Subexponential asymptotics of asymptotically block-Toeplitz and upper block-Hessenberg Markov chains

This paper studies the subexponential asymptotics of the stationary distribution vector of an asymptotically block-Toeplitz and upper block-Hessenberg (atUBH) Markov chain in discrete time. The atUBH Markov chain is a kind of the upper block-Hessenberg (UBH) one and is a generalization of the M/G/1-type one. The atUBH Markov chain typically arises from semi-Markovian retrial queues, as the queue-length process, its embedded process, or appropriately time-scaled versions of these processes. In this paper, we present subexponential and locally subexponential asymptotic formulas for the stationary distribution vector. We then extend the locally subexponential asymptotic formula to a continuous-time version of the atUBH Markov chain by uniformization and change of time scale. This extension expands the applicability of the locally subexponential asymptotic formula.

This research was supported in part by JSPS KAKENHI Grant Number JP21K11770.

Authors and Affiliations

1. Graduate School of Management, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji-shi, Tokyo, 192–0397, Japan

   Hiroyuki Masuyama

Corresponding author

Correspondence to Hiroyuki Masuyama.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix A: Proofs of lemmas

This section consists of three subsections, and they are devoted to the proofs of Lemmas 2.3, 3.2, and 3.4, respectively.

1.1 A.1 Proof of Lemma 2.3

First, after some preparation, we prove (2.17) and then (2.18). With these equations, we provide the proofs of (2.19)–(2.21).

To prove (2.17) (and thus the other three equations), we express \(\varvec{G}_n\) by using the powers of a submatrix of \(\varvec{P}\). Let \(\varvec{P}_n\), \(n\in {\mathbb {N}}\), denote a submatrix of \(\varvec{P}\) such that

(A.1)

Let \(\varvec{P}_n^{(m)}(k,\ell )\), \(k,\ell ,m \in {\mathbb {Z}}_+\), denote the \((k,\ell )\)-th block matrix of \((\varvec{P}_n)^m\) partitioned in the same way as \(\varvec{P}_n\). Since \(\varvec{P}_n\) contains the probabilities of transitions avoiding below level n, the elements of \((\varvec{P}_n)^m\) have the following probabilistic interpretation:

for \(k,\ell ,m \in {\mathbb {Z}}_+\) and \(i,j\in {\mathbb {M}}\). This interpretation and (2.3) yield

Substituting (A.2) into (2.4) leads to

Next, we express \(\varvec{G}\) in such a form as the expression (A.3) of \(\varvec{G}_n\). To this end, let

where the second equality is due to (A.1) and the condition (iii) of Assumption 2.1, and where the convergence of this limit is uniform over all the elements. It follows from (A.4) and (2.10) that

It also follows from the probabilistic interpretation of \(\varvec{G}\) (see the proof of Lemma 2.1) that

where \(\varvec{P}_{\infty }^{(m)}(k,\ell )\), \(k,\ell ,m \in {\mathbb {Z}}_+\), denotes the \((k,\ell )\)-th block matrix of \((\varvec{P}_{\infty })^m\).

We prove (2.17) based on the expressions (A.3) and (A.5) of \(\varvec{G}_n\) and \(\varvec{G}\). Since \(\varvec{P}_n\) is substochastic, so is \((\varvec{P}_n)^m\). Therefore, it follows from (A.4) and the dominated convergence theorem that

Furthermore, applying the dominated convergence theorem to (A.3) and then using (A.5), (A.6), and (1.2) (see Remark 2.1), we obtain

which shows that (2.17) holds.

Next, we prove (2.18). It follows from (2.1) and (2.5) that

Applying the dominated convergence theorem to this equation and then using (1.2), (2.13), and (2.17), we obtain

which shows that (2.18) holds.

We move on to the proof of (2.19). Equation (2.14) implies that there exists some \(d \in {\mathbb {N}}\) such that \(\Vert \varvec{\varPhi }^{d} \Vert _{\infty } < 1\), or equivalently,

Equation (2.18) yields \(\lim _{n\rightarrow \infty }(\varvec{\varPhi }_n)^{d}=\varvec{\varPhi }^{d}\). Therefore, there exists some \(\xi \in (0,1)\) and \(N:=N_{\xi } \in {\mathbb {N}}\) such that

Using (2.2), (A.7), and \(\varvec{\varPhi }_n\varvec{e} \le \varvec{e}\), we obtain

which leads to

Therefore, applying the dominated convergence theorem to (2.2) and then using (2.18) and (2.14), we have

which shows that (2.19) holds.

