Quasi-Stationary Asymptotics for Perturbed Semi-Markov Processes in Discrete Time

We consider a discrete time semi-Markov process where the characteristics defining the process depend on a small perturbation parameter. It is assumed that the state space consists of one finite communicating class of states and, in addition, one absorbing state. Our main object of interest is the asymptotic behavior of the joint probabilities of the position of the semi-Markov process and the event of non-absorption as time tends to infinity and the perturbation parameter tends to zero. The main result gives exponential expansions of these probabilities together with a recursive algorithm for computing the coefficients in the expansions. An application to perturbed epidemic SIS models is discussed.


Introduction
The aim of this paper is to present a detailed asymptotic analysis of the long time behavior of non-linearly perturbed discrete time semi-Markov processes with absorption.
We consider a discrete time semi-Markov process ξ (ε) (n), on a finite state space, depending on a small perturbation parameter ε ≥ 0 in the sense that its transition probabilities are functions of ε. It is assumed that these functions are continuous at ε = 0 so that the process ξ (ε) (n) for ε > 0 can be interpreted as a perturbation of the process ξ (0) (n). Furthermore, we assume that for ε small enough, the state space can be partitioned into one communicating class of states {1, . . . , N} and one absorbing state 0. The absorption time, that is, the first hitting time of state 0 for the semi-Markov process ξ (ε) (n), is denoted by μ (ε) 0 . Our main object of interest is the asymptotic behavior of the probabilities P (ε) ij (n) = P i {ξ (ε) (n) = j, μ (ε) 0 > n}, i, j = 0, as n → ∞ and ε → 0.
It turns out that the forms of the asymptotic results depend on if one-step absorption probabilities vanish asymptotically or if some of them are non-zero in the limit. In the former case the absorption time μ (ε) 0 → ∞ in probability as ε → 0 and we get so-called pseudostationary asymptotics for the probabilities P (ε) ij (n). In the latter case, μ (ε) 0 is stochastically bounded as ε → 0 and we get so-called quasi-stationary asymptotics for the probabilities P (ε) ij (n). In the present paper we give a unified treatment of both cases. Our perturbation conditions are formulated in terms of the following mixed powerexponential moments for transition probabilities, p (ε) ij (ρ (0) , r) = ∞ n=0 n r e ρ (0) n Q (ε) ij (n), r = 0, 1, . . . , i, j = 0, where Q (ε) ij (n) are the transition probabilities for the semi-Markov process and ρ (0) is a non-negative constant determined by the distribution of first return time to the initial state for the limiting semi-Markov process. In the pseudo-stationary case ρ (0) = 0 and then the moments in Eq. 1 reduce to usual power moments.
We allow for smooth non-linear perturbations which means that the moments in Eq. 1 may be non-linear functions of ε which for r = 0, . . . , k can be expanded in asymptotic power series with respect to ε.
As it turns out, the asymptotics of the probabilities P (ε) ij (n) depends on the balance between the rate at which the time n → ∞ and the perturbation ε → 0. If we write n = n (ε) as a function of ε, this balance is characterized by the following relation, ε r n (ε) → λ r ∈ [0, ∞), for some 1 ≤ r ≤ k. ( Under the assumptions mentioned above and some additional Cramér type conditions on moments of transition times and a non-periodicity condition for the limiting semi-Markov process, we obtain the following which is our main result: For any n (ε) → ∞ as ε → 0 in such a way that relation (2) holds, we have P i {ξ (ε) (n (ε) ) = j, μ (ε) 0 > n (ε) } exp(−(ρ (0) + c 1 ε + · · · + c r−1 ε r−1 )n (ε) ) → π (0) ij e λ r c r as ε → 0, i, j = 0.
Relation (3) is supplemented with (i) an explicit expression for the constant π (0) ij , (ii) an equation from which ρ (0) can be found at least numerically, and (iii) a recursive algorithm for computing the coefficients c 1 , . . . , c r as rational functions of coefficients in expansions of the moments in Eq. 1.
In the pseudo-stationary case, the asymptotic relation (3) takes a simpler form. In this case, ρ (0) = 0 and the constants π (0) ij do not depend on the initial state i and are given by the stationary probabilities of the limiting semi-Markov process.
