Appendix A: Finiteness of Φ
In this appendix we prove that \(\varPhi(\bar{Q})\) is finite when the one-letter marginal of \(\bar{Q}\) has a finite relative entropy w.r.t. \(\bar{\mu}\). Lemmas A.1–A.3 below were used in Sect. 4.1, in Step 3 of the proof of Theorem 3.3.
Lemma A.1
Fix
δ>0 and
\(\bar{\mu}\in\mathcal{P}(\bar{E})\)
satisfying (2.4). Then, for all
\(\bar{Q}\in \mathcal{P}^{\mathrm{inv}}(\widetilde{\bar{E}}{}^{\mathbb{N}})\)
with
\(h(\pi _{1,1}\bar{Q}\mid\bar{\mu} )<\infty \), there are constants
γ∈(δ
−1,∞) and
\(K(\delta,\gamma ,\bar{\mu} )\in(0,\infty)\)
such that
$$ \bigl|\varPhi(\bar{Q})\bigr|<\gamma h(\pi_{1,1}\bar{Q}\mid\bar{ \mu})+ K(\delta, \gamma ,\bar{\mu}). $$
(A.1)
Proof
The proof comes in 3 steps.
1. Abbreviate
$$ f(\bar{\omega}_1) = \frac{d(\pi_{1,1} \bar{Q})}{d\bar{\mu}}( \bar{\omega}_1), \quad\bar{\omega}_1 \in\bar{E}. $$
(A.2)
Fix γ∈(δ
−1,∞). For \(m\in\mathbb{N}\) and \(l\in\mathbb{Z}\), define
$$ \begin{aligned}[c] A_{m} &= \bigl\{\bar{ \omega}_1\in\bar{E}\colon\ m-1\leq\gamma \log f( \bar{ \omega}_1)<m\bigr\}, \\ A_{0} &= \bigl\{\bar{\omega}_1\in\bar{E}\colon\ 0\leq f( \bar{\omega }_1)<1\bigr\}, \\ B_{l} &= \{\bar{\omega}_1\in\bar{E}\colon\ l-1\leq \bar{\omega }_1<l\}. \end{aligned} $$
(A.3)
Note that the A
m
’s and the B
l
’s are pairwise disjoint, and that
$$ \bar{E}= A_{0} \cup \biggl[\bigcup_{m\in\mathbb{N}} A_{m} \biggr], \qquad \bar{E}_+\cup\{0\}=\bigcup_{l\in\mathbb{N}} B_{l}, \qquad\bar{E}_-= \bigcup_{l\in-\mathbb{N}_0} B_{l}, $$
(A.4)
where \(\bar{E}_{+}\) and \(\bar{E}_{-}\) denote the set of positive and negative real numbers in \(\bar{E}\). Also note that
$$ \begin{aligned}[b] \varPhi( \bar{Q}) &= \int_{ \bar{E}} \bar{\omega}_1(\pi_{1,1} \bar{Q}) (d \bar { \omega}_1) \leq\int_{ \bar{E}} (0\vee\bar{ \omega}_1)f( \bar{\omega}_1) \bar{\mu}(d \bar{ \omega}_1) \\ &=\sum_{m\in\mathbb{N}_0} \int_{A_m } (0\vee \bar{\omega}_1)\;f( \bar{\omega}_1)\bar{\mu}(d \bar{ \omega}_1) = \mathrm{I} +\mathrm{II}+\mathrm{III}, \end{aligned} $$
(A.5)
where
$$ \begin{aligned}[c] \mathrm{I} &= \int_{A_0\cap[\bigcup_{l\in\mathbb{N}} B_l] } \bar{\omega}_1 f(\bar{\omega}_1)\bar{\mu} (d\bar{ \omega}_1) \leq\sum_{l\in\mathbb{N}}l \mathbb{P}_{\bar{\omega}}(B_l ), \\ \mathrm{II} &= \sum _{m\in\mathbb{N}} \int_{A_m \cap[\bigcup_{l=1}^{m-1} B_l]} \bar{ \omega}_1f( \bar{\omega}_1)\bar{\mu}(d \bar{\omega }_1), \\ \mathrm{III} &= \sum_{m\in\mathbb{N}} \int _{A_m\cap[\bigcup_{l\in\mathbb {N}_0}B_{m+l}]} \bar{\omega}_1f( \bar{ \omega}_1)\bar{\mu}(d \bar{\omega}_1) \leq\sum _{m\in\mathbb{N}} e^{m/\gamma}\sum _{l\in\mathbb{N}_0} (m+l) \mathbb{P}_{\bar{\omega}}(B_{m+l}). \end{aligned} $$
(A.6)
The term I follows from the restriction of the \(\bar{\mu}\)-integral to the set \(A_{0}\cap\bar{E}_{+}\). The terms II and III follow from the restrictions to the sets \(\bigcup _{m\in\mathbb{N}}[A_{m}\cap \bigcup_{l=1}^{m-1} B_{l}]\) and \(\bigcup_{m\in\mathbb{N}}[A_{m}\cap\bigcup _{l\in\mathbb{N}_{0}} B_{m+l}]\). The bound on I uses that f<1 on A
0 and \(\bar{\omega}_{1}<l\) on B
l
. The bound on III follows from the fact that f<e
m/γ on A
m
and \(\bar{\omega}_{1}<m+l\) on B
m+l
. It follows from (A.6) that
$$ \begin{aligned}[b] \mathrm{I} +\mathrm{III} &\leq2 \sum_{m\in\mathbb{N}} e^{m/\gamma}\sum_{l\in \mathbb{N}_0} (m+l) \mathbb{P}_{\bar{\omega} }(B_{m+l}) \\ & \leq2 \sum_{m\in\mathbb{N}} e^{m/\gamma}\sum _{l\in\mathbb {N}_0} (m+l) \mathbb{P} _{\bar{\omega}}(\bar{\omega} _1\geq m+l-1) \\ & \leq2 C(\delta) \sum_{m\in\mathbb{N}} e^{m/\gamma}\sum _{l\in \mathbb{N}_0} (m+l) e^{-\delta(m+l-1)} \\ & \leq2 C(\delta)\,e^{\delta } \sum_{m\in\mathbb{N}} e^{-m(\delta -1/\gamma )}\sum_{l\in\mathbb{N}_0} (m+l) e^{-\delta l} =k_+(\delta,\gamma,\bar{\mu})<\infty, \end{aligned} $$
(A.7)
where the third inequality uses (2.4). Moreover, use that \(\bar{\omega} _{1}<m-1\leq\gamma \log f\) on \(A_{m} \cap\bigcup_{l=1}^{m-1} B_{l}\), to estimate
$$ \begin{aligned}[b] \mathrm{II} &\leq\gamma \sum_{m\in\mathbb{N}} \int_{\bar{\omega}_1\in A_{m} \cap[\bigcup _{l=1}^{m-1} B_{l}]} f(\bar{\omega}_1)\log f(\bar{ \omega}_1)\bar{\mu}(d \bar{\omega }_1) \\ &\leq \gamma \sum_{m\in\mathbb{N}} \int_{\bar{\omega}_1\in A_{m}} f( \bar{\omega}_1)\log f(\bar{\omega}_1)\bar{\mu}(d \bar{\omega }_1) \\ &= \gamma \int_{ \bar{E}\backslash A_{0}} f(\bar{ \omega}_1)\log f(\bar{\omega}_1)\bar{\mu}(d \bar{ \omega }_1) \\ &=\gamma h(\pi_{1,1} \bar{Q}\mid\bar{ \mu})-\gamma \int_{ A_{0}} f(\bar{\omega}_1)\log f( \bar{\omega}_1)\bar{\mu}(d \bar{\omega }_1) \\ &\leq \gamma h(\pi_{1,1} \bar{Q}\mid\bar{\mu})+\gamma e^{-1}< \infty, \end{aligned} $$
(A.8)
where the third inequality uses that flogf≥−e
−1 on A
0, and the second equality that
$$ \begin{aligned}[b] h(\pi_{1,1} \bar{Q}|\bar{\mu}) &= \int_{\bar{E}\backslash A_{0}} f( \bar{\omega}_1)\log f(\bar{\omega}_1)\bar{\mu}(d \bar{\omega }_1)+ \int_{ A_{0}} f(\bar{ \omega}_1)\log f(\bar{\omega}_1)\bar{\mu}(d \bar{ \omega }_1)\\&<\infty. \end{aligned} $$
(A.9)
Put
$$ K_+(\delta,\gamma ,\bar{\mu})=k_+(\delta,\gamma , \bar{\mu })+\gamma e^{-1}. $$
(A.10)
2. Similarly, we have
$$ \begin{aligned}[b] \varPhi(\bar{Q}) &= \int_{ \bar{E}} \bar{\omega}_1(\pi_{1,1} \bar{Q}) (d \bar { \omega}_1) \geq\int_{ \bar{E}} (0\wedge\bar{ \omega}_1)f( \bar{\omega }_1)\bar{\mu}(d \bar{ \omega}_1) \\ &=\sum_{m\in\mathbb{N}_0} \int_{A_m } (0 \wedge\bar{\omega}_1) f( \bar{\omega}_1)\bar{ \mu}(d \bar{\omega}_1) = \mathrm{I}' +\mathrm{II}'+\mathrm{III}', \end{aligned} $$