Finally, we provide the proof of (2.20) and (2.21). Since \(\sum _{\ell =0}^{\infty }\varvec{A}_n(\ell )\varvec{e} \le \varvec{e}\) for all \(n \in {\mathbb {Z}}_+\), it follows from (2.6), (2.7), and (A.8) that, for all \(n \in {\mathbb {Z}}_+\),

Combining this and the condition (ii) of Assumption 2.1 leads to

which shows that (2.21) holds. Furthermore, applying the dominated convergence theorem to (2.7) and then using (1.2), (2.17), and (2.19), we obtain

where the last equality is due to (2.15). The above equation shows that (2.20) holds. The proof has been completed.

1.2 A.2 Proof of Lemma 3.2

We provide the proof of (3.10) and omit those of (3.11) and (3.12) because we can prove the latter two in the same way as the first one.

The proof of (3.10) is as follows. Based on (2.11), we fix \(\varepsilon >0\) arbitrarily and \(m_{*}:= m_{*} (\varepsilon ) \in {\mathbb {Z}}_{\ge \tau }\) such that, for all \(m \in {\mathbb {Z}}_{\ge m_*}\),

where the size of \(\varvec{e}\) is equal to M. Recall here that \(\varvec{G}\) is stochastic and thus \(\varvec{G}^m \le \varvec{e}\varvec{e}^{\top }\) for all \(m \in \mathbb {Z_+}\). Using this inequality, (3.6), and \(Y \in {\mathcal {L}}\), we have

which leads to

In addition, since \(\{\overline{\varvec{A}}_n(k);k\in {\mathbb {Z}}_+\}\) is nonincreasing, we obtain

and

Substituting (A.9) into (A.11) divided by \({\mathbb {P}}(Y > k)\), we have

and

It follows from (A.12), (3.6), and \(Y \in {\mathcal {L}}\) that

Letting \(\varepsilon \downarrow 0\) in (A.13) yields

Substituting this into (A.10) results in (3.10).

1.3 A.3 Proof of Lemma 3.4

We provide the proof of (3.18) and omit those of (3.19) and (3.20) because we can prove the latter two in the same way as the first one (see Remark A.1).

To prove (3.18), we derive an inequality that estimates \(\left| \varvec{A}_n(k+i) - \varvec{A}_n(k)\right|\) for \(i=1,2,\dots ,\tau \) in terms of \({\mathbb {P}}(V=k)\), where \(\tau \) is the period of the single recurrent class of \(\varvec{G}\) (see Lemma 2.1). It follows from (3.15) that for any \(\varepsilon > 0\) there exists some \(K:=K_{\varepsilon } \in {\mathbb {Z}}_+\) such that, for all \(k \in {\mathbb {Z}}_{\ge K}\),

Remark A.1

The proofs of (3.19) and (3.20) require the following instead of (A.14):

which follows from (A.14) and (1.2).

In what follows, we prove (3.18) with the inequality (A.14). Fix \(n \in {\mathbb {Z}}_+\) and \(\varepsilon > 0\) arbitrarily. Fix \(K \in {\mathbb {Z}}_+\) such that (A.14) holds for all \(k \in {\mathbb {Z}}_{\ge K}\). Furthermore, fix \(m_{*} \in {\mathbb {Z}}_{\ge \tau }\) such that (A.9) holds for all \(m \in {\mathbb {Z}}_{\ge m_*}\). It then follows from \(\varvec{G} \le \varvec{e}\varvec{e}^{\top } \), (3.16), and \(V \in {\mathcal {L}}\) that

and thus

Using (A.14), we obtain

Substituting (A.9) into (A.16) divided by \({\mathbb {P}}(V > k)\) and then noting that the size of \(\varvec{e}\) in (A.9) is equal to M, we have

It follows from (A.17), (3.16), and \(V \in {\mathcal {L}}\) that

Letting \(\varepsilon \downarrow 0\) in (A.18) yields

Following the above argument, we can obtain

Combining these results leads to

Substituting this into (A.15) and then using \({\mathbb {P}}(V_\mathrm{de} = k) = {\mathbb {P}}(V > k)/{\mathbb {E}}[V]\), we obtain (3.18).