In order to prove (3) we use the theory of perturbed discrete time renewal equations developed in Gyllenberg and Silvestrov (1994), Englund and Silvestrov (1997), and Silvestrov and Petersson (2013). However, the results cannot be applied directly. This is because the conditions for semi-Markov processes are naturally formulated in terms of its transition probabilities while the application of the renewal theory requires conditions for some non-local characteristics of the semi-Markov process to hold. To prove that the conditions we formulate for semi-Markov processes are sufficient for the conditions required for the results from renewal theory we use techniques from Gyllenberg and Silvestrov (2008). In particular, we need to calculate the coefficients in the expansions of mixed powerexponential moments for first return times based on the coefficients in the expansions of the moments in Eq. 1. This analysis makes up a substantial part of the proof of the main result and may also have applications beyond the scope of this paper.
The asymptotic relation (3) is proved for continuous time semi-Markov processes in Silvestrov (1999, 2008). In Gyllenberg and Silvestrov (2008) the result is also extended to the case of initial transient states.
Expansions of the type given in Eq. 3 and similar types of exponential expansions have also been given for ruin probabilities in perturbed risk models, see for example Silvestrov (2000, 2008), Englund (2001), Blanchet and Zwart (2010), Ni (2011, 2014), and Petersson (2014b. In the pseudo-stationary case, many of the existing results in the literature are concerned with an asymptotic analysis of absorption times or other types of first hitting times in various types of Markov and semi-Markov processes, see for example Keilson (1966), Latouche and Louchard (1978), Latouche (1991), Avrachenkov and Haviv (2004), Drozdenko (2007), and Jung (2013).
In the quasi-stationary case, almost all papers in the literature are concerned with models without perturbations. In particular, a great deal of attention has been given the study of so-called quasi-stationary distributions, see for example Darroch and Seneta (1965), Seneta and Vere-Jones (1966), Cheong (1970), Flaspohler and Holmes (1972), Collet et al. (2013), andvan Doorn andPollett (2013). For models with perturbations, asymptotic expansions of quasi-stationary distributions are given in Gyllenberg and Silvestrov (2008) for continuous time regenerative processes and semi-Markov processes, and in Petersson (2014a) for discrete time regenerative processes.
One of the most extensively studied models of perturbed stochastic processes is the model of linearly or analytically perturbed Markov chains. In particular, asymptotic expansions of stationary distributions have been given for so-called nearly uncoupled Markov chains. For some results and more references related to this line of research, we refer to Simon and Ando (1961), Schweitzer (1968), Stewart (1991), Hassin and Haviv (1992), Yin andZhang (1998, 2003), Altman et al. (2004), and Avrachenkov et al. (2013).
For more references related to pseudo-stationary and quasi-stationary asymptotics we refer to the extensive bibliography given in Gyllenberg and Silvestrov (2008).
Let us now briefly outline the structure of the paper. In Section 2, we present in detail the model of perturbed discrete time semi-Markov processes and formulate our main result. An application to perturbed epidemic SIS models is discussed in Section 3. In Section 4, we derive systems of linear equations for moments of first hitting times and give a necessary and sufficient condition for these moments to be finite. Some solidarity properties for moments of first hitting times are proved in Section 5 and asymptotic power series expansions for mixed power-exponential moments of first hitting times are constructed in Section 6. In Section 7, we prove a solidarity property of periodicity which is needed in order to apply the renewal theory and, in Section 8, we present exponential expansions for perturbed discrete time regenerative processes. Finally, using the results from Sections 4-8, the proof of the main result is given in Section 9.