(A.11)
where
$$ \begin{aligned}[c] \mathrm{I}' &= \int_{A_0\cap[\bigcup_{l\in-\mathbb{N}_0} B_l] } \bar{\omega }_1f(\bar{\omega}_1)\bar{\mu} (d \bar{ \omega}_1) \geq\sum_{l\in-\mathbb{N}_0}(l-1) \mathbb{P}_{\bar{\omega }}(B_l ), \\ \mathrm{II}' &= \sum _{m\in\mathbb{N}} \int_{A_m \cap[\bigcup_{l=-m+1}^{0} B_l]} \bar{ \omega}_1f(\bar{\omega}_1)\;\bar{\mu}(d \bar{\omega }_1), \\ \mathrm{III}' &= \sum_{m\in\mathbb{N}} \int_{A_m\cap[\bigcup_{l\in-\mathbb {N}_0}B_{l-m}] } \bar{\omega}_1f(\bar{ \omega}_1)\;\bar{\mu}(d \bar{\omega}_1) \geq\sum _{m\in\mathbb{N}} e^{m/\gamma} \sum _{l\in-\mathbb{N}_0} (l-m-1) \mathbb{P}_{\bar{\omega}}(B_{l-m}). \end{aligned} $$
(A.12)
The bounds on I′ and III′ use that \(\bar{\omega}_{1}\geq l-1\) on B
l
and f<e
m/γ on A
m
. Note that
$$ \begin{aligned}[b] \mathrm{I}'+\mathrm{III}' &\geq2 \sum _{m\in\mathbb{N}_0} e^{m/\gamma}\sum _{l\in -\mathbb{N}_0} (l-m-1) \mathbb{P} _{\bar{\omega}}(B_{l-m}) \\ & \geq2\sum_{m\in\mathbb{N}_0} e^{m/\gamma}\sum _{l\in-\mathbb {N}_0} (l-m-1) \mathbb{P} _{\bar{\omega}}(\bar{\omega} _1\leq l-m) \\ &\geq2 C(\delta)\sum_{m\in\mathbb{N}_0} e^{m/\gamma}\sum _{l\in -\mathbb{N}_0} (l-m-1) e^{\delta(l-m)} =k_-(\delta,\gamma , \bar{\mu})>-\infty. \end{aligned} $$
(A.13)
Also use that \(\bar{\omega}_{1}\geq-m\geq-[\gamma \log f+1]\) on \(~A_{m} \cap\bigcup _{l=-m+1}^{0} B_{l}\), to estimate
$$ \begin{aligned}[b] \mathrm{II}' &\geq-\sum _{m\in\mathbb{N}} \int_{\bar{\omega}_1\in A_{m} \cap[\bigcup _{l=-m+1}^{0} B_{l}]} f( \bar{ \omega}_1)\bigl[\gamma \log f( \bar{\omega}_1)+1\bigr] \bar{\mu }(d \bar{\omega}_1) \\ &\geq-\gamma \sum _{m\in\mathbb{N}} \int_{ A_{m}} f( \bar{ \omega}_1)\log f( \bar{\omega}_1)\,\bar{\mu}(d \bar { \omega}_1)-1 \\ &= -\gamma \int_{ \bar{E}\backslash A_{0}} f( \bar{ \omega}_1)\log f( \bar{\omega}_1)\,\bar{\mu}(d \bar { \omega}_1) -1 \\ &=-\gamma h(\pi_{1,1} \bar{Q}\mid\bar{ \mu})+\gamma \int_{ A_{0}} f( \bar{\omega}_1)\log f( \bar{\omega}_1)\,\bar{\mu}(d \bar {\omega}_1)-1 \\ &\geq-\bigl[\gamma h(\pi_{1,1} \bar{Q}\mid\bar{\mu})+\gamma e^{-1}+1\bigr]>-\infty. \end{aligned} $$
(A.14)
3. Put \(K_{-}(\delta,\gamma ,\bar{\mu})=1+\gamma e^{-1}-k_{-}(\delta,\gamma ,\bar{\mu})\). Then the claim follows with \(K(\delta,\gamma ,\bar{\mu})=K_{+}(\delta,\gamma ,\bar{\mu })\vee K_{-}(\delta,\gamma ,\bar{\mu})\). □
For the sake of completeness we state the follow finiteness results for \(\varPhi_{\hat{\beta}, \hat{h}}\) that were proved in [4], Appendix A.
Lemma A.2
Fix
\(\hat{\beta}, \hat{h},g>0\). Then
\(\hat{\omega}\)-a.s. there exists a
\(K(\hat{\omega},\hat{\beta}, \hat{h} ,g)<\infty\)
such that, for all
\(N\in\mathbb{N}\)
and for all sequences 0=k
0<k
1<⋯<k
N
<∞,
$$ -g k_N + \sum_{i=1}^N \log\phi_{\hat{\beta}, \hat{h}} (\hat {\omega} _{(k_{i-1},k_i]} ) \leq K(\hat{\omega}, \hat{\beta}, \hat{h},g)N, $$
(A.15)
where
\(\hat{\omega}_{(k_{i-1},k_{i}]}\)
is the word cut out from
\(\hat {\omega}\)
by the
ith excursion interval (k
i−1,k
i
].
Lemma A.3
Fix
\(\hat{\beta}, \hat{h}>0\), \(\rho\in\mathcal{P}(\mathbb{N})\)
and
\(\hat{\mu}\in\mathcal{P}(\mathbb{R})\)
satisfying (2.2) and (2.4). Then, for all
\(\hat{Q}\in\mathcal{P}^{\mathrm{inv}}(\widetilde {\hat{E}}^{\mathbb{N}})\)
with
\(h(\pi_{1} \hat{Q}\mid q_{\rho,\hat{\mu}})<\infty\), there are finite constants
C>0, \(\gamma>2\hat{\beta}/C\)
and
\(K=K(\hat{\beta}, \hat{h},\rho,\hat{\mu},\gamma)\)
such that
$$ \varPhi_{\hat{\beta}, \hat{h}}(\hat{Q})\leq\gamma\; h(\pi_1 \hat {Q}|q_{\rho,\hat{\mu}})+K. $$
(A.16)
Appendix B: Application of Varadhan’s Lemma
In this appendix we prove (4.18) and the claim above it. This was used in Sect. 4 to complete the proof of Theorem 4.1.
Lemma B.1
For every
\(\hat{\beta},\hat{h}>0\)
and
\(\bar{\beta}\geq0\),
$$ s^\mathrm{que}(\hat{\beta},\hat{h},\bar{\beta};g) = S^\mathrm {que}(\hat{\beta},\hat{h},\bar{\beta};g) \quad \forall g\in \mathbb{R}, $$
(B.1)
where
\(s^{\mathrm{que}}(\hat{\beta},\hat{h},\bar{\beta};g)\)
is the
ω-a.s. constant limit defined in (4.16), and
\(S^{\mathrm{que}}(\hat {\beta} ,\hat{h} ,\bar{\beta};g)\)
is as in (4.4). In particular, the map
\(g\mapsto S^{\mathrm{que}}(\hat {\beta} ,\hat{h} ,\bar{\beta};g)\)
is finite on (0,∞) and infinite on (−∞,0).
Proof
Throughout the proof \(\hat{\beta},\hat{h}>0\), \(\bar{\beta}\geq0\) and \(\bar{h}\in\mathbb{R}\) are fixed. The proof comes in 3 steps, where we establish the equality in (B.1) for the cases g<0, g=0 and g>0 separately.
Step 1. For g<0 the proof of (B.1) is given in two steps.
1a. In this step we show that \(S^{\mathrm{que}}(\hat{\beta},\hat {h},\bar{\beta} ;g)=\infty\) when g<0. Fix \(L\in\mathbb{N}\) and let \(Q^{L}=(q^{L}_{\hat{\mu}\otimes\bar {\mu} })^{\otimes\mathbb{N}}\), with
$$ q^L_{\hat{\mu}\otimes\bar{\mu}}(d\omega _1,\ldots,d\omega _n) =\delta_{Ln}(\hat{\mu}\otimes\bar{\mu})^{\otimes n}(d \omega _1,\ldots,d\omega _n) $$
(B.2)
and \((\omega _{1},\ldots,\omega _{n})=((\hat{\omega}_{1},\bar{\omega }_{1}),\ldots,(\hat{\omega}_{n},\bar{\omega}_{n}))\in E^{n}\). It follows from (4.4) that
$$ \begin{aligned}[b] S^\mathrm{que}(\hat{\beta},\hat{h},\bar{\beta};g) &\geq\bar{\beta}\varPhi \bigl(Q^L\bigr)+\varPhi_{\hat{\beta},\hat{h}}\bigl(Q^L \bigr)-I^{\mathrm{ann}}\bigl(Q^L\bigr)-gL\\& \geq-\log2-gL+\log\rho(L). \end{aligned} $$
(B.3)
The second inequality uses that Φ(Q
L)=0, I
ann(Q
L)=−logρ(L) and \(\varPhi_{\hat{\beta},\hat{h}}(Q^{L})\geq-\log2\). Letting L→∞ and using that ρ has a polynomial tail by (2.2), we get the claim.