Appendix B: Proof of Theorem 3.1

This section consists of two subsections: Section B.1 contains the main body of the proof of Theorem 3.1; Section B.2 provides those of the lemmas for this theorem.

1.1 B.1 Main body of the proof

We first present upper and lower bounds for the R-matrix \(\varvec{R}_n(k)\) of the atUBH Markov chain. Using them, we derive upper and lower bounds for the stationary distribution vector \(\varvec{\pi }\) together with the convolution forms of the bounds. We then present tail asymptotic results on the components of the convolution forms. Finally, with these preliminary results, we complete the proof of Theorem 3.1.

The following lemma presents upper and lower bounds for \(\varvec{R}_n(k)\).

Lemma B.1

Suppose that Assumptions 2.1 and 3.1 are satisfied. For any \(\varepsilon \in (0,1)\), there then exists some \(n_0:=n_0(\varepsilon ) \in {\mathbb {Z}}_+\) such that \(\lim _{\varepsilon \rightarrow \infty }n_0(\varepsilon ) = \infty \) and the following holds:

where

and

Proof

See Appendix B.2.1. \(\square \)

The upper and lower bounds for \(\varvec{R}_n(k)\) in Lemma B.1 enable us to derive those for \(\varvec{\pi }\), but we need some definitions to describe the derivation. In what follows, unless otherwise stated, fix \(\varepsilon \in (0,1)\) arbitrarily and fix \(n_0=n_0(\varepsilon ) \in {\mathbb {Z}}_+\) such that \(\lim _{\varepsilon \rightarrow \infty }n_0(\varepsilon ) = \infty \) and (B.1) holds. Let \(\varvec{\pi }_{[0,n_0]}\) denote a \(1 \times (M_0 + n_0 M)\) vector such that

Furthermore, let \(\varvec{R}_{[0,n_0]}(k)\), \(k\in {\mathbb {Z}}_+\), denote an \((M_0 + n_0 M) \times M\) matrix such that

We then define \(\varvec{\pi }_{\varepsilon }^+:=(\varvec{\pi }_{\varepsilon }^+(0),\varvec{\pi }_{\varepsilon }^+(1),\dots )\) and \(\varvec{\pi }_{\varepsilon }^-:=(\varvec{\pi }_{\varepsilon }^-(0),\varvec{\pi }_{\varepsilon }^-(1),\dots )\) as nonnegative vectors such that

where combining these equations with (B.2a), (B.3a), and (B.6a) yields

The upper and lower bounds for \(\varvec{\pi }\) are given in the following.

Lemma B.2

If Assumptions 2.1 and 3.1 are satisfied, then

where the matrix sequences \(\{(\varvec{R}_{\varepsilon }^+)^{*m}(k);k\in {\mathbb {Z}}_+\}\) and \(\{(\varvec{R}_{\varepsilon }^-)^{*m}(k);k\in {\mathbb {Z}}_+\}\) are the m-fold convolutions of \(\{\varvec{R}_{\varepsilon }^+(k);k\in {\mathbb {Z}}_+\}\) and \(\{\varvec{R}_{\varepsilon }^-(k);k\in {\mathbb {Z}}_+\}\), respectively.

Proof

See Appendix B.2.2. \(\square \)

Lemmas B.3 and B.4 present the tail asymptotic results on the components of the upper and lower bounds for \(\varvec{\pi }\).

Lemma B.3

Suppose that Assumptions 2.1, 3.1, and 3.2 are satisfied. If \(Y \in {\mathcal {L}}\), then

Proof

See Appendix B.2.3.\(\square \)

Lemma B.4

Suppose that Assumptions 2.1, 3.1, and 3.2 are satisfied. If \(Y \in {\mathcal {L}}\), then, for any \(\varepsilon > 0\),

Proof

See Appendix B.2.4. \(\square \)

We are now ready to prove Theorem 3.1. Applying Proposition D.2 (i) to (B.11) of Lemma B.2 yields

Using (B.5), (B.6), and Lemma B.3, we rewrite the terms involved with \(\varvec{\pi }_{[0,n_0]}\):