Exponential Expansions for Perturbed Semi-Markov Processes
In this section, we define perturbed discrete time semi-Markov processes and present the main result of this paper.
Since the transition probabilities of the Markov chain (η (ε) n , κ (ε) n ) do not depend on the current value of the second component, it follows that η (ε) n is itself a (homogeneous) Markov chain. Its transition probabilities are given by and it is called an embedded Markov chain for the corresponding semi-Markov process. It is sometimes convenient to write the transition probabilities of the Markov chain (η (ε) Let us also define random variables for first hitting times. Let ν j is the first hitting time of the embedded Markov chain into state j and μ (ε) j is the first hitting time of the semi-Markov process into state j . Note that ν (ε) j and μ (ε) j are both possibly improper random variables taking values in the set N ∪ {∞}. Throughout the paper, we will use the notation Here, and in what follows, we write P i (A (ε) ) = P(A (ε) | η (ε) 0 = i) for any event A (ε) . Corresponding notation for conditional expectations will also be used.
In order to consider the semi-Markov process ξ (ε) (n), for ε > 0, as a perturbation of the semi-Markov process ξ (0) (n), the following continuity condition will be used: Furthermore, we will assume that {1, . . . , N} is a communicating class of states for sufficiently small ε. This is implied by condition A together with the following condition: ij > 0, for all i, j = 0. Transitions to state 0 may, or may not be possible, both for the limiting process and the perturbed process.
We define moment generating functions for conditional transition times by An important role for our results is played by the root of the following characteristic equation, where φ i ), ρ ∈ R, and i = 0 can be chosen arbitrarily. It will be shown that this equation has a unique non-negative solution ρ (ε) for sufficiently small ε that does not depend on i. In order to do this, we will need the following condition: C: There exists β > 0 such that: ii (β i ) ∈ (1, ∞), for some i = 0 and β i ≤ β. Let us define the following mixed power-exponential moment generating functions for transition probabilities: In order to construct exponential asymptotic expansions in our main result, we will need the following perturbation condition: The period of the (possibly improper) distribution g (ε) ii (n) is defined by In particular, d i = 1 means that g (ε) ii (n) is non-periodic. In order to guarantee non-periodicity of g (0) ii (n), we will assume that the following condition holds: jj (n) is a non-periodic distribution for some j = 0.

Let us define
We are now ready to present the main result of this paper. The proof is given in Section 9.
Theorem 1 Assume that conditions A-E hold. Then: (i) For ε sufficiently small, there exists a unique non-negative root ρ (ε) of the characteristic equation (5) which does not depend on the choice of initial state i. Moreover, we have the asymptotic expansion where the coefficients a 1 , . . . , a k can be calculated from explicit recursive formulas given in Theorem 2 and in the proof of Theorem 1. (ii) For any non-negative integer valued function n (ε) → ∞ as ε → 0 in such a way that ε r n (ε) → λ r ∈ [0, ∞) for some 1 ≤ r ≤ k, we have It is interesting to note that in the pseudo-stationary case, π ij does not depend on i. Indeed, in this case ρ (0) = 0 and μ (0) 0 = ∞ almost surely, so we get That is, π (0) ij is the quotient of the expected number of visits to state j during an excursion starting from state i and the expected length of this excursion for the limiting semi-Markov process. It is known that this quantity does not depend on state i. Moreover, in this case π (0) j = π (0) ij , j = 1, . . . , N, are the stationary probabilities for the limiting semi-Markov process.