1b. In this step we show that \(s^{\mathrm{que}}(\hat{\beta},\hat {h},\bar{\beta} ;g)=\infty\) when g<0. The proof follows from a moment estimate. We start by showing that, for each \(\bar{\beta}\in\mathbb{R}\),
$$ \limsup_{N\to\infty} \frac{1}{N} \log E^\ast_0 \bigl(e^{N\bar{\beta }\varPhi (R^{\bar{\omega} }_N)} \bigr) \leq \bar{M}(-\bar{\beta}) $$
(B.4)
(recall (2.4)). Indeed, for any \(\bar{\beta}\in\mathbb {R}\), by the Markov inequality,
$$ \begin{aligned}[b] \mathbb{P}_{\bar{\omega}} \biggl(\frac{1}{N} \log E^\ast_0 \bigl(e^{N\bar{\beta}\varPhi(R^{\bar{\omega} }_N)} \bigr) \geq \bar{M}(-\bar{\beta})+\epsilon \biggr) &= \mathbb{P}_{\bar{\omega}} \bigl( E^\ast_0 \bigl(e^{N\bar{\beta }\varPhi(R^{\bar{\omega}}_N)} \bigr) \geq e^{N( \bar{M}(-\bar{\beta})+\epsilon)} \bigr) \\ &\leq e^{-N \bar{M}(-\bar{\beta})}e^{-\epsilon N} \mathbb{E}_{\bar{\omega}} \bigl(E^\ast_0 \bigl(e^{N\bar{\beta }\varPhi(R^{\bar{\omega}}_N)} \bigr) \bigr) \\ &=e^{-N \bar{M}(-\bar{\beta})}e^{-\epsilon N} E_0^\ast \bigl[\mathbb{E}_{\bar{\omega}} \bigl(e^{\bar{\beta}\sum _{i=1}^N\bar{\omega} _{k_{i-1}}} \bigr) \bigr]\\& =e^{-\epsilon N}. \end{aligned} $$
(B.5)
The claim therefore follows from the Borel-Cantelli lemma. Note from (B.5) that (B.4) holds if we replace \(E^{\ast}_{0}\) with \(E^{\ast}_{g}\), g≥0.
Let τ
i
be the length of the i-th word, let \(L\in\mathbb{N}\), and put
$$ k_N=\sum_{i=1}^N \tau_i\quad\mbox{and}\quad k_N(L) =\sum _{i=1}^N [\tau_i1_{\{\tau_i<L\}}+ L 1_{\{\tau_i\geq L\} } ]. $$
(B.6)
For any −∞<q<0<p<1 with p
−1+q
−1=1 and g<0, it follows from (4.13) that
$$ \begin{aligned}[b] e^{\bar{h}N}F_N^{\hat{\beta},\hat{h},\bar{\beta},\bar{h},\omega }(g) &=E^\ast_0 \bigl(\exp \bigl[-gk_N+N \bigl( \varPhi_{\hat{\beta},\hat {h}}\bigl(R_N^{\omega }\bigr)+\bar{\beta} \varPhi\bigl(R_N^{\omega }\bigr) \bigr) \bigr] \bigr), \\ &\geq \biggl(\frac{1}{2} \biggr)^NE^\ast_0 \bigl(\exp \bigl[-gk_N(L)+N\bar{\beta} \varPhi\bigl(R_N^{\omega } \bigr) \bigr] \bigr) \\ &\geq \biggl(\frac{1}{2} \biggr)^NE^\ast_0 \bigl(e^{-g pk_N(L)} \bigr)^{1/p} E^\ast_0 \bigl(e^{Nq\bar{\beta}\varPhi(R_N^{\omega })} \bigr)^{1/q} \\ & = \biggl(\frac{1}{2} \biggr)^N\,\mathcal{N}_L(pg)^{N/p} E^\ast_0 \bigl(e^{Nq\bar{\beta}\,\varPhi(R_N^{\omega })} \bigr)^{1/q}, \end{aligned} $$
(B.7)
where
$$ \mathcal{N}_L(g)=\sum_{n=1}^{L-1} \rho(n)e^{-ng}+e^{-Lg}\sum_{n\geq L} \rho(n). $$
(B.8)
The first inequality in (B.7) uses that \(\varPhi_{\hat{\beta },\hat{h} }(Q)\geq -\log2\), k
N
≥k
N
(L) and g<0. The second inequality follows from the reverse Hölder inequality with the above choice of p and q. Note that \(\mathcal{N}_{L}(g)\) is finite for \(g\in\mathbb{R}\) and \(\lim_{L\to\infty}\mathcal{N}_{L}(g)=\mathcal{N}(g)\). It therefore follows from (4.16), (B.4) and (B.7) that
$$ \begin{aligned}[b] s^{\mathrm{que}}(\hat{\beta},\hat{h}, \bar{\beta};g) &=\limsup_{N\to\infty}\frac{1}{N} \log \bigl(e^{N \bar{h}} F_N^{\hat{\beta},\hat{h},\bar{\beta},\bar{h},\omega }(g) \bigr) \\ &\geq-\log2+ \frac{1}{p}\log\mathcal{N}_L(pg)+\frac{1}{q} \bar{M}(-\bar {\beta}q). \end{aligned} $$
(B.9)
Letting L→∞, we get from (3.6) that \(s^{\mathrm{que}}(\hat{\beta} ,\hat{h} ,\bar{\beta};g) =\infty\), since \(\mathcal{N}(pg)=\infty\) for g∈(−∞,0).
Step 2. In this step, which is divided into 2 substeps, we consider the case g>0.
2a. Lower bound. For M>0, define
$$ \varPhi^{-M}(Q)=\int _{\bar{E}}(\bar{\pi}_{1,1}Q) (d\bar{\omega }_1)\bigl[\bar{\omega}_1\vee(-M)\bigr]. $$
(B.10)
Note that Φ
−M is lower semi-continuous and that
$$ \bar{\beta}\varPhi^{-M}(Q)+ \varPhi_{\hat{\beta},\hat{h}}(Q)\leq\bar {\beta}\varPhi(Q)+\varPhi_{\hat{\beta},\hat{h}}(Q) - \bar{\beta}\int_{\bar{\omega}_1<-M} \bar{\omega}_1\,(\bar{\pi}_{1,1} Q) (d\bar{\omega}_1). $$
(B.11)
Therefore, for any p,q>1 with 1/p+1/q=1, it follows from the Hölder inequality that
$$ \begin{aligned}[b] &\frac{1}{N} \log E_g^\ast \bigl(e^{N (\varPhi_{\hat{\beta},\hat {h}}(R_N^\omega ) +\bar{\beta}\varPhi^{-M}(R^\omega _N) )} \bigr) \\ & \quad \leq \frac{1}{pN} \log E_g^\ast \bigl(e^{pN [\varPhi_{\hat{\beta },\hat{h} }(R_N^\omega ) +\bar{\beta}\,\varPhi(R^\omega _N) ]} \bigr) + \frac{1}{qN} \log E_g^\ast \bigl(e^{-q\,\bar{\beta}\sum_{i=1}^N \bar{\omega}_{k_{i}} 1_{\{\bar{\omega}_{k_{i}}<-M\}}} \bigr). \end{aligned} $$
(B.12)
The rest of the proof consists of taking the appropriate limits and showing that the left-hand side of (B.12) is bounded from below by \(S^{\mathrm{que}}(\hat{\beta}, \hat{h},\bar{\beta};g)\), while the second term in the right-hand side tends to zero and the first term tends to \(s^{\mathrm{que}}(\hat{\beta},\hat{h},\bar{\beta};g)\).
Let us start with the second term in the right-hand side of (B.12). Note from (3.6) that
$$ \begin{aligned}[b] &\limsup_{N\to\infty} \frac{1}{qN}\log E_g^\ast \bigl(e^{-q\, \bar{\beta}\sum_{i=1}^N \bar{\omega}_{k_{i}}1_{\{\bar{\omega }_{k_{i}<-M\}}}} \bigr) \\ & \quad =-\frac{1}{q}\log\mathcal{N}(g)+ \limsup_{N\to\infty} \frac {1}{qN}\log E_0^\ast \bigl(e^{-gk_N-q\, \bar{\beta}\sum_{i=1}^N \bar{\omega}_{k_{i}}1_{\{\bar{\omega }_{k_{i}<-M\}}}} \bigr) \\ & \quad \leq-\frac{1}{q}\log\mathcal{N}(g)+ \frac{1}{2q}\log\mathcal{N}(2g) + \limsup_{N\to\infty}\frac{1}{2qN}\log E_0^\ast \bigl(e^{-2q\, \bar{\beta}\sum_{i=1}^N \bar{\omega}_{k_{i}}1_{\{\bar{\omega }_{k_{i}<-M\}}}} \bigr) \\ & \quad \leq-\frac{1}{q}\log\mathcal{N}(g)+ \frac{1}{2q}\log\mathcal{N}(2g)+ \frac{1}{2q} \log \int _{\bar{E}}e^{-2q\bar{\beta}\bar{\omega}_1 1_{\{\bar{\omega }_{1}<-M\}}}\bar{\mu}(d\bar{\omega}_1). \end{aligned} $$
(B.13)
The first inequality uses the Cauchy-Schwarz inequality, the second inequality uses (B.4). Note from (2.4) that the above bound tends to zero upon when M→∞ followed by q→∞.