With (B.2), we also rewrite the term \(\sum _{\ell =0}^{\infty } \sum _{m=0}^{\infty } (\varvec{R}_{\varepsilon }^+)^{*m}(\ell )\) of (B.16) as

where \(\varvec{R} = \sum _{\ell =1}^{\infty } \varvec{R}(\ell ) \ge \varvec{O}\) and \(\varvec{S}_{\varepsilon }=\sum _{\ell =1}^{\infty }\varvec{S}_{\varepsilon }(\ell ) \ge \varvec{O}\). Note that the spectral radius of \(\varvec{R}\) is less than one (see Lemma 2.2 (i)). Thus, fix \(\varepsilon > 0\) sufficiently small such that the spectral radius of \(\varvec{R} + \varepsilon \varvec{S}_{\varepsilon }\) is less than one. We then have

Furthermore, from (B.19), Lemma B.4, and Proposition D.1 (i), we obtain

Substituting (B.17)–(B.20) into (B.16) yields

Recall here that \(\varvec{\pi }(n) > \varvec{0}\) for all \(n \in {\mathbb {Z}}_+\) and that \(\sup _{n\in {\mathbb {Z}}_+}\varvec{c}_n^{\mathrm{A}} < \infty \) and \(\varvec{c}_n^\mathrm{A} \ne \varvec{0}\) for some \(n\in {\mathbb {N}}\) (see Assumption 3.2). Therefore,

Furthermore, letting \(\varepsilon \downarrow 0\) in (B.21) and then using (2.16) and \(\lim _{\varepsilon \downarrow 0}n_0(\varepsilon ) = \infty \) (see Lemma B.1), we have

From (2.21) and \(\sum _{n=0}^{\infty } \varvec{\pi }(n) \varvec{e}=1\), we obtain

Therefore, using the dominated convergence theorem, we have

Substituting this into (B.23) yields

Similarly, we can show that

Combining (B.10) with (B.24) and (B.25) and then using \(Y \in {\mathcal {S}}\subset {\mathcal {L}}\), we have

which shows that (3.13) holds. Finally, it follows (B.22) and \(\varvec{\varpi } > \varvec{0}\) that the right-hand side of (3.13) is finite and positive. The proof of Theorem 3.1 has been completed.

1.2 B.2 Proofs of the lemmas for Theorem 3.1

This subsection provides the proofs of Lemmas B.1–B.4 for Theorem 3.1 in Sections B.2.1–B.2.4, respectively.

1.2.1 B.2.1 Proof of Lemma B.1

Lemma 2.3 yields

where the convergence is uniform over \((k,m) \in {\mathbb {N}}\times \mathbb {Z_+}\). It thus follows from (2.7) and (B.26) that for any \(\varepsilon \in (0,1)\) there exists some \(n_1:=n_1(\varepsilon ) \in {\mathbb {N}}\) such that \(\lim _{\varepsilon \downarrow 0} n_1(\varepsilon ) = \infty \) and, for all \(n \in {\mathbb {Z}}_{\ge n_1}\),

It also follows from (3.3) and (3.5) that for any \(\varepsilon \in (0,1)\) there exists some integer \(n_0:=n_0(\varepsilon ) \ge n_1\) such that, for all \(n \in {\mathbb {Z}}_{\ge n_0}\),

Applying (B.28) to (B.27) and then using (2.15) and (B.4), we obtain the following: For all \(n \in {\mathbb {Z}}_{\ge n_0}\) and \(k \in {\mathbb {N}}\),

and, similarly,

Finally, we have \(\lim _{\varepsilon \downarrow 0} n_0(\varepsilon ) \ge \lim _{\varepsilon \downarrow 0} n_1(\varepsilon ) = \infty \).

1.2.2 B.2.2 Proof of Lemma B.2

We can readily obtain the convolution forms (B.11) and (B.12) from (B.7) and (B.8), respectively, proceeding as in the proof of Theorem 1 in [16]. Thus, we prove by induction the upper and lower bounds (B.10) for \(\varvec{\pi }(k+n_0)\). Recall that (see (B.2a), (B.3a), and (B.9)),

Therefore, combining (B.7) and (B.8) with (B.6) leads to

These equations, together with (2.8), yield

which shows that (B.10) holds for \(k=1\). As the inductive assumption, suppose that (B.10) holds for \(k=k_*\), that is,

where \(k_*\) is some positive integer. Applying the left inequality in (B.30) to (B.29) with \(k=k_{*}+1\) and then using (B.1) with \(n=\ell +n_0\), we obtain

where the last equality is due to (2.8). Similarly, we can show that

As a result, (B.10) has been proved.