Perturbed Epidemic SIS Models
In this section, we present an example of an epidemic model where the results of the present paper can be used.
Epidemic SIS (Susceptible-Infectious-Susceptible) models provide natural examples of processes having a state space with the structure considered in this paper. Such models describe the evolution of an infectious disease in a population where individuals who recover from the disease do not develop immunity and can be infected again. This means that the disease can persist for a long time in the population before the epidemic disappears. Epidemic SIS models have been studied in, for example, Weiss and Dishon (1971), Cavender (1978), Longini (1980), Kryscio and Lefévre (1989), Nåsell (1996), and Allen and Burgin (2000).
The model under consideration in this section, which is a small modification of the classical Reed-Frost model, describes the number of infected individuals in a homogeneously mixed population of constant size N . At each time n = 0, 1, . . . , it is assumed that any individual in the population is either susceptible or infected/infectious. Suppose that there are i infected individuals, and thus N − i susceptible individuals, at time n. Then, each infected individual, independently of one another, has infectious contact with each of the susceptible individuals, independently, with probability p. The infected individuals at time n + 1 are those susceptible individuals who got infected at time n. All other individuals, including those that were infectious at time n, are susceptible.
Let us also assume that the probability of infectious contact p = p (ε) depends on a small perturbation parameter ε ≥ 0 in such a way that p (ε) → p (0) as ε → 0, where p (0) > 0. An example of a specific choice of p (ε) will be given below. For each ε ≥ 0, we let I (ε) n , n = 0, 1, . . . , be the number of infected individuals at time n for the model corresponding to p (ε) . The population size N is assumed to be fixed and not dependent of ε.
It follows from the description above that I (ε) n , for each ε ≥ 0, is a homogeneous discrete time Markov chain on the state space X = {0, 1, . . . , N}. In order to determine its onestep transition probabilities p (ε) ij , notice that if I (ε) n = i, then any of the N − i susceptible individuals at time n avoids infection with probability (1 − p (ε) ) i . Since each of these individuals are infected independently of one another, it follows that I (ε) n+1 has a binomial distribution with parameters N − i and 1 Note that state N can only be reached if we start with all individuals infected. If this state is excluded, then, for each ε ≥ 0, the Markov chain I (ε) n has a state space consisting of one communicating class of states {1, . . . , N − 1} and one absorbing state 0. Thus, this is an example of the quasi-stationary case.
The model described above can be embedded into the model of non-linearly perturbed semi-Markov processes studied in the present paper. Indeed, let the transition probabilities (4) be defined by Then, according to the definitions in Section 2, we have that I (ε) n is a discrete time semi-Markov process with times between successive moments of jumps being identically equal to one.
Furthermore, it follows from Eqs. 6 and 9 that moment functionals used in the perturbation condition D take the form Thus, condition D holds if we have the following asymptotic expansions for the one-step transition probabilities, Specifically, let us assume that p (ε) = p 0 + ε, where p 0 > 0. In this case, the perturbation parameter ε represents an increase of the probability of infectious contact compared to some reference value p 0 . It then follows from Eq. 8 that p (ε) ij are non-linear functions of ε which can be represented in the form given by Eq. 10. Explicit formulas for the coefficients p[1], . . . , p[k] can be derived, but these are quite involved and are not given here.
The main result of the present paper can be used to get approximations of the probabilities 0 is the first time when there are no infected individuals in the population. Indeed, let us define, for n = 0, 1, . . . , i, j = 0, It then follows from Theorem 1, that if n (ε) is a non-negative integer valued function such that n (ε) → ∞ as ε → 0 in such a way that ε r n (ε) → λ r ∈ [0, ∞), then the approximations P (ε) ij,r (n), r = 1, . . . , k, have zero asymptotic relative error, i.e., P The advantage of the approximations (11) is the explicit form of P (ε) ij,r (n) as a function of n and ε. The computationally expensive part, which is to calculate ρ (0) andπ (0) ij , only needs to be performed for a single value of the perturbation parameter, i.e., for ε = 0. Adding more terms in the expansion, that is, choosing a larger value of r, gives a more detailed form of the approximations. It is, however, important to remember that the qualities of the approximations rely on an asymptotic result and for fixed values of n and ε it is not necessarily the case that addition of higher order coefficients improve the approximations. Moreover, the asymptotic correctness of the approximations depend on the balance between the rates at which n → ∞ and ε → 0 as described above. In order to study the qualities of the approximations in this example, some numerical experiments could yield some valuable insight, but this is beyond the scope of the present paper.

