Asymptotic Expansions for Stationary Distributions of Nonlinearly Perturbed Semi-Markov Processes. 2

Asymptotic expansions with explicit upper bounds for remainders are given for stationary distributions of nonlinearly perturbed semi-Markov processes with finite phase spaces. The corresponding algorithms are based on a special technique of sequential phase space reduction, which can be applied to processes with an arbitrary asymptotic communicative structure of phase spaces.


Introduction
This is Part 2 of the paper, Silvestrov and Silvestrov (2017). In what follows, we continue counting of sections, lemmas, theorems and relations from Part 1.
In Part 1, algorithms for constructing of asymptotic expansions, with remainders of a standard form o(·), for stationary distributions of nonlinearly perturbed semi-Markov processes have been given. In Part 2, we present algorithms for construction the above asymptotic expansions in a more advanced form, with explicit upper bounds for remainders.
We consider models, where the phase space is one class of communicative states, for embedded Markov chains of pre-limiting perturbed semi-Markov processes, while it can possess an arbitrary communicative structure, i.e., can consist of one or several closed classes of communicative states and, possibly, a class of transient states, for the limiting embedded Markov chain.
The initial perturbation conditions are formulated in the forms of Taylor and Laurent asymptotic expansions with explicit upper bounds for remainders, respectively, for transition probabilities (of embedded Markov chains) and expectations of sojourn times, for perturbed semi-Markov processes.
The algorithms are based on special time-space screening procedures for sequential phase space reduction and algorithms for re-calculation of asymptotic expansions with explicit upper bounds for remainders, which constitute perturbation conditions for the semi-Markov processes with reduced phase spaces.
The final asymptotic expansions for stationary distributions of nonlinearly perturbed semi-Markov processes are given in the form of Taylor asymptotic expansions, with explicit upper bounds for remainders.
The algorithms presented in the paper have an universal character. They can be applied to perturbed semi-Markov processes with an arbitrary asymptotic communicative structure and are computationally effective due to recurrent character of computational procedures.
The short survey of works in the area is given in the introduction to Part 1 of this paper. A comprehensive bibliography of works in the area can be found in the recent paper by Silvestrov and Silvestrov (2016a).
Part 2 of the paper includes six sections and two appendices.
In Section 8, we present explicit formulas for computing parameters of explicit power upper bounds remainders of expansions obtained as results of multiplication by constant, summation, multiplication and division operations with asymptotic Laurent expansions (Lemmas 9-12).
In Section 9, we get Laurent asymptotic expansions with explicit upper bounds for remainders for transition characteristics of nonlinearly perturbed semi-Markov processes. The method is based on re-computing of asymptotic expansions and upper bounds for remainders for transition characteristics of perturbed semi-Markov processes (obtained by application of the time-space procedure of one-state reduction of phase space described in Section 9 and the above operational rules for Laurent asymptotic expansions) via the corresponding expansions for transition characteristics of initial semi-Markov processes (Theorems 6 and 7).
In Section 10, we present recurrent algorithms for computing asymptotic expansions with explicit upper bounds for remainders for expected hitting times of nonlinearly perturbed reduced semi-Markov processes, obtained as the result of sequential repetition of the above one-step time-space screening procedure of one-state reduction of phase space (Theorem 8).
In Section 11, we, finally, get asymptotic expansions with explicit upper bounds for remainders for stationary distributions of nonlinearly perturbed reduced semi-Markov processes (Theorem 9). This theorem presents results which can be interpreted as high order analogs for results concerned explicit upper estimates for rates of convergence in simple asymptotic relations about convergence of stationary probabilities to their limiting values, for perturbed semi-Markov processes. Such upper estimates are replaced by asymptotic expansions for stationary probabilities with explicit upper bounds for remainders, in Theorem 9.
In Appendix A, we give short proofs of Lemmas 1-4 and 9-12, which present operational rules for Laurent asymptotic expansions.
In Appendix B, we present an example illustrating theoretical results obtained in the present paper.
We would like to conclude the introduction with the remark that the present paper is a shorten version of the report Silvestrov and Silvestrov (2016b), where one can find some additional details of proofs, comments and references.