For the first term in the right-hand side of (B.12) we proceed as follows. Note from Lemma A.2 that \(\hat{\omega}\)-a.s.
$$ -gk_N+pN\varPhi_{\hat{\beta},\hat{h}}\bigl(R_N^x \bigr)\leq NK(\hat{\omega },p,\hat{\beta},\hat{h},g), $$
(B.14)
where we use that
$$ -gk_N+pN\varPhi_{\hat{\beta},\hat{h}}\bigl(R_N^\omega \bigr) =p \biggl(-\frac{g}{p}k_N+N\varPhi_{\hat{\beta},\hat{h}} \bigl(R_N^\omega \bigr) \biggr). $$
(B.15)
Therefore, for any 1<p<∞, it follows from (B.4) and (B.14) that \(\hat{\omega}\)-a.s.
$$ \begin{aligned}[b]& \limsup_{N\to\infty} \frac{1}{pN}\log E_g^\ast \bigl(e^{Np [\varPhi_{\hat{\beta},\hat {h}}(R_N^\omega ) +\bar{\beta}\,\varPhi(R^\omega _N) ]} \bigr) \\ & \quad =-\frac{1}{p}\log\mathcal{N}(g) +\limsup_{N\to\infty} \frac{1}{pN}\log E_0^\ast \bigl(e^{-gk_N+Np [\varPhi_{\hat{\beta},\hat {h}}(R_N^\omega ) +\bar{\beta}\varPhi(R^\omega _N) ]} \bigr) \\ & \quad \leq\frac{1}{p} \bigl(K(p,\hat{\beta},\hat{h},g)+ \bar{M}(-p\bar {\beta})- \log\mathcal{N}(g) \bigr)<\infty. \end{aligned} $$
(B.16)
Next, for −∞<r<0<s<1 with r
−1+s
−1=1, it follows from the argument leading to (B.9) that
$$ \begin{aligned}[b]& \limsup_{N\to\infty} \frac{1}{pN}\log E_g^\ast \bigl(e^{Np [\varPhi_{\hat{\beta},\hat {h}}(R_N^\omega ) +\bar{\beta}\varPhi(R^\omega _N) ]} \bigr) \\ & \quad \geq\log\frac{1}{2}+\frac{1}{p}\mathcal{N}(g) + \frac{1}{sp} \mathcal{N}(sg) +\frac{1}{pr} \bar{M}(-pr\bar{\beta}) >-\infty, \end{aligned} $$
(B.17)
since g∈(0,∞). Define
$$ S(p) = \limsup_{N\to\infty}\frac{1}{N}\log E_g^\ast \bigl(e^{Np [\varPhi_{\hat{\beta},\hat {h}}(R_N^\omega ) + \bar{\beta}\varPhi(R^\omega _N) ]} \bigr). $$
(B.18)
By (B.16)–(B.17), the map p↦S(p) is convex and finite on (0,∞), and hence continuous on (0,∞). It therefore follows from (4.15) that the left-hand side of (B.16) converges to \(s^{\mathrm{que}} (\hat{\beta},\hat{h},\bar{\beta};g)-\log\mathcal{N}(g)\) as p↓1. It follows from (B.16)–(B.17) that this limit is finite, which proves the finiteness of the map \(g\mapsto s^{\mathrm{que}} (\hat{\beta},\hat{h},\bar{\beta};g)\) on (0,∞).
Finally, we turn to the left-hand side of the inequality in (B.12). For any ϵ>0 and \(Q\in\mathcal{C}^{\mathrm{fin}}\cap\mathcal{R}\), note from the lower semi-continuity of the map \(Q\mapsto\bar{\beta}\varPhi^{-M}(Q)+\varPhi_{\hat{\beta},\hat {h}}(Q)\) that the set
$$\begin{aligned} A_\epsilon(Q)= \bigl\{Q'\in\mathcal{P}^{\mathrm{inv}}\bigl( \widetilde {E}^\mathbb{N}\bigr)\colon\ \bar{\beta}\varPhi^{-M} \bigl(Q'\bigr)+\varPhi_{\hat{\beta},\hat{h}}\bigl(Q'\bigr) \geq\bar{\beta}\varPhi^{-M}(Q)+\varPhi_{\hat{\beta},\hat {h}}(Q)-\epsilon \bigr\} \end{aligned}$$
(B.19)
is open. This implies that
$$ \begin{aligned}[b]& \limsup_{N\to\infty} \frac{1}{N}\log E_g^\ast \bigl(e^{N [\varPhi_{\hat{\beta},\hat {h}}(R_N^\omega ) + \bar{\beta}\varPhi^{-M}(R^\omega _N) ]} \bigr) \\ & \quad \geq\liminf_{N\to\infty}\frac{1}{N} \log E_g^\ast \bigl(e^{N [\varPhi_{\hat{\beta},\hat {h}}(R_N^\omega ) + \bar{\beta}\varPhi^{-M}(R^\omega _N) ]}1_{A_\epsilon (Q)} \bigl(R_N^\omega \bigr) \bigr) \\ & \quad \geq\bar{\beta} \varPhi^{-M}(Q)+\varPhi_{\hat{\beta},\hat{h}}(Q) -\epsilon-\inf _{Q'\in A_\epsilon(Q)}I_g^{\mathrm{que}}\bigl(Q' \bigr) \\ & \quad \geq\bar{\beta}\varPhi^{-M}(Q)+\varPhi_{\hat{\beta},\hat {h}}(Q)-I_g^{\mathrm{que}}(Q)- \epsilon \\ & \quad \geq\bar{\beta}\varPhi(Q)+\varPhi_{\hat{\beta},\hat{h}}(Q)-I_g^{\mathrm{que}}(Q)- \epsilon \\ & \quad = \bar{\beta}\varPhi(Q)+\varPhi_{\hat{\beta},\hat{h}}(Q)-gm_Q-I^{\mathrm{ann}}(Q)- \log\mathcal{N} (g)-\epsilon. \end{aligned} $$
(B.20)
The second inequality uses Theorem 3.3, the third inequality uses that Q∈A
ϵ
(Q), the fourth inequality follows from the fact that Φ≤Φ
−M, while the equality follows from Lemma 3.9. It therefore follows from (B.12)–(B.13), (B.20) and the comment below (B.18) that, after taking the supremum over \(\mathcal{C}^{\mathrm{fin}}\cap \mathcal{R}\) followed by M→∞, ϵ→0 and p↓1,
$$ \begin{aligned}[] s^{\mathrm{que}}(\hat{\beta},\hat{h},\bar{\beta};g)& \geq\sup _{Q\in \mathcal{C}^{\mathrm{fin}}\cap\mathcal{R}} \bigl[\bar{\beta}\varPhi(Q)+\varPhi_{\hat{\beta},\hat {h}}(Q)-gm_Q-I^{\mathrm{ann}}(Q) \bigr]\\& = S^{\mathrm{que}}(\hat{\beta},\hat{h},\bar{\beta};g). \end{aligned} $$
(B.21)
2b. Upper bound. Let \(\chi(\hat{\omega}_{I_{i}})=\log\phi_{\hat{\beta},\hat {h}}(\hat{\omega}_{I_{i}})\). For M>0, define
$$ \begin{aligned}[c] \varPhi^M(Q)&=\int _{\bar{E}}(\bar{\pi}_{1,1}Q) (d\bar{\omega}_1) (\bar {\omega}_1\wedge M), \\ \varPhi_{\hat{\beta},\hat{h}}^{M}(Q) &=\int_{\widetilde{\hat{E}}}(\hat{\pi}_{1}Q) (d\hat{\omega }_{I_1}) \bigl(\chi(\hat{\omega} _{I_1})\wedge M\bigr). \end{aligned} $$
(B.22)
Note that Φ
M and \(\varPhi_{\hat{\beta},\hat{h}}^{M}\) are upper semi-continuous and
$$ \begin{aligned}[b]& \bar{\beta}\varPhi(Q)+ \varPhi_{\hat{\beta},\hat{h}}(Q)-\int_{\hat {\omega}_1\geq M} \bar{ \omega}_1(\bar{\pi}_{1,1}Q) (d\bar{\omega}_1) - \int_{\chi\geq M}(\hat{\pi}_{1}Q) (d\hat{ \omega}_{I_1})\chi(\hat {\omega}_{I_1})\\& \quad \leq \bar{\beta} \varPhi^M(Q)+ \varPhi_{\hat{\beta},\hat{h}}^{M}(Q). \end{aligned} $$
(B.23)
Therefore, for any −∞<q<0<p<1 with q
−1+p
−1=1, the reverse Hölder inequality gives
$$ \begin{aligned}[b] &\frac{1}{N} \log E^\ast_g \bigl(e^{N [\bar{\beta}\varPhi^M(R_N^\omega )+\varPhi ^M_{\hat{\beta},\hat{h} }(R_N^\omega ) ]} \bigr) \\ &\quad\geq \frac{1}{qN} \log E^\ast_g \bigl(e^{-q [\bar{\beta }\sum _{i=1}^N\bar{\omega}_{k_{i}} 1_{\{\bar{\omega}_{k_{i}\geq M}\}} + \sum_{i=1}^N\chi(\hat{\omega}_{I_i})1_{\{\chi(\hat{\omega }_{I_i})\geq M\}} ]} \bigr) \\ &\qquad{}+ \frac{1}{pN} \log E^\ast_g \bigl(e^{pN [\bar {\beta} \varPhi (R_N^\omega ) +\varPhi_{\hat{\beta},\hat{h}}(R_N^\omega ) ]} \bigr). \end{aligned} $$
(B.24)
The rest of the proof for the upper bound follows after showing that the left-hand side of (B.24) gives rise to the desired upper bound, while the right-hand side gives rise to \(s^{\mathrm{que}}(\hat{\beta},\hat{h},\bar{\beta};g)\) after taking appropriate limits.