Moments of First Hitting Times
In this section, we consider moment generating functions of first hitting times. First, a system of linear equations for these functions is derived and then, a necessary and sufficient condition for them to be finite is given.
Moment generating functions of first hitting times are defined by Alternatively, this can be written as We also define moment generating functions for transition probabilities, By conditioning on (η Throughout the paper, we will use the convention 0 · ∞ = 0. With this convention, relation (12) holds for all ρ ∈ R and i, j = 0, even in the case where some of the moment generating functions involved take infinite values. In this case relation (12) may take the form ∞ = ∞.
In what follows, it will sometimes be more convenient to work with matrices. For each j = 0, we define column vectors and where the elements are given by Using Eqs. 13, 14 and 15, we can write (12) in the following matrix notation, The vectors and matrices above are allowed to have entries with the value ∞. By remarks given above, this means that relation (16) holds for all ρ ∈ R.
We will now derive an alternative representation for the vector (ε) j (ρ) of moment generating functions.
Let us for each Since all elements of the matrices on the right hand side are non-negative, it follows that j A (ε) (ρ) is well defined and has elements that take values in the set [0, ∞]. As will be shown next, the elements of j A (ε) (ρ) can be given a probabilistic interpretation. Let j = 0 be fixed. We define random variables by Notice that δ (ε) From Eqs. 17, 18 and 19 it follows that Let us now derive an alternative formula for (ε) j (ρ).
The indicator function can be written as Note that for all i, j = 0, From Eqs. 18 and 21-24 it follows that for all i, j = 0, Now, using Eq. 20 we can write (25) in the following matrix notation, This representation will now be used to prove the following lemma which gives a necessary and sufficient condition for (ε) j (ρ) to be finite.
First note that then the following relation holds, Iterating (29) gives for n = 1, 2, . . . , Since (I − j P (ε) (ρ)) −1 < ∞ it follows from Eq. 30 that we necessarily have Letting n → ∞ in Eq. 30 and using (31), it follows that From Eqs. 26 and 32 we conclude that We supplement Lemma 1 with a corresponding result for the moment generating functions φ The following lemma gives a necessary and sufficient condition for (ε) j (ρ) to be finite. The proof is analogous to the proof of Lemma 1 and is therefore omitted.

Solidarity Properties for Moments of First Hitting Times
In this section, a solidarity lemma for moment generating functions of first hitting times is proved.
Let us introduce the following moment generating functions, Before giving the solidarity lemma, we first prove an auxiliary lemma which gives a connection between φ (ε) ii (ρ) and φ (ε) jj (ρ).
Lemma 3 Let i = 0 be fixed. Assume that we for some ε ≥ 0 and ρ ∈ R have: kj > 0, for all k, j = 0.
Then, the following relation holds for all j = i, Proof By using the regenerative property of the semi-Markov process we can for any j = 0, i write the following relations for moment generating functions, (37) Recall that we use the convention 0 · ∞ = 0, so relations (34)-(37) hold for all ρ ∈ R.
It has now been shown that all generating functions in Eqs. 34-37 are finite and these relations can now be used to prove that (33) holds.
The next lemma is essential for the proof of our main result. The form of part (b) of condition C implies that the results of this lemma can be considered as solidarity properties for moments of first hitting times.

Lemma 4 Assume that conditions
Now, for any j = 0, i we can write We also notice that under condition B, It follows from Eqs. 48, 49 and 50 that j φ ii (ρ ) < 1.

Power Series Expansions for Moments of First Hitting Times
In this section, it is shown how mixed power-exponential moments of first hitting times can be expanded in power series with respect to the perturbation parameter. We first derive recursive systems of linear equations for these moments. Then, some properties of asymptotic matrix expansions are presented. Finally, we construct the desired