Laurent Asymptotic Expansions with Explicit Upper Bounds for Remainders
In this section, we present so-called operational rules for Laurent asymptotic expansions with explicit upper bounds for remainders. The corresponding proofs and comments are given in Appendix A.
Let A(ε) be a real-valued function defined on an interval (0, ε 0 ], for some 0 < ε 0 ≤ 1, and given on this interval by a Laurent asymptotic expansion, where We refer to such Laurent asymptotic expansion as a It is useful to note that there is no sense to consider, it seems, a more general case of upper bounds for the remainder o A (ε k A ), with parameter δ A > 1. Indeed, let us define The following proposition supplements Lemma 1.
Let us consider four Laurent asymptotic expansions, The following lemma presents operational rules for computing parameters of upper bounds for remainders of Laurent asymptotic expansions.

Lemma 10
The above asymptotic expansions have the following operational rules for computing remainders: with parameters h C , k C and coefficients c r , r = h C , . . . , k C given in proposition (i) of Lemma 2, and parameters δ C , G C and ε C given by the formulas: with parameters h C , k C and coefficients c r , r = h C , . . . , k C given in proposition (ii) of Lemma 2, and parameters δ C , G C and ε C given by formulas: with parameters h C , k C and coefficients c r , r = h C , . . . , k C given in proposition (iii) of Lemma 2, and parameters δ C , G C and ε C given by formulas: , and C(ε) = 1 B(ε) , ε ∈ (0, ε 0 ] is a pivotal (h C , k C , δ C , G C , ε C )-expansion with parameters h C , k C and coefficients c r , r = h C , . . . , k C given in proposition (iv) of Lemma 2, and parameters δ C , G C and ε C given formulas: -expansion with parameters h D , k D and coefficients d r , r = h D , . . . , k D given in proposition (v) of Lemma 2, and parameters δ D , G D , ε D given by formulas: Also, let us denote F A m = max h Am ≤i≤k Am |a i,m |, for m = 1, . . . , N.
The summation and multiplication rules for computing of upper bounds for remainders given in propositions (ii) and (iii) of Lemma 10 possess the communicative property, but do not possess the associative and distributional properties.
Lemma 10 let us get an effective low bound for parameter δ A for any (h A , k A , δ A , G A , ε A )-expansion A(ε) obtained as the result of a finite sequence of operations (described in Lemma 10) performed over expansions from some finite set of such expansions.
The following lemma summarize these properties of Laurent asymptotic expansions with explicit upper bounds for remainders.

Lemma 12
The summation and multiplication operations for Laurent asymptotic expansions defined in Lemma 2 possess the following algebraic properties, which should be understood as equalities for the corresponding parameters of upper bounds for their remainders: . . . , N, accord-ing to the rules presented in Lemmas 2 and 10, then