1.2.3 B.2.3 Proof of Lemma B.3

To prove (B.13), it suffices to show that

due to (B.6) and \(Y \in {\mathcal {L}}\).

In what follows, we provide the proof of (B.31). Lemma 2.3 implies that, for \((m,n) \in ({\mathbb {Z}}_+)^2\),

where the convergence is uniform over \((m,n) \in ({\mathbb {Z}}_+)^2\). Therefore, for any fixed \(\delta > 0\) (independently of n), the following inequalities hold for all sufficiently large k:

Applying this inequality to (2.7), we obtain, for all sufficiently large k,

From (B.32a), we have

Applying (3.6) and (3.10) to (B.33), we obtain

Similarly, we have

Letting \(\delta \downarrow 0\) in (B.34) and (B.35) leads to (B.31).

1.2.4 B.2.4 Proof of Lemma B.4

It follows from (2.15) and (B.4) that

Applying (3.11) to (B.36) yields

Furthermore, applying (3.7), (3.8), and (3.12) to (B.37) yields

Therefore, we have

Combining these with (B.2) and (B.3) leads to (B.14) and (B.15), respectively.

Appendix C: Proof of Theorem 3.2

This section outlines the proof of Theorem 3.2, which proceeds as in that of Theorem 3.1. However, instead of Lemmas B.3 and B.4 for Theorem 3.1, we require Lemmas C.1 and C.2.

Lemma C.1

Suppose that Assumptions 2.1, 3.1, and 3.3 are satisfied. If \(V_\mathrm{de} \in {\mathcal {L}}_{[1]}\) (or equivalently, \(V \in {\mathcal {L}}\)), then

Proof

To prove (C.1), it suffices to show that

Fix \(\delta > 0\) arbitrarily and independently of n. Recall that (B.32) holds for all sufficiently large k. Applying (3.16) and (3.18) to (B.32), we obtain

Letting \(\delta \downarrow 0\) in the above inequalities, we have (C.2). \(\square \)

Lemma C.2

Suppose that Assumptions 2.1, 3.1, and 3.3 are satisfied. If \(V_\mathrm{de} \in {\mathcal {L}}_{[1]}\), then, for any \(\varepsilon > 0\),

Proof

As in the proof of Lemma B.4, we can readily prove Lemma C.2, though we use Assumption 3.3 and Lemmas 3.3 and 3.4 instead of Assumption 3.2 and Lemmas 3.1 and 3.2, respectively. The details are omitted. \(\square \)

Using Lemmas C.1, C.2, and B.2, and proceeding as in the proof of Theorem 3.1 (see Appendix B), we can show that Theorem 3.2 holds. To save space, we omit the details.

Appendix D: Convolution of matrix sequences with subexponential tails

The following are fundamental results on the asymptotics of the convolution of matrix sequences associated with subexponential tails.

Proposition D.1

Suppose that \(\{\varvec{M}(k);k\in {\mathbb {Z}}_+\}\) is a sequence of finite-dimensional nonnegative square matrices such that \(\varvec{M}:= \sum _{k=0}^{\infty }\varvec{M}(k) < \infty \) and \(\sum _{n=0}^{\infty }\varvec{M}^n = (\varvec{I} - \varvec{M})^{-1} < \infty \).

1. (i)

   If there exist some \(U \in {\mathcal {S}}\) and nonnegative matrix \(\widetilde{\varvec{M}} < \infty \) such that

   $$\begin{aligned} \limsup _{k\rightarrow \infty }{\overline{\varvec{M}}(k) \over {\mathbb {P}}(U>k)} \le \widetilde{\varvec{M}}, \end{aligned}$$

   then

   $$\begin{aligned} \limsup _{k\rightarrow \infty } {\overline{\sum _{n=0}^{\infty }\varvec{M}^{*n}}(k) \over {\mathbb {P}}(U>k)} \le (\varvec{I} - \varvec{M})^{-1}\widetilde{\varvec{M}} (\varvec{I} - \varvec{M})^{-1}. \end{aligned}$$