It follows from [4], Step 2 in the proof of Lemma B.1, that
$$ \frac{g}{q} k_N + \sum_{i=1}^N \chi(\hat{\omega}_{I_i})1_{\{\chi(\hat {\omega}_{I_i})\geq M\} }\leq0 $$
(B.25)
for M large enough. Hence, for M large enough, it follows from (B.4), (B.25) and q<0 that
$$ \begin{aligned}[b] &\limsup_{N\to\infty} \frac{1}{qN}\log E^\ast_g \bigl(e^{-q [\bar{\beta}\sum_{i=1}^N\bar{\omega}_{k_{i}} 1_{\{\bar{\omega}_{k_{i}\geq M}\}}+\sum_{i=1}^N\chi(\hat{\omega}_{I_i}) 1_{\{\chi(\hat{\omega}_{I_i})\geq M\}} ]} \bigr) \\ & \quad =-\frac{1}{q}\log\mathcal{N}(g)+\limsup_{N\to\infty} \frac {1}{qN}\log E^\ast_0 \bigl(e^{-q [\frac{g}{q} k_N+\sum_{i=1}^N\chi(\hat{\omega}_{I_i}) 1_{\{\chi(\hat{\omega}_{I_i})\geq M\}}+\bar{\beta}\sum _{i=1}^N\bar{\omega}_{k_{i}} 1_{\{\bar{\omega}_{k_{i}\geq M}\}} ]} \bigr) \\ & \quad \geq\frac{1}{q} \biggl(\log \int_{\mathbb{R}}e^{-q\bar{\beta}\bar{\omega}_1 1_{\{\bar{\omega }_{1}\geq M\}}} \bar{\mu}(d\bar{\omega}_1)-\log \mathcal{N} (g) \biggr), \end{aligned} $$
(B.26)
which tends to zero as M→∞ followed by q→−∞. Furthermore, it follows from (B.16)–(B.18) and the remark below (B.18) that
$$ s^{\mathrm{que}}(\hat{\beta},\hat{h},\bar{\beta};g)-\log \mathcal{N}(g) =\lim_{p \uparrow1}\limsup_{N\to\infty} \frac{1}{pN}\log E^\ast_g \bigl(e^{pN [\bar{\beta}\varPhi(R_N^\omega )+\varPhi _{\hat{\beta},\hat{h}}(R_N^\omega ) ]} \bigr). $$
(B.27)
Since \(\bar{\beta}\varPhi^{M}+\varPhi^{M}_{\hat{\beta},\hat{h}}\) is upper semi-continuous, it follows from Dembo and Zeitouni [7], Lemma 4.3.6, and Theorem 3.3 that
$$ \begin{aligned}[b]& \limsup_{N\to\infty} \log E^\ast_g \bigl(e^{N [\bar{\beta}\varPhi^M(R_N^\omega )+\varPhi^M_{\hat {\beta},\hat{h}}(R_N^\omega ) ]} \bigr)\\& \quad \leq\sup _{Q\in\mathcal{P}^{\mathrm{inv}}(\widetilde{E}^{\otimes\mathbb{N}})} \bigl[\bar{\beta}\varPhi^M(Q)+ \varPhi^M_{\hat{\beta},\hat {h}}(Q)-I_g^{\mathrm{que}}(Q) \bigr] \\ & \quad =\sup_{Q\in\mathcal{R}} \bigl[\bar{\beta}\varPhi^M(Q)+ \varPhi^M_{\hat {\beta},\hat{h} }(Q)-I_g^{\mathrm{que}}(Q) \bigr] \\ & \quad =\sup_{Q\in\mathcal{C}^{\mathrm{fin}}\cap\mathcal{R}} \bigl[\bar {\beta}\varPhi^M(Q)+ \varPhi ^M_{\hat{\beta},\hat{h} }(Q)-gm_Q-I^{\mathrm{ann}}(Q) \bigr]-\log\mathcal{N}(g) \\ & \quad \leq\sup_{Q\in\mathcal{C}^{\mathrm{fin}}\cap\mathcal{R}} \bigl[\bar {\beta} \varPhi(Q)+\varPhi _{\hat{\beta},\hat{h} }(Q)-gm_Q-I^{\mathrm{ann}}(Q) \bigr]-\log\mathcal{N}(g). \end{aligned} $$
(B.28)
The first equality uses that \(I_{g}^{\mathrm{que}}(Q)=\infty\) for \(Q\notin \mathcal{R}\) (recall (3.14)) and the fact that \(\hat{\beta }\varPhi ^{M}+\varPhi ^{M}_{\hat{\beta},\hat{h}}\leq M(1+\bar{\beta})\), the second equality uses (3.9) and the fact that \(I_{g}^{\mathrm{que}} =I_{g}^{\mathrm{ann}}\) on \(\mathcal{R}\). (The removal of Q’s with m
Q
=I
ann(Q)=∞ again follows from \(\hat{\beta}\varPhi^{M}+\varPhi^{M}_{\hat{\beta},\hat {h}}\leq M(1+\bar{\beta})\)), the last inequality uses that Φ
M(Q)≤Φ(Q) and \(\varPhi^{M}_{\hat{\beta},\hat {h}}(Q)\leq \varPhi_{\hat{\beta} ,\hat{h}}(Q)\). Therefore, combining (B.24)–(B.28) and letting M→∞ and p↑1 in the appropriate order, we conclude the proof of the upper bound.
Step 3. For g=0 we show that
$$ \lim_{g\downarrow0}s^{\mathrm{que}}(\hat{\beta}, \hat{h},\bar{\beta };g)=\lim_{g\downarrow 0}S^{\mathrm{que}}(\hat{\beta}, \hat{h},\bar{\beta};g) =S^{\mathrm{que}}(\hat{\beta},\hat{h},\bar{ \beta};0)=S_*^{\mathrm{que}}(\hat {\beta},\hat{h},\bar{\beta}) $$
(B.29)
(recall (4.8)). The first equality follows from Steps 1 and 2 of the proof. The second uses that the map \(g\mapsto S^{\mathrm{que}}(\hat{\beta},\hat {h},\bar{\beta};g)\) is decreasing and lower semi-continuous on [0,∞).
Furthermore, note from (4.4) and (4.8) that
$$ \begin{aligned}[b] S_*^{\mathrm{que}}(\hat{\beta}, \hat{h},\bar{\beta}) &=\sup_{Q\in\mathcal{C}^{\mathrm{fin}}} \bigl[\bar{\beta}\varPhi(Q) + \varPhi_{\hat{\beta}, \hat{h}}(Q)-I^{\mathrm{que}}(Q) \bigr] \\ &\geq\sup _{Q\in\mathcal{C}^{\mathrm{fin}}\cap\mathcal{R}} \bigl[\bar {\beta}\varPhi(Q) +\varPhi_{\hat{\beta}, \hat{h}}(Q)-I^{\mathrm{que}}(Q) \bigr] \\ &= S^{\mathrm{que}}(\hat{\beta},\hat{h},\bar{\beta};0). \end{aligned} $$
(B.30)
Here we use that I
que=I
ann on \(\mathcal{R}\). The rest of the proof will follow once we show the reverse of (B.30). To do so we proceed as follows:
For \(L\in\mathbb{N}\) and −∞<q<0<p<1, with p
−1+q
−1=1, it follows from (4.15) and the reverse Hölder inequality that
$$ \begin{aligned}[b] & s^{\mathrm{que}}(\hat{\beta}, \hat{h},\bar{\beta};g)-\log\mathcal {N}(g)\\& \quad = \limsup_{N\rightarrow \infty } \frac{1}{N} \log E_g^\ast \bigl(e^{N [\bar{\beta}\varPhi(R_N^\omega )+\varPhi _{\hat{\beta} ,\hat{h} }(R_N^\omega ) ]} \bigr) \\ & \quad \geq\limsup_{N\rightarrow\infty}\frac{1}{pN} \log E_g^\ast \bigl(e^{Np [\bar{\beta}\varPhi^{-L}(R_N^\omega )+\varPhi _{\hat{\beta},\hat{h} }(R_N^\omega ) ]} \bigr)\\& \qquad {}+ \limsup _{N\rightarrow\infty}\frac{1}{qN}\log E_g^\ast \bigl(e^{Nq\bar{\beta}\sum_{i=1}^N \bar{\omega}_{k_i}1_{\{\bar{\omega}_{k_i}<-L\}}} \bigr) \\ & \quad \geq- \frac{1}{p}\log\mathcal{N}(g)+ \frac{1}{p}\sup _{Q\in\mathcal {C}^{\mathrm{fin}}\cap \mathcal{R}} \bigl[p \bigl(\bar{\beta}\varPhi^{-L}(Q)+ \varPhi_{\hat{\beta},\hat{h}}(Q) \bigr)-gm_Q-I^{\mathrm{ann}}(Q) \bigr] \\ & \qquad {}+\frac{1}{q}\log\int_\mathbb{R}e^{\bar{\beta}q \bar{\omega}_11_{\{\bar{\omega}_{1}<-L\}}}\bar{\mu}(d\bar{\omega}_1) \\ & \quad =\frac{1}{p} s_{L}^{\mathrm{que}}(\hat{ \beta},\hat{h},\bar{\beta},p;g)+ \frac{1}{q}\log\int_\mathbb{R} e^{\bar{\beta}q \bar{\omega}_1 1_{\{\bar{\omega}_{1}<-L\}}}\bar{\mu}(d\bar{\omega}_1)- \frac{1}{p}\log \mathcal{N}(g). \end{aligned} $$
(B.31)
Here, Φ
−L is defined in (B.10). The second inequality uses Steps 1 and 2 of the proof, particularly (B.4) and the remark below (B.5).
Below we show that, for any p∈[0,∞) and \(L\in\mathbb{N}\),
$$ \begin{aligned}[b] \lim_{g\downarrow0} s_{L}^{\mathrm{que}}( \hat{\beta},\hat{h},\bar {\beta},p;g)&\geq S_{\ast }^{\mathrm{que}}(\hat{ \beta},\hat{h},\bar{\beta},p)\\& =\sup_{Q\in\mathcal{C}^{\mathrm{fin}}} \bigl[p \bigl(\bar{\beta}\varPhi(Q)+ \varPhi_{\hat{\beta},\hat{h}}(Q) \bigr)-I^{\mathrm{que} }(Q) \bigr]. \end{aligned} $$
(B.32)
Therefore, upon taking limits in the order g→0, L→∞ and lim inf
p→1, it follows from (B.31), (B.32) and the lower semi-continuity of the map \(p\mapsto S_{\ast}^{\mathrm{que}}(\hat{\beta},\hat{h},\bar{\beta},p)\) on [0,∞) that
$$ \lim_{g\downarrow0} s^{\mathrm{que}}(\hat{\beta},\hat{h},\bar{\beta };g)\geq S_{\ast}^{\mathrm{que}}(\hat{\beta},\hat{h},\bar{\beta}). $$
(B.33)
The following lemma, which is proved in [4], Appendix C, will be used in the proof of (B.32).