Asymptotic Expansions with Explicit Upper Bounds for Remainders for Transition Characteristics
Let us recall the perturbed semi-Markov processes η (ε) (t), t ≥ 0, with phase space X = {1, . . . , N} and transition probabilities Q (ε) ij (t), t ≥ 0, i, j ∈ X, introduced in Part I of the paper. These processes depend on a perturbation parameter ε ∈ (0, ε 0 ], for some 0 < ε 0 ≤ 1. We also recall transition probabilities of the corresponding embedded Markov chains, We assume that condition A, introduced in Part I, holds for semi-Markov processes η (ε) (t). In particular, we recall the transition sets Y i , i ∈ X (which include states j ∈ X with non-zero probabilities p ij (ε) and guarantee ergodicity of the processes η (ε) (t)) introduced in this condition.
However, we replace the perturbation condition D by the following stronger condition, in which the corresponding Taylor asymptotic expansions are given in the form with explicit upper bounds for remainders: Also, we replace the perturbation condition E by the following stronger condition, in which the corresponding Laurent asymptotic expansions are given in the form with explicit upper bounds for remainders: We would like to re-call the comments given in Section 10, which explain slightly unusual formulations of lemmas and theorems, which includes references to descriptions of algorithms given in their proofs.
Lemma 13 Let conditions A and D hold. Then, for every r ∈ X, the pivotal (l − rr ,l + rr )expansion for the non-absorption probabilityp rr (ε) given in Lemma 8 * is, also, a (l − rr ,l + rr ,δ rr ,Ḡ rr ,ε rr )-expansion, with parametersδ rr ,Ḡ rr andε rr , which can be computed according to the algorithm described below, in the proof of the lemma.
Proof Let r ∈ Y r . First, propositions (i) of Lemmas 3 and 11 (the multiple summation rule) should be applied to the sum j ∈Y + rr p rj (ε). Second, propositions (i) (the multiplication by constant −1) and (ii) (the summation with constant 1) of Lemmas 2 and 10 should be applied to the asymptotic expansion for probability p rr (ε) given in condition D , in order to get the asymptotic expansion for function 1 − p rr (ε). Third, Lemmas 1 and 9 should be applied to the asymptotic expansion for functionp rr (ε) given in two alternative forms by relation (31). This yields the corresponding pivotal the (l − rr ,l + rr )-expansion for probabilitiesp rr (ε), given in Lemma 8, and proves that this expansion is a (l − rr ,l + rr ,δ rr ,Ḡ rr ,ε rr )-expansion, with parameters computed in the process of realization of the above algorithm. The case r / ∈ Y r is trivial, since, in this case, probabilityp rr (ε) ≡ 1.
Let us recall formula (19) for the transition probabilities r p ij (ε), i, j ∈ r X = X \ {r} of the reduced embedded Markov chain r η (ε) n , introduced in Part I, Let us introduce parameter, δ • = min j ∈Y i ,i∈X δ ij . Obviously, inequalities δ ij ≥ δ • , j ∈ Y i , i ∈ X hold for parameters δ ij appearing in upper bounds for the remainders of asymptotic expansions in condition D .
Theorem 6 Conditions A and D , assumed to hold for the Markov chains η (ε) n , also hold for the reduced Markov chains r η (ε) n , for every r ∈ X. Also, for every j ∈ r Y i , i ∈ r X, r ∈ X, the pivotal ( r l − ij , r l + ij )-expansion for the transition probability r p ij (ε) given in Theorem 2 is a ( r l − ij , r l + ij , r δ ij , r G ij , r ε ij )-expansion appearing in condition D for the Markov chains r η (ε) n . Parameters r δ ij , r G ij and r ε ij can be computed using the algorithm described below, in the proof of the theorem. The inequalities Proof Condition A holds for the Markov chains r η (ε) n by Lemma 6, with the same parameter ε 0 as for the Markov chains η (ε) n and with the transition sets r Y i , i ∈ r X given by relation (20).
Let us prove that condition D holds for the Markov chains r η (ε) n , with the same parameter ε 0 as for the Markov chains η (ε) n and the transition sets r Y i , i ∈ r X given by relation (20). Let j, r ∈ Y i ∩ Y r . First, propositions (v) (the division rule) of Lemmas 2 and 10 should be applied to the quotient p rj (ε) 1−p rr (ε) . Second, propositions (iii) (the multiplication rule) of Lemmas 2 and 10 should be applied to the product p ir (ε) · p rj (ε) 1−p rr (ε) . Third, propositions (ii) (the summation rule) of Lemmas 2 and 10 should be applied to sum r p ij (ε) . The asymptotic expansions for probabilities p ir (ε), p rj (ε), and p ij (ε), given in condition D , and probability 1 − p rr (ε), given in Lemmas 8 and 13, should be used. This yields the corresponding pivotal ( r l − ij , r l + ij )-expansions for transition probabilities r p ij (ε), j ∈ r Y i , i ∈ r X, r ∈ X, given in Theorem 2, and proves that these expansions are ( r l − ij , r l + ij , r δ ij , r G ij , r ε ij )-expansions, with parameters computed in the process of realization of the above algorithm.
In these cases, the above algorithm is readily simplified. Thus, condition D holds for the reduced Markov Let us recall formula (22) for the expectations of sojourn times r e ij (ε), for i, j ∈ r X = X \ {r} for the reduced semi-Markov process r η (ε) Let us introduce parameter, δ * = min j ∈Y i ,i∈X (δ ij ∧δ ij ). Obviously, inequalities δ ij ,δ ij ≥ δ * , j ∈ Y i , i ∈ X hold for parameters δ ij andδ ij appearing in upper bounds for the remainders of asymptotic expansions in conditions D and E .