   In addition, if there exist some \(U \in {\mathcal {S}}_{[1]}\) and nonnegative matrix \(\widetilde{\varvec{M}}' < \infty \) such that

   $$\begin{aligned} \limsup _{k\rightarrow \infty }{\varvec{M}(k) \over {\mathbb {P}}(U=k)} \le \widetilde{\varvec{M}}', \end{aligned}$$

   then

   $$\begin{aligned} \limsup _{k\rightarrow \infty } {\sum _{n=0}^{\infty }\varvec{M}^{*n}(k) \over {\mathbb {P}}(U = k)} \le (\varvec{I} - \varvec{M})^{-1}\widetilde{\varvec{M}}' (\varvec{I} - \varvec{M})^{-1}. \end{aligned}$$
2. (ii)

   Even replacing “\(\displaystyle \limsup _{k\rightarrow \infty }\)" and “\(\le \)" with “\(\displaystyle \liminf _{k\rightarrow \infty }\)" and “\(\ge \)", respectively, in the statement (i), we have a true statement.

3. (iii)

   Even replacing “\(\displaystyle \limsup _{k\rightarrow \infty }\)" and “\(\le \)" with “\(\displaystyle \lim _{k\rightarrow \infty }\)" and “\(=\)", respectively, in the statement (i), we have a true statement.

Remark D.1

Proposition D.1 is a straightforward extension of the combination of [30, Lemma 6] and [23, Proposition A.2.6].

Proposition D.2

Suppose that \(\{\varvec{M}(k);k\in {\mathbb {Z}}_+\}\) and \(\{\varvec{N}(k);k\in {\mathbb {Z}}_+\}\) are finite-dimensional nonnegative matrix sequences such that their convolution \(\varvec{M} * \varvec{N}(k)\) is well-defined. Furthermore, suppose that \(\varvec{M}:= \sum _{k=0}^{\infty }\varvec{M}(k) < \infty \) and \(\varvec{N}:= \sum _{k=0}^{\infty }\varvec{N}(k) < \infty \). Under these conditions, the following hold:

1. (i)

   If there exist some \(U \in {\mathcal {S}}\) and nonnegative matrices \(\widetilde{\varvec{M}} < \infty \) and \(\widetilde{\varvec{N}} < \infty \) such that

   $$\begin{aligned} \limsup _{k\rightarrow \infty }{\overline{\varvec{M}}(k) \over {\mathbb {P}}(U> k)} \le \widetilde{\varvec{M}}, \qquad \limsup _{k\rightarrow \infty }{\overline{\varvec{N}}(k) \over {\mathbb {P}}(U > k)} \le \widetilde{\varvec{N}}, \end{aligned}$$

   then

   $$\begin{aligned} \limsup _{k\rightarrow \infty } {\overline{\varvec{M} *\varvec{N}}(k) \over {\mathbb {P}}(U > k)} \le \widetilde{\varvec{M}} \varvec{N} + \varvec{M} \widetilde{\varvec{N}}. \end{aligned}$$

   In addition, if there exist some \(U \in {\mathcal {S}}_{[1]}\) and nonnegative matrices \(\widetilde{\varvec{M}}' < \infty \) and \(\widetilde{\varvec{N}}' < \infty \) such that

   $$\begin{aligned} \limsup _{k\rightarrow \infty }{\varvec{M}(k) \over {\mathbb {P}}(U = k)} \le \widetilde{\varvec{M}}', \qquad \limsup _{k\rightarrow \infty }{\varvec{N}(k) \over {\mathbb {P}}(U = k)} \le \widetilde{\varvec{N}}', \end{aligned}$$

   then

   $$\begin{aligned} \limsup _{k\rightarrow \infty } {\varvec{M} *\varvec{N}(k) \over {\mathbb {P}}(U = k)} \le \widetilde{\varvec{M}}' \varvec{N} + \varvec{M} \widetilde{\varvec{N}}'. \end{aligned}$$
2. (ii)

   Even replacing “\(\displaystyle \limsup _{k\rightarrow \infty }\)" and “\(\le \)" with “\(\displaystyle \liminf _{k\rightarrow \infty }\)" and “\(\ge \)", respectively, in the statement (i), we have a true statement.

3. (iii)

   Even replacing “\(\displaystyle \limsup _{k\rightarrow \infty }\)" and “\(\le \)" with “\(\displaystyle \lim _{k\rightarrow \infty }\)" and “\(=\)", respectively, in the statement (i), we have a true statement.

Remark D.2

Proposition D.2 is a straightforward extension of the combination of [19, Proposition A.3] and [23, Proposition A.2.5].

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