Lemma B.2
Suppose that
E
is finite. Then for every
\(Q\in\mathcal{P}^{\mathrm{inv}}(\widetilde{E}^{\mathbb{N}})\)
there exists a sequence (Q
n
) in
\(\mathcal{R}^{\mathrm{fin}}\)
such that w-lim
n→∞
Q
n
=Q
and lim
n→∞
I
ann(Q
n
)=I
que(Q), where
\(\mathcal{R}^{\mathrm{fin}} = \{Q\in\mathcal{R}\colon\ m_{Q}<\infty\}\)
and w-lim means weak limit.
In our case \(E=\mathbb{R}^{2}\).
For the rest of the proof we proceed as in [4], Appendix C. For \(M\in\mathbb{N}\), let
$$ D^\ast_M= \{ -M,-M+1/M, \ldots,M-1/M,M \} $$
(B.34)
be a grid of spacing 1/M in [−M,M], which represents a finite set of letters approximating \(\mathbb{R}\). Put \(D_{M}= D^{\ast}_{M}\times D^{\ast}_{M}\) and let \(\widetilde{D_{M}}=\bigcup_{n\in\mathbb{N}}(D^{\ast}_{M})^{n}\) be the set of finite words drawn from \(D^{\ast}_{M}\). Furthermore, let \(\hat{T}_{M}: \mathbb{R}\rightarrow D^{\ast}_{M}\) and \(\bar{T}_{M}: \mathbb{R}\rightarrow D^{\ast}_{M}\) be the letter maps
$$ \begin{aligned}[c] \hat{T}_M(x)&= \left\{ \begin{array}{l@{\quad}l} M &\mbox{for}\ x\in[M,\infty),\\ \lceil xM\rceil/M &\mbox{for} \ x\in(-M,M),\\ -M &\mbox{for} \ x\in(-\infty,-M], \end{array} \right. \\ \bar{T}_{M}(x)&= \left\{ \begin{array}{l@{\quad}l} M &\mbox{for} \ x\in[M,\infty),\\ \lfloor xM\rfloor/M &\mbox{for} \ x\in(-M,M),\\ -M &\mbox{for} \ x\in(-\infty,-M]. \end{array} \right. \end{aligned} $$
(B.35)
Thus, \(\hat{T}_{M}\) moves points upwards on (−∞,M), while \(\bar{T}_{M}\) does the opposite on (−M,∞). Let \(T_{M}:\mathbb{R}^{2}\rightarrow D_{M}\) be such that \(T_{M}((x,y))= (\hat{T}_{M}(x), \bar{T}_{M}(y))\), for all \((x,y)\in\mathbb{R}^{2}\). This map naturally extends to \(\widetilde{\mathbb{R}^{2}}\), \(\widetilde{\mathbb{R}^{2}}^{\otimes\mathbb{N}}\) and to the sets of probability measures on them. Furthermore, put \(\hat{\mu}_{M}=\hat{\mu}\circ\hat{T}_{M}^{-1}\), \(\bar{\mu}_{M}=\bar{\mu}\circ \bar{T}^{-1}_{M}\),
$$ \begin{aligned}[c] \varPhi_{\hat{\beta},\hat{h},M}\bigl(Q^M \bigr)&=\int_{\widetilde{D^\ast_M}}\hat{\pi}_1Q^M(dx) \log \phi^M_{\hat{\beta},\hat{h}}(x), \\ \varPhi_{L,M}\bigl(Q^M\bigr)&=\int_{D^\ast_M} \bar{\pi}_{1,1}Q^M(dy_1) \phi^M_{L}(y_1), \end{aligned} $$
(B.36)
where
$$ \begin{aligned}[c] \phi^M_{\hat{\beta},\hat{h}}(x)&= \left\{ \begin{array}{l@{\quad}l} \phi_{\hat{\beta},\hat{h}}(x)&\mbox{for}\ x=(x_1,\ldots,x_m)\in (D_M^\ast \setminus\{M\})^m,\\ \frac{1}{2} &\mbox{otherwise,} \end{array} \right . \\ \phi^M_{L}(y_1)&= \left\{ \begin{array}{l@{\quad}l} y_1&\mbox{for}\ y_1\in D_M^\ast\cap(-L,M],\\ -L &\mbox{otherwise.} \end{array} \right . \end{aligned} $$
(B.37)
Here, \(\hat{\pi}_{1}\) is the projection onto the first word in the word sequence formed by the copolymer disorder \(\hat{\omega}\), and \(\bar{\pi}_{1,1}\) is the projection onto the first letter of the first word in the word sequence formed by the pinning disorder \(\bar{\omega}\).
Next, note from (B.31) that
$$ s_{L}^{\mathrm{que}}(\hat{\beta},\hat{h},\bar{ \beta},p;g)- \log \mathcal{N}(g) =\limsup_{N\rightarrow\infty} \frac{1}{N}\log E_g^\ast \bigl(e^{Np [\bar{\beta}\varPhi^{-L}(R_N^\omega )+\varPhi_{\hat{\beta},\hat{h}}(R_N^\omega ) ]} \bigr). $$
(B.38)
For the rest of the proof we assume that L≤M. Consider a combined model with disorder taking values in \(D_{M}^{\ast}\) instead of \(\mathbb{R}\), and with \(\hat{\mu}\) and \(\bar{\mu}\) replaced by \(\hat{\mu}_{M}\) and \(\bar{\mu}_{M}\), respectively. In this set-up, if in (B.38) we replace \(\varPhi_{\hat{\beta},\hat{h}}\) and Φ
−L by \(\varPhi _{\hat{\beta},\hat{h}, M}\) and Φ
L,M
respectively, then we obtain
$$ \begin{aligned}[b] & s_{L,M}^{\mathrm{que}}(\hat{\beta},\hat{h},\bar{ \beta},p;g)- \log \mathcal{N}(g)\\& \quad =\limsup_{N\rightarrow\infty} \frac{1}{N}\log E_g^\ast \bigl(e^{Np [\bar{\beta}\varPhi_ {L,M}(R_{N}^{M,\omega })+\varPhi_{\hat{\beta},\hat{h}, M}(R_{N}^{M,\omega }) ]} \bigr). \end{aligned} $$
(B.39)
Here, \(R_{N}^{M,\omega }\) is the empirical process of N-tuple of words cut out from the i.i.d. sequence of letters drawn from D
M
according to the marginal law \(\hat{\mu}_{M}\otimes\bar{\mu}_{M}\). Note from (B.10) that
$$ \begin{aligned}[c] \varPhi^{-L}(Q)&=\int _{\mathbb{R}}\bar{\pi}_{1,1}Q(dy_1) \phi_L(y_1), \\ \phi_L(y_1)&= \left\{ \begin{array}{l@{\quad}l} y_1 &\mbox{for} \ y_1\in(-L,\infty)\\ -L &\mbox{otherwise.} \end{array} \right . \end{aligned} $$
(B.40)
It therefore follows from (B.38)–(B.39) that
$$ s_{L}^{\mathrm{que}}(\hat{\beta},\hat{h},\bar{ \beta},p;g)\geq s_{L,M}^{\mathrm{que}}(\hat{\beta},\hat{h},\bar{ \beta},p;g), $$
(B.41)
since, for each \(x,y\in\mathbb{R}\), \(\phi_{L}(y)\geq\phi^{M}_{L}(\bar{T}_{M}(y))\) and \(\phi_{\hat{\beta},\hat{h}}(x)\geq\phi^{M}_{\hat{\beta},\hat{h}}(\hat{T}_{M}(x))\) by (B.35), (B.37) and (B.40).