Theorem 7
Conditions A -C, D and E , assumed to hold for the semi-Markov processes η (ε) (t), also hold for the reduced semi-Markov processes r η (ε) (t), for every r ∈ X. Also, for every j ∈ r Y i , i ∈ r X, r ∈ X, the pivotal ( r m − ij , r m + ij )-expansion for expectation r e ij (ε) given in Theorem 3 is a ( r m − ij , r m + ij , rδij , rĠij , rεij )-expansion appearing in condition E for the semi-Markov processes r η (ε) (t). Parameters rδij , rĠij and rεij can be computed using the algorithm described below, in the proof of the theorem. The Proof Conditions A and D hold for the semi-Markov processes r η (ε) (t), respectively, by Lemma 6 and Theorem 6, with the same parameter ε 0 as for the semi-Markov processes η (ε) (t), and the transition sets r Y i , i ∈ r X given by relation (20). Also conditions B and C hold for processes r η (ε) (t), by Lemma 7.

Asymptotic Expansions with Explicit Upper Bounds for Remainders for Expected Hitting Times
As in Part I, letr i,N = r i,1 , . . . , r i,N = r i,1 , . . . , r i,N−1 , i be a permutation of the sequence 1, . . . , N such that r i,N = i, and letr i,n = r i,1 , . . . , r i,n , n = 1, . . . , N be the corresponding chain of growing sequences of states from space X.

Theorem 8 Let conditions A-C, D and E hold for the semi-Markov processes η (ε) (t).
Then, for every i ∈ X, the pivotal ( Proof Let us assume that p (ε) i = 1. Denote asr i,0 η (ε) (t) = η (ε) (t) the initial semi-Markov process. Let us exclude state r i,1 from the phase space of semi-Markov processr i,0 η (ε) (t) using the time-space screening procedure described in Section 10. Letr i,1 η (ε) (t) be the corresponding reduced semi-Markov process. The above procedure can be repeated. The state r i,2 can be excluded from the phase space of the semi-Markov processr i,1 η (ε) (t). Let r i,2 η (ε) (t) be the corresponding reduced semi-Markov process. By continuing the above procedure for states r i,3 , . . . , r i,n , we construct the reduced semi-Markov processr i,n η (ε) (t).
The processr i,n η (ε) (t) has the phase spacer i,n X = X \ {r i,1 , r i,2 , . . . , r i,n }. The transition probabilitiesr i,n p i j (ε), i , j ∈r i,n X and the expectations of sojourn times r i,n e i j (ε), i , j ∈r i,n X are determined for the processr i,n η (ε) (t) by the transition probabilities and the expectations of sojourn times for the processr i,n−1 η (ε) (t), via relations (32) and (33).
By Theorems 2, 3, 6 and 7, the semi-Markov processesr i,n η (ε) (t) satisfy conditions A -C, D and E . The transition setsr i,n Y i , i ∈r i,n X, for the processr i,n η (ε) (t), are determined by the transition setsr i,n−1 Y i , i ∈r i,n−1 X, for the processr i,n−1 η (ε) (t), via relation (20). For every j ∈r i,n Y i , i ∈r i,n X, the pivotal (r i,n l − i j ,r i,n l + i j )-expansion for transition probabilityr i,n p i j (ε), given in Theorem 2, is, by Theorem 6, a (r i,n l − i j ,r i,n l + i j ,r i,n δ i j ,r i,n G i j , r i,n ε i j )-expansion, with parametersr i,n δ i j ,r i,n G i j andr i,n ε i j given in this theorem. Analogously, for every j ∈r i,n Y i , i ∈r i,n X, the pivotal (r i,n m − i j ,r i,n m + i j )-expansion for expectationr i,n e i j (ε), given in Theorem 3, is, by Theorem 7, a (r i,n m − i j ,r i,n m + i j ,r i,nδi j , r i,nĠi j ,r i,nεi j )-expansion, with parametersr i,nδi j ,r i,nĠi j andr i,nεi j given in this theorem. Also, by Theorem 7, the inequalitiesr i,nδi j ≥ δ * , j ∈r i,n Y i , i ∈r i,n X hold.
The algorithm described above can be repeated, for every i ∈ X.

