentials with non-summable variations

A Hoeffding-type inequality gives finite sample bounds for deviations of additive functionals from their mean. When the variation rate of the potential is summable, we have an exponential inequality [Reference Gallesco, Gallo and TakahashiGGT14, Reference Chazottes, Gallo and TakahashiCGT20, Reference MartonMar98]. Nevertheless, the rate of concentration for potentials when the variation rate is not summable is an open question. Using our result, we can obtain the following stretched exponential inequality for sums of random variables.

Proposition 8.2. Assume that the alphabet A is finite and $\phi $ satisfies Assumption ( $\mathcal {A}$ ) with $\delta \in (1/2,1]$ . Let $\mu $ be compatible with $\phi $ . For all $\delta '<2\delta -1$ , $n\geq 1$ , $t\geq 0$ , and all functions $h:A \to \mathbb {R}$ , we have

$$ \begin{align*} \mu\bigg[\bigg|\frac{1}{n}\sum_{i=1}^n(h(\eta_i)-\mathbb{E}[h(\eta_i)])\bigg|\geq t\bigg]\leq 2\exp\bigg\{\!-\frac{C_{10}n^{\delta'}t^2}{R(h)^2}\bigg\}, \end{align*} $$

where $R(h):=\max _{a\in A}h(a)-\min _{a\in A}h(a)$ and $C_{10}$ is a constant that depends on C, $\delta $ , and $\delta '$ .

Proof This is a consequence of Theorem 3.2 of [Reference Chazottes, Collet, Külske and RedigCCKR07] and Corollary 3.3. In order to apply Theorem 3.2 of [Reference Chazottes, Collet, Külske and RedigCCKR07], we need to estimate the terms $\|\overline {D}\|_{\ell ^2(\mathbb {N})}^2$ and $\|\delta f\|^2_{\ell ^2(\mathbb {N})}$ (for $f(x_1,\ldots ,x_n)=({1}/{n})\sum _{i=1}^n h(x_i)$ ) there. We have

$$ \begin{align*} \|\overline{D}\|_{\ell^2(\mathbb{N})}^2\leq \bigg(1+\sum_{i=1}^n \sup_{x,y\in \mathcal{X}_-}\mathbb{P}^{x,y}[\hat{\eta}_i\neq \hat{\omega}_i]\bigg)^2. \end{align*} $$

Now, for $\delta '<2\delta -1$ , using Corollary 3.3, we obtain

(12) $$ \begin{align} \|\overline{D}\|_{\ell^2(\mathbb{N})}^2\leq c_1 n^{1-\delta'} \end{align} $$

for some positive constant $c_1$ depending on $C, \delta $ , and $\delta '$ .

For a given function $f: A^{n}\to \mathbb {R}$ , we define the oscillation of f at site $i\in \{1,\ldots ,n\}$ by

$$ \begin{align*} \delta_if:= \sup_{x_j=x'_j, j \neq i}|f(x_1,\ldots,x_n)-f(x'_1,\ldots,x'_n)|. \end{align*} $$

Now, taking $f(x_1,\ldots , x_{n})= ({1}/{n})\sum _{i=1}^{n}h(x_i)$ , we have $\delta _i f={R(h)}/{n}$ for $i\in \{1,\ldots ,n\}$ . Thus, we obtain

(13) $$ \begin{align} \|\delta f\|^2_{\ell^2(\mathbb{N})}=\sum_{i=1}^{n}\bigg(\frac{R(h)}{n}\bigg)^2= \frac{R(h)^2}{n}. \end{align} $$

Finally, using (12) and (13) in Theorem 3.2 of [Reference Chazottes, Collet, Külske and RedigCCKR07], we obtain Proposition 8.2.

8.3 Poisson autoregression model

As a second application of our results, we consider a model with countable infinite alphabet called Poisson autoregression, which is popular in applications [Reference Kedem and FokianosKF05]. Only the Markovian case of these models were studied in the literature. We will show how we can choose the parameters of non-Markovian Poisson autoregression models to satisfy the Assumption ( $\mathcal {A}$ ) and thus apply the results of §3.

Consider an absolutely converging sequence $(\beta _i)_{i\geq 1}$ and a sequence of non-negative integers $(\gamma _i)_{i\geq 1}$ such that $S:=\sum _{i=1}^{\infty }|\beta _i|\gamma _i<\infty $ . Consider $A=\mathbb {Z}_+$ and the potential $\phi $ defined for all $x\in \mathcal {X}_-$ by

$$ \begin{align*}\phi(x)=-\lambda(x_{-\infty}^{-1}) + x_0\log \lambda(x_{-\infty}^{-1}) - \sum_{k = 0}^{x_0}\log(k), \end{align*} $$

where

$$ \begin{align*}\lambda(x_{-\infty}^{-1})=\exp\bigg\{\sum_{i=1}^{\infty}\beta_i (x_{-i}\wedge \gamma_i)\bigg\}. \end{align*} $$

For this model, we obtain

(14) $$ \begin{align} \chi^2_k(\phi)=\sup_{a\in \mathcal{X}} \sup_{x,y\in \mathcal{X}_-}(e^{\lambda(a_{-k}^{-1}y)({\lambda(a_{-k}^{-1}x)}/({\lambda(a_{-k}^{-1}y)})-1)^2}-1). \end{align} $$

Now, since $e^{-S} \leq \lambda (x_{-\infty }^{-1})\leq e^S$ and the exponential function is locally bi-lipschitz, using (14), we have that

$$ \begin{align*} c_1^{-1} \bigg(\sum_{i=k+1}^{\infty}|\beta_i|\gamma_i\bigg)^2 \leq \chi^2_k(\phi)\leq c_1 \bigg(\sum_{i=k+1}^{\infty}|\beta_i|\gamma_i\bigg)^2, \end{align*} $$

where $c_1$ is a positive constant that depends only on S.

Finally, choosing the sequences $(\beta _i)_{i\geq 1}$ and $(\gamma _i)_{i\geq 1}$ such that

$$ \begin{align*}\frac{c_2^{-1}}{i^{({3+\varepsilon})/{2}}}\leq |\beta_i|\gamma_i \leq \frac{c_2}{i^{({3+\varepsilon})/{2}}}\end{align*} $$

for some $\varepsilon>0$ and $c_2\geq 1$ , we obtain

$$ \begin{align*} \frac{c_3^{-1}}{k^{1+\varepsilon}}\leq \chi^2_k(\phi)\leq \frac{c_3}{k^{1+\varepsilon}}, \end{align*} $$

where $c_3$ is a positive constant.

Finally, for this model, we mention that the existence of a shift-invariant probability measure compatible with $\phi $ is obtained by applying Theorem 5.1 of [Reference Johansson, Öberg and PollicottJÖP07] with $K=e^{2\sinh S}$ and $\pi $ equal to the Poisson law with parameter $e^S$ . Assumption ( $\mathcal {A}$ ) implies the square summability of the variation, which guarantees the uniqueness of the shift-invariant probability measure [Reference Johansson, Öberg and PollicottJÖP07, Corollary 4.2].