In the sequel we will write, respectively, \(\mathcal{C}^{\mathrm{fin}}_{M}\), \(\mathcal{R}_{M}\), \(I^{\mathrm{que}}_{M}\), \(I^{\mathrm{ann}}_{M}\) associated with D
M
and \(\hat{\mu}_{M}\otimes\bar{\mu}_{M}\) as the analogues of \(\mathcal{C}^{\mathrm{fin}}\), \(\mathcal{R}\), I
que, I
ann associated with \(\mathbb{R}^{2}\) and \(\hat{\mu}\otimes\bar{\mu}\). It follows from Steps 1 and 2 above that
$$ \begin{aligned}[b] & s_{L,M}^{\mathrm{que}}(\hat{\beta},\hat{h},\bar{ \beta},p;g)\\ & \quad =\sup_{Q^M\in\mathcal{C}_M^{\mathrm{fin}}\cap\mathcal{R} _M} \bigl[ p\bar{\beta}\varPhi_{L,M} \bigl(Q^M\bigr)+p\varPhi_{\hat{\beta},\hat{h}, M}(Q_M)-gm_{Q_M}-I^{\mathrm{ann}}_M \bigl(Q^M\bigr) \bigr]. \end{aligned} $$
(B.42)
Note from Lemma B.2 that for any \(Q^{M}\in\mathcal {C}^{\mathrm{fin}}_{M}\) with \(I_{M}^{\mathrm{que}}(Q^{M})<\infty\) there exists a sequence \((Q^{M}_{n})\) in \(\mathcal{C}_{M}^{\mathrm{fin}}\cap\mathcal{R}_{M}\) such that \(\mathrm{w}\mbox{-}\mathrm{lim}_{n\rightarrow \infty }\,Q^{M}_{n}=Q^{M}\) and \(\lim_{n\rightarrow\infty} I_{M}^{\mathrm{ann}}(Q^{M}_{n})=I_{M}^{\mathrm{que}}(Q^{M})\), because D
M
is finite. Therefore
$$ \begin{aligned}[b] &\lim_{g\downarrow0}s_{L,M}^{\mathrm{que}}( \hat{\beta},\hat{h},\bar {\beta} ,p;g)\\ & \quad =s_{L,M}^{\mathrm{que}}(\hat{ \beta},\hat{h},\bar{\beta},p;0) \\ & \quad =\sup_{Q^M\in\mathcal{C}_M^{\mathrm{fin}}\cap\mathcal{R}_M} \bigl[ p\bar{\beta}\varPhi_{L,M} \bigl(Q^M\bigr)+p\varPhi_{\hat{\beta},\hat{h}, M}\bigl(Q^M \bigr)-I^{\mathrm{ann}}_M\bigl(Q^M\bigr) \bigr] \\ & \quad =\sup_{Q^M\in\mathcal{C}_M^{\mathrm{fin}}} \bigl[ p\bar{\beta}\varPhi_{L,M} \bigl(Q^M\bigr)+p\varPhi_{\hat{\beta},\hat{h}, M}\bigl(Q^M \bigr)-I^{\mathrm{que}}_M\bigl(Q^M\bigr) \bigr]. \end{aligned} $$
(B.43)
The first equality uses the second equality of (B.29) and the remark below it. The third equality uses the above remark about Lemma B.2, the boundedness and the continuity of the map \(Q^{M}\mapsto p\bar{\beta}\varPhi_{L,M}(Q^{M})+p\varPhi_{\hat{\beta},\hat{h}, M}(Q_{M})\) on \(\mathcal{C}^{\mathrm{fin}}_{M}\), and the fact that \(I_{M}^{\mathrm{que}}=I_{M}^{\mathrm{ann}}\) on \(\mathcal{C}_{M}^{\mathrm{fin}}\cap\mathcal {R}_{M}\). Note also that those Q
M’s with \(I_{M}^{\mathrm{que}}(Q^{M})=\infty\) do not contribute to the above supremum. For each \(Q\in\mathcal{P}^{\mathrm{inv}}(\widetilde{\mathbb{R}^{2}}^{\otimes \mathbb{N}})\), define \([Q]^{M}=Q\circ T_{M}^{-1}\in \mathcal{P}^{\mathrm{inv}}(\widetilde{D_{M}}^{\otimes\mathbb{N}})\). Since T
M
is a projection map we have that \(I^{\mathrm{que}}_{M}([Q]^{M})\leq I^{\mathrm{que}}(Q)\). It therefore follows from (B.43) that
$$ \begin{aligned}[b] \lim_{g\downarrow0}s_{L,M}^{\mathrm{que}}( \hat{\beta},\hat{h},\bar {\beta},p;g) &=\sup_{Q\in\mathcal{C}^{\mathrm{fin}}} \bigl[ p\bar{\beta}\varPhi_{L,M}\bigl([Q]^M\bigr)+p\varPhi_{\hat{\beta},\hat{h}, M} \bigl([Q]^M\bigr)-I^{\mathrm{que}}_M \bigl([Q]^M\bigr) \bigr] \\ &\geq\sup_{Q\in\mathcal{C}^{\mathrm{fin}}} \bigl[ p\bar{\beta}\varPhi_{L,M} \bigl([Q]^M\bigr)+p\varPhi_{\hat{\beta},\hat{h}, M}\bigl([Q]^M \bigr)-I^{\mathrm{que}}(Q) \bigr]. \end{aligned} $$
(B.44)
Moreover, for any \(Q\in\mathcal{P}^{\mathrm{inv}}(\widetilde{\mathbb {R}^{2}}^{\otimes\mathbb{N}})\) and \(y\in\mathbb{R}\),
$$\mathop{\mathrm{w}\mbox{-}\mathrm{lim}}_{M\rightarrow\infty}[Q]^M=Q, \qquad \lim_{M\rightarrow\infty}\phi^M_{\hat{\beta},\hat{h}}\bigl(\hat{T}_M(y)\bigr)= \phi_{\hat{\beta},\hat{h}}(y) \quad \mbox{and} \quad \lim_{M\rightarrow\infty}\phi ^M_{L}\bigl(\bar{T}_M(y)\bigr)= \phi_{L}(y).$$
Fatou’s lemma therefore tells us that
$$\liminf_{M\rightarrow\infty} \varPhi_{\hat{\beta},\hat{h}, M}\bigl([Q]^M\bigr)\geq\varPhi_{\hat{\beta},\hat {h}}(Q) \quad \mbox{and} \quad \liminf_{M\rightarrow\infty} \varPhi_{L, M}\bigl([Q]^M\bigr)\geq\varPhi^{-L}(Q).$$
These, together with (B.41) and (B.43), imply that
$$ \begin{aligned}[b] \lim_{g\downarrow0}s_{L}^{\mathrm{que}}( \hat{\beta},\hat{h},\bar{\beta },p;g)&\geq \lim_{g\downarrow0}s_{L,M}^{\mathrm{que}}( \hat{\beta},\hat{h},\bar {\beta},p;g) \\ &\geq\sup_{Q\in\mathcal{C}^{\mathrm{fin}}} \bigl[ p\bar{\beta}\varPhi^{-L}(Q)+p \varPhi_{\hat{\beta},\hat{h}}(Q)-I^{\mathrm{que}}(Q) \bigr] \\ &\geq\sup_{Q\in\mathcal{C}^{\mathrm{fin}}} \bigl[ p\bar{\beta}\varPhi(Q)+p \varPhi_{\hat{\beta},\hat{h}}(Q)-I^{\mathrm{que}}(Q) \bigr] \\ &=S_\ast^{\mathrm{que}}(\hat{\beta},\hat{h},\bar{\beta},p). \end{aligned} $$
(B.45)
The last inequality uses that Φ
−L≥Φ. □
Appendix C: Proof of Lemma 6.3
In this appendix we prove Lemma 6.3. To do so we need another lemma, which we state and prove in Sect. C.1. In Sect. C.2 we use this lemma to prove Lemma 6.3.
3.1 C.1 A Preparatory Lemma
Lemma C.1
For every
\(\hat{\beta},\bar{\beta}\geq0\)
and
\(\hat{h}\geq \hat{h}^{\mathrm{ann}}_{c}(\hat{\beta})\),
$$ S^{\mathrm{que}}(\hat{\beta},\hat{h},\bar{\beta};0) \leq\sup _{\bar {Q}\in\bar{\mathcal{C}}^{\mathrm{fin}}} \bigl[\bar{\beta}\varPhi(\bar{Q})+\varXi_{\hat{\beta}, \hat {h}}(r_{\bar{Q}})- \bar{I}^{\mathrm{que}}(\bar{Q} ) \bigr], $$
(C.1)
where
$$ \varXi_{\hat{\beta}, \hat{h}}(r_{\bar{Q}}) =\sum_{n\in\mathbb{N}}r_{\bar{Q}}(n) \log \biggl(\frac{1}{2} \bigl[1+e^{n[\hat{M}(2\hat{\beta})-2\hat{\beta}\hat{h}]} \bigr] \biggr) $$
(C.2)
and
\(r_{\bar{Q}}\)
is the word length distribution under
\(\bar{Q}\).