Asymptotic Expansions with Explicit Upper Bounds for Remainders for Stationary Distributions
Let us recall the pivotal (n − i , n + i )-expansion for stationary probability π i (ε) of nonlinearly perturbed semi-Markov process η (ε) (t) given, under conditions A-E, in Theorem 5. This asymptotic expansion has the following form, for i ∈ X, According Theorem 5, the above asymptotic expansion is invariant with respect to the choice of sequence statesr i,N−1 = (r i,1 , . . . , r i,N−1 ) used in the corresponding algorithm, for every i ∈ X.
The following theorem is the main new result in Part II of this paper.

Theorem 9 Let conditions A -C, D and E hold for the semi-Markov processes η (ε) (t).
Then, for every i ∈ X, the pivotal (n − i , n + i )-expansion (34) for the stationary probability π i (ε), given in Theorem 5 and obtained as the result of sequential exclusion of states r i,1 , . . . , r i,N−1 from the phase space X of the processes η (ε) (t), is a (n − i , n + i ,r i,N−1 δ i ,r i,N−1 G i ,r i,N−1 ε i )-expansion. Parametersr i,N−1 δ i ,r i,N−1 G i andr i,N−1 ε i can be computed using the algorithm described below, in the proof of the theorem. Also, inequalityr i,N−1 δ * i ≥ δ * holds making it possible to rewrite function π i (ε) as the piv- Proof Let us choose an arbitrary state i ∈ X. First, proposition (i) (the multiple summation rule) of Lemmas 3 and 11 should be applied to the pivotal (m − i , m + i )-expansion for the expectation e i (ε) = j ∈Y i e ij (ε) given by relation (29) The explicit upper bounds for remainders in the asymptotic expansions given in Theorem 9 have a clear and informative power-type form. An useful property of these upper bounds is that they are uniform with respect to the perturbation parameter. The recurrent algorithm for finding these upper bounds is computationally effective.
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Appendix A: Operational Rules for Laurent Asymptotic Expansions
First of all, we would like to refer to some books, for example, Markushevich (1985) and Hörmander (1990), where one can find basic facts about Laurent series, mainly, connected with representation problems for analytical functions. We, however, are interested in simpler objects and problems that are Laurent asymptotic expansions and operational rules for such expansions, with remainders given in the standard form o(·) or with explicit power type upper bounds. We present the corresponding explicit formulas in eight lemmas and group their proofs in pairs for Lemmas 1 and 9, . . ., 4 and 12. The former lemmas in these pairs give formulas for computing parameters, coefficients and remainders, while the latter lemmas give explicit formulas for computing explicit upper bounds for remainders in the corresponding expansions. Some known or obvious details of the proofs are ommited.

A.1
The formulas given in Lemmas 1 and 9 are quite obvious.

