Corrigendum: ``ERROR BOUNDS FOR LAST-COLUMN-BLOCK-AUGMENTED TRUNCATIONS OF BLOCK-STRUCTURED MARKOV CHAINS'' Vol. 60, No. 3, 2017, pp. 271-320

Journal of the Operations Research Society of Japan ⃝ The Operations Research Society of Japanc Vol. 61, No. 2, April 2018, pp. 236–238

Corrigendum:

“ERROR BOUNDS FOR LAST-COLUMN-BLOCK-AUGMENTED TRUNCATIONS OF BLOCK-STRUCTURED MARKOV CHAINS”

Vol. 60, No. 3, 2017, pp. 271–320

Hiroyuki Masuyama

Kyoto University

(Received November 9, 2017)

Section 2.2 of Masuyama [2] presents a computable and nontrivial lower bound ϕ(β)_K,N for the factor ϕ(β)_K of the error bounds given in Theorems 2.1, 2.2 and 2.4. The author stated that the lower bound ϕ(β)_K,N exists because (see [2, Equation (2.66)])

lim N→∞↑ ϕ (β) K,N = ϕ (β) K , (1)

where the symbol ↑ represents “convergence from below”. However, the proof of (1), pre-sented in [2], is not complete. Thus, this corrigendum presents a complete proof of (1).

It follows from [1, Section 2.2, Proposition 2.14] that, for all t≥ 0 and (k, i; ℓ, j) ∈ F2_, lim

N→∞ ↑ [exp{QFNt}](k,i;ℓ,j)= p

(t)_{(k, i; ℓ, j),}

where [exp{Q_FNt}]_(k,i;ℓ,j) denotes the (k, i; ℓ, j)th element of exp{Q_FNt}. Therefore, by the monotone convergence theorem, we have, for all (k, i; ℓ, j)∈ F2_,

lim N→∞ ↑

∫ _∞ 0

βe−βt[exp{Q_FNt}]_(k,i;ℓ,j)dt = ∫ _∞

0

βe−βtp(t)(k, i; ℓ, j)dt > 0. (2) Using [2, Equations (2.3) and (2.59)], we rewrite (2) as

lim N→∞↑ ϕ

(β)

FN(k, i; ℓ, j) = ϕ(β)(k, i; ℓ, j) > 0, ∀(k, i; ℓ, j) ∈ F2. (3) Although ϕ(β)_FN(k, i; ℓ, j) is defined for (k, i; ℓ, j)∈ (FN)2 _{(see [2, Equation (2.59)]), we set}

ϕ(β)_FN(k, i; ℓ, j) = 0, (k, i)∈ F \ FN or (ℓ, j)∈ F \ FN. (4) It then follows from (3) and [2, Equation (2.65)] that {ϕ(β)_K,N; N = K, K + 1, . . .} is nonde-creasing and thus

lim N→∞ϕ (β) K,N = sup N≥K ϕ(β)_K,N = sup N≥K sup (ℓ,j)∈FN min (k,i)∈FK ϕ(β)_FN(k, i; ℓ, j) = sup N≥K sup (ℓ,j)∈F min (k,i)∈FK ϕ(β)_FN(k, i; ℓ, j), (5) 236

Corrigendum 237

where the last equality holds due to (4). Note here that the order of double supremum is interchangeable (see the lemma below), i.e.,

sup N≥K sup (ℓ,j)∈F min (k,i)∈FK ϕ(β)_FN(k, i; ℓ, j) = sup (ℓ,j)∈F sup N≥K min (k,i)∈FK ϕ(β)_FN(k, i; ℓ, j). (6)

Substituting (6) into (5), and using (3), we obtain lim N→∞ϕ (β) K,N = sup (ℓ,j)∈F sup N≥K min (k,i)∈FK ϕ(β)_FN(k, i; ℓ, j) = sup (ℓ,j)∈F lim N→∞(k,i)min∈FK ϕ(β)_FN(k, i; ℓ, j) = sup (ℓ,j)∈F min (k,i)∈FK lim N→∞ϕ (β) FN(k, i; ℓ, j) = sup (ℓ,j)∈F min (k,i)∈FK ϕ(β)(k, i; ℓ, j) = ϕ(β)_K ,

where the last equality follows from [2, Equation (2.10)]. As a result, we have proved that (1) holds.

We close this corrigendum by providing the lemma, which enables us to interchange the order of double supremum.

Lemma (Interchanging the Order of Double Supremum) Let {an,m; n, m∈ N} de-note a sequence of real numbers, where N = {1, 2, 3, . . . }. We then have

sup (n,m)∈N2 an,m = sup n∈N sup m∈N an,m = sup m∈N sup n∈N an,m.

Proof. By symmetry, it suffices to prove that sup (n,m)∈N2 an,m = sup n∈N sup m∈N an,m. (7) If sup (n,m)∈N2 an,m > sup n∈Nm∈Nsupan,m, then, for some (n′, m′) ∈ N2_{, we have a}

n′,m′ > supn∈Nsupm∈Nan,m whereas, by definition, an′,m′ ≤ supm_∈Nan′,m ≤ supn_∈Nsupm∈Nan,m, which yields a contradiction. On the other hand, if sup (n,m)∈N2 an,m < sup n∈N sup m∈N an,m, then sup i∈N sup j∈N ai,j ≤ sup i∈N sup j∈N sup (n,m)∈N2 an,m = sup (n,m)∈N2 an,m < sup n∈N sup m∈N an,m,

which also yields a contradiction. Consequently, (7) holds. 2

238 H. Masuyama

References

[1] W.J. Anderson: Continuous-Time Markov Chains: An Applications-Oriented Approach (Springer, New York, 1991).

[2] H. Masuyama: Error bounds for last-column-augmented truncations of block-structured Markov chains. Journal of the Operations Research Society of Japan, 60 (2017), 271–320.

Hiroyuki Masuyama

Graduate School of Informatics Kyoto University

Kyoto 606-8501, Japan

E-mail: [email protected]