Proof
Throughout the proof, \(\hat{\beta}>0\), \(\bar{\beta}\geq0\) and \(\hat{h}\geq\hat{h}^{\mathrm{ann}}_{c}(\hat{\beta})\) are fixed. Put
$$ S^\omega _N = E_0^\ast \bigl(e^{N [\bar{\beta}\varPhi (R_N^\omega ) +\varPhi_{\hat{\beta}, \hat{h}}(R_N^\omega ) ]} \bigr) = E_0^\ast \bigl(e^{N [\bar{\beta}\varPhi(R_N^{\bar{\omega}}) +\varPhi_{\hat{\beta}, \hat{h}}(R_N^{\hat{\omega}}) ]} \bigr). $$
(C.3)
Note from (B.5) and the Borel-Cantelli lemma that, for every ϵ>0 and \(\bar{\omega}\)-a.s., there exists an \(N_{0}=N_{0}(\bar{\omega },\epsilon)<\infty\) such that
$$ E_0^\ast \bigl(e^{N\bar{\beta}\varPhi(R_N^\omega )} \bigr) = E_0^\ast \bigl(e^{N\bar{\beta}\varPhi(R_N^{\bar{\omega}})} \bigr) \leq e^{N [\bar{M}(-\bar{\beta})+\epsilon]} \quad\forall N\geq N_0. $$
(C.4)
Therefore, \(\bar{\omega}\)-a.s. and for all N≥N
0,
$$ \begin{aligned}[b] \mathbb{E}_{\hat{\omega}} \bigl(S^\omega _N \bigr) &=\sum_{0=k_0<k_1<\cdots<k_N<\infty}\prod _{i=1}^N \rho(k_i-k_{i-1}) \,e^{\bar{\beta}\bar{\omega}_{k_i}} \frac{1}{2} \bigl[1+\mathbb{E}_{\hat{\omega}} \bigl(e^{-2\hat{\beta}\sum _{k=k_{i-1}+1}^{k_i}(\hat{\omega}_k+\hat{h} )} \bigr) \bigr] \\ &=\sum_{0=k_0<k_1<\cdots<k_N<\infty} \prod_{i=1}^N \rho (k_i-k_{i-1}) \, e^{\bar{\beta}\bar{\omega}_{k_i}} \frac{1}{2} \bigl(1+e^{(k_i-k_{i-1})[\hat{M}(2\hat{\beta})-2\hat{\beta}\hat {h}]} \bigr) \\ &=\sum _{0=k_0<k_1<\cdots<k_N<\infty} \Biggl(\prod_{i=1}^N \rho(k_i-k_{i-1}) \Biggr)\, \bigl(e^{N[\bar{\beta}\varPhi(R_N^\omega )+ \varXi_{\hat{\beta},\hat {h}}(r_{R_N^\omega })]} \bigr) \\ &=E_0^\ast \bigl(e^{N[\bar{\beta}\varPhi(R_N^\omega )+\varXi_{\hat {\beta}, \hat{h}}(r_{R_N^\omega })]} \bigr) <\infty. \end{aligned} $$
(C.5)
Finiteness follows from (C.4) and the fact that \(\varXi_{\hat {\beta}, \hat{h} }\leq 0\) if \(\hat{h}\geq \hat{h}^{\mathrm{ann}}_{c}(\hat{\beta})\). Therefore, for every δ>0, \(\bar{\omega} \)-a.s. and N≥N
0, we have
$$ \mathbb{P}_{\hat{\omega}} \biggl(\frac{1}{N}\log S^\omega _N\geq\frac{1}{N}\log\mathbb{E}_{\hat{\omega}}\bigl( S^\omega _N\bigr)+\delta \biggr) =\mathbb{P}_{\hat{\omega}} \bigl( S^\omega _N\geq\mathbb{E}_{\hat {\omega}}\bigl( S^\omega _N\bigr) e^{N\delta } \bigr)\leq e^{-N\delta}. $$
(C.6)
From the Borel-Cantelli lemma we therefore obtain that ω-a.s.
$$ \begin{aligned}[b] s^{\mathrm{que}}(\hat{\beta},\hat{h},\bar{\beta};0)&= \limsup_{N\rightarrow\infty}\frac{1}{N}\log S^\omega _N \leq\limsup_{N\rightarrow\infty}\frac{1}{N}\log\mathbb{E}_{\hat {\omega}} \bigl(S^\omega _N\bigr)+\delta \\ &=\sup_{\bar{Q}\in\bar{\mathcal{C}}^{\mathrm{fin}}} \bigl[\bar{\beta }\varPhi(\bar{Q})+ \varXi_{\hat{\beta} ,\hat{h} }(r_{\bar{Q}}) - \bar{I}^{\mathrm{que}}(\bar{Q}) \bigr]+\delta. \end{aligned} $$
(C.7)
The equality uses Steps 1 and 2 in the proof of Lemma B.1 and the observation that \(\varXi_{\hat{\beta}, \hat{h}}\) is independent of ω (i.e., only pinning disorder is present), and \(-\log2\leq\varXi_{\hat{\beta},\hat{h}}\leq0\) for \(\hat{h}\geq \hat{h}_{c}^{\mathrm{ann}}(\hat{\beta} )\), where we use (3.15) instead of (3.14). Finally, let δ↓0. □
For \(\hat{\beta}=0\)
$$\lim_{g\downarrow0}s^{\mathrm{que}}(0,\hat{h},\bar{\beta};g)\leq s^{\mathrm{que}}(0,\hat{h},\bar{\beta};0)\leq S^{\mathrm{que}}(0,\hat{h},\bar{\beta};0)=\lim_{g\downarrow0}s^{\mathrm{que}}(0,\hat{h},\bar{\beta};g). $$
The first inequality uses that the map \(g\mapsto s^{\mathrm{que}}(0,\hat{h},\bar{\beta};g)\) is non-inceasing and the second uses (C.7) (since \(\Xi_{0,\hat{h}}\equiv0\)) and the fact that \(S^{\mathrm{que}}=S^{\mathrm{que}}_{\ast}\) (this was proved in Appendix B). This proves the equality (4.18) at g=0 for \(\hat{\beta}=0\).
3.2 C.2 Proof of Lemma 6.3
Proof
Throughout the proof, \(\bar{\beta}\geq0\) and \(\hat{\beta}>0\) are fixed and \(\hat{h}\geq \hat{h}^{\mathrm{ann}}_{c}(\hat{\beta})\). Note from (4.6) and (4.7) that \(\varPhi_{\hat {\beta} ,\infty }\equiv-\log2\). Therefore, replacing \(\bar{\beta}\bar{\varPhi}+\varPhi_{\hat{\beta}, \hat{h}}\) by \(\bar{\beta}\bar{\varPhi}-\log 2\) in (4.8), we get
$$ \begin{aligned}[b] S^{\mathrm{que}}(\hat{\beta},\infty,\bar{\beta};0) &=S_\ast^{\mathrm{que}}(\hat{\beta},\infty,\bar{\beta})=s^{\mathrm{que}}( \hat{\beta},\infty,\bar{\beta} ;0) \\ &=\limsup_{ N\rightarrow\infty}\frac{1}{N} \log E^\ast_0 \bigl(e^{N[\log\frac{1}{2}+\bar{\beta}\varPhi(R^{\bar{\omega }}_N)]} \bigr) \\ &=- \log2+\sup_{\bar{Q}\in\bar{\mathcal{C}}^{\mathrm{fin}}} \bigl[\bar{\beta}\varPhi(\bar{Q})-\bar{I}^{\mathrm{que}}(\bar{Q}) \bigr] \\ &=-\log2+\bar{h}_c^{\mathrm{que}}( \bar{\beta}). \end{aligned} $$
(C.8)
The fourth equality follows from the proof of Lemma B.1, while the last equality uses [6], Theorem 1.3. Next, note that
$$ S^{\mathrm{que}}\bigl(\hat{\beta},\infty^-,\bar{\beta};0\bigr) = \lim _{\hat{h}\uparrow\infty}S^{\mathrm{que}}(\hat{\beta},\hat {h},\bar{\beta};0) \geq S^{\mathrm{que}}(\hat{\beta},\infty,\bar{\beta};0), $$
(C.9)
since the map \(\hat{h}\mapsto S^{\mathrm{que}}(\hat{\beta},\hat{h},\bar {\beta};0)\) is non-increasing. For \(\hat{h} \geq\hat{h}^{\mathrm{ann}}_{c}(\hat{\beta})\) it follows from (C.1) that
$$ \begin{aligned}[b] S^{\mathrm{que}}(\hat{\beta}, \hat{h},\bar{\beta};0) &\leq\sup_{\bar {Q}\in\bar{\mathcal{C}}^{\mathrm{fin}}} \bigl[\bar{\beta} \varPhi(\bar{Q})+\varXi_{\hat{\beta}, \hat {h}}(r_{\bar{Q}})-\bar{I}^{\mathrm{que}}( \bar{Q} ) \bigr] \\ &=\sup_{{\scriptstyle r\in\mathcal{P}(\mathbb{N});\atop+\scriptstyle {m_r<\infty}}}\sup_{{\scriptstyle \bar {Q}\in\bar{\mathcal{C}} ^{\mathrm{fin}};\atop+\scriptstyle {r_{\bar{Q}}=r}}} \bigl[\bar{\beta}\varPhi(\bar{Q})+\varXi_{\hat{\beta}, \hat {h}}(r)-\bar{I}^{\mathrm{que}}( \bar{Q} ) \bigr] \\ &\leq\sup_{{\scriptstyle r\in\mathcal{P}(\mathbb{N});\atop+\scriptstyle {m_r<\infty}}} \Bigl(\varXi_{\hat{\beta},\hat{h}}(r) +\sup_{\bar{Q}\in\bar{\mathcal{C}}^{\mathrm{fin}}} \bigl[\bar{\beta}\varPhi(\bar{Q})-\bar{I}^{\mathrm{que}}(\bar{Q}) \bigr] \Bigr) \\ &\leq\log \biggl[\frac{1}{2} \bigl(1+e^{[\hat{M}(2\hat{\beta})-2\hat {\beta}\hat{h}]} \bigr) \biggr] +\sup_{\bar{Q}\in\bar{\mathcal{C}}^{\mathrm{fin}}} \bigl[\bar{ \beta}\varPhi(\bar{Q})-\bar{I}^{\mathrm{que}}(\bar{Q}) \bigr] \\ &=\log \biggl[ \frac{1}{2} \bigl(1+e^{[\hat{M}(2\hat{\beta})-2\hat {\beta}\hat{h}]} \bigr) \biggr] +\bar{h}^{\mathrm{que}}_c( \bar{\beta}). \end{aligned} $$
(C.10)
The third inequality uses that, for \(\hat{h}\geq\hat{h}^{\mathrm{ann}}_{c}(\hat{\beta})\), \(\varXi_{\hat{\beta},\hat{h}}(r)\leq\log [\frac{1}{2} (1+e^{[\hat{M}(2\hat{\beta} )-2\hat{\beta}\hat{h} ]} ) ]\) for all \(r\in\mathcal{P}(\mathbb{N})\). Therefore
$$ \lim_{\hat{h}\uparrow\infty} S^{\mathrm{que}}(\hat{\beta},\hat {h},\bar{ \beta};0) \leq-\log2 +\bar{h}^{\mathrm{que}}_c(\bar{ \beta})=S^{\mathrm{que}}(\hat{\beta },\infty,\bar{\beta};0). $$
(C.11)
□