A.2 The same relates to formulas in propositions (i) (the multiplication by a constant rule)
of Lemmas 2 and 10.
Proposition (ii) (the summation rules) of Lemmas 2 and 10 can be obtained by simple accumulation of coefficients for different powers of ε and terms accumulated in the corresponding remainders, and, then, by using obvious upper bounds for absolute values of sums of terms accumulated in the corresponding remainders.
Proposition (iii) (the multiplication rule) of Lemma 2 can be proved by multiplication of the corresponding asymptotic expansions A(ε) and B(ε) and accumulation of coefficients for powers ε l for l = h C , . . . , k C in their product, where Obviously, o C (ε k C ) ε k C → 0 as ε → 0. It should be noted that the accumulation of coefficients for powers ε l can be made in Eq. 35 only up to the maximal value l = k C =

(k A + h B ) ∧ (k B + h A ), because of the presence in the expression for remainder o C (ε k C ) terms b h B ε h B o A (ε k A ) and a h A ε h A o B (ε k B ).
Also, relation (36) readily implies relations (a)-(c), which determines parameters δ C , G C , ε C andδ C ,G C ,ε C in proposition (iii) of Lemma 10, in particular, the following inequalities take place, for ε ∈ (0, ε C ], Parameter ε C ∈ (0, 1] is taken to nonnegative powers in all terms appearing in the sums on the right hand side of the first inequality in relation (37). This makes it possible to estimate these sums from above by the corresponding simpler expressions replacing parameter ε C by 1 and absolute values of coefficients a i and b j by their maximum values.
Analogous approach is used below. The assumptions of proposition (iv) in Lemma 2 imply that ε −h B B(ε) → b h B = 0 as ε → 0. This relation implies that there exists 0 < ε 0 ≤ ε 0 such that B(ε) = 0 for ε ∈ (0, ε 0 ], and, thus, function C(ε) = 1 B(ε) is well defined for ε ∈ (0, ε 0 ]. Note that h B ≤ k B . The assumptions of proposition (iv) of Lemma 2 imply that, h B , or, equivalently, that the following representation takes place, The latter two relations prove proposition (iv) of Lemma 2, for the case h B = k B . Indeed, these relations mean that function C(ε) = 1 B(ε) can be represented in the form of (h C , k C )- This is formula (c) from proposition (iv) of Lemma 2, for the case h B = k B . Note that, in the case h B = k B , the above asymptotic expansion for function C(ε) can not be extended.
ε on the right hand side in the latter relation has an uncertain asymptotic behavior as ε → 0.
Let us now assume that h B + 1 ≤ k B . In this case, the assumptions of proposition (iv) of Lemma 2 and the above asymptotic relations imply that or, equivalently, that the following representation takes place, The latter two relations prove proposition (iv) of Lemmas 2, for the case h B + 1 = k B . Indeed, these relations mean that function C(ε) can be represented in the form of (h C , k C )- )) ≡ 1. This relation yields formula, This is formula (c) from proposition (iv) of Lemma 2, for the case h B + 1 = k B . Note that, in the case h B + 1 = k B , the above asymptotic expansion for function C(ε) can not be extended.
on the right hand side in the latter relation has an uncertain asymptotic behavior as ε → 0.
We can repeat the above arguments for the general case h B +n = k B , for any n = 0, 1, . . . and to prove that, in the case h B + n = k B , function C(ε) can be represented in the form of (h C , k C )-expansion with parameters h C = −h B , k C = k B − 2h B = −h B + n = h C + n and coefficients c h C , . . . , c k C given in proposition (iv) of Lemma 2. Moreover, identity B(ε) · C(ε) ≡ 1, 0 < ε ≤ ε 0 , let us find the corresponding remainder o C (ε k C ) from the following relation, (40) Proposition (iii) of Lemma 2, applied to the product on the left hand side in relation (40), permits to represent this product in the form of (h, k)-expansion with parameters By canceling coefficient for ε l on the left and right hand sides in relation (40), for l = 0, . . . , k B − h B , and then, by solving Eq. 40 with respect to the remainder o C (ε k C ), we get the formula for this remainder given in proposition (iv) of Lemma 2, The assumptions made in proposition (iv) of Lemma 2, imply that B(ε) = 0 and the following inequality holds for 0 < ε ≤ ε C , where ε C is given in proposition (iv) of Lemma 2, The existence of ε 0 declared in proposition (iv) of Lemma 2 is obvious. For example, one can choose ε 0 = ε C . It is also useful to note that formulas given in proposition (iv) of Lemma 2 imply that ε C = ε B ∧ε B ∈ (0, ε 0 ], since ε B ∈ (0, ε 0 ] andε B ∈ (0, ∞).
The assumptions made in proposition (iv) of Lemma 2 and inequality (42) imply that the following inequality holds, for 0 < ε ≤ ε C , Inequality (43) By equating coefficients for powers ε l for l = h D , . . . , k D on the left and right hand sides of the third equality in relation (44), we get formulas (b) for coefficients d h d , . . . , d k D given in proposition (v) of Lemma 2.
Proposition (iii) of Lemma 2, applied to the product on the right hand side in Eq. 44, permits to represent this product in the form of (h, k)-expansion with parameters Inequality (42), the assumptions made in proposition (v) of Lemma 10 and relation (45) finally imply that the following inequality holds, for 0 < ε ≤ ε D given in relation (c) of this proposition, Inequality ( A.4 The first two identities for Laurent asymptotic expansions given in proposition (i) of Lemma 4 are obvious. The third identity given in this proposition follows in an obvious way from proposition (i) of Lemma 2. By applying propositions (iii) and (iv) of Lemma 2 to the product C(ε) = A(ε) · A(ε) −1 , we get parameters and coefficients c n = I(n = 0), n = 0, . . . , k C . Also, relations (40) and (41) imply that the elimination identity A(ε) · A(ε) −1 ≡ 1 holds, since the remainder of Laurent asymptotic expansion for function A(ε) −1 is given by formula (c) from proposition (iv) of Lemma 2. Propositions (ii) and (iii) of Lemma 4 in the parts concerned commutative property of summation and multiplication operations follow from, respectively, propositions (ii) and (iii) of Lemma 2.
Let (B+C) . These relations and Lemma 1 imply equalities for the corresponding coefficients and remainders, for the asymptotic expansions of functions (A(ε) + B(ε)) + C(ε) and A(ε) + (B(ε) + C(ε)). The above remarks prove proposition (ii) of Lemma 4 in the part concerned with the associative property of summation operation for Laurent asymptotic expansions.
Let (B·C) . These relations and Lemma 1 imply equalities for the corresponding coefficients and remainders, for the asymptotic expansions of functions (A(ε) · B(ε)) · C(ε) and A(ε) · (B(ε) · C(ε)). The above remarks prove proposition (iii) of Lemma 4 in the part concerned with the associative property of multiplication operation for Laurent asymptotic expansions.
We would like also to explain an unexpected, in some sense, asymptotic behavior of stationary probabilities π i (ε), in the above example. As a matter of fact, states 1 and 2 are asymptotically absorbing states with non-absorption probabilities of different order, respectively, O(ε 2 ) and O(ε). While, state 3 is a transient asymptotically non-absorbing state. This, seems, should cause convergence of the stationary probability π 1 (ε) to 1 and the stationary probabilities π 2 (ε) and π 3 (ε) to 0 as ε → 0, with different rates of convergence. This, however, does not take place, and all three probabilities converge to non-zero limits. This is because of the expected sojourn times e 1 (ε), e 2 (ε) and e 3 (ε) have orders, respectively, O(ε), O(1) and O(ε −1 ). These expectations compensate absorption effects for states 1, 2 and 3.
In the above example, computations of explicit upper bounds for remainders in the asymptotic expansions for stationary probabilities π 1 (ε), π 2 (ε) and π 3 (ε) can also be realized in the case, where conditions D and E hold instead of conditions D and E. We, however, omit this presentation, in order to escape overloading the paper by technical numerical computations.
In conclusion, we would like also to mention that results related to applications of asymptotic expansions for expectations of hitting times and stationary distributions for perturbed semi-Markov processes and Markov chains to the asymptotical analysis of perturbed queuing and control systems, reliability models, stochastic networks and modes of bio-stochastic systems can be found in the works referred in Part 1 of this paper. Also, a survey and a comprehensive bibliography of works in related areas can be found in paper Silvestrov and Silvestrov (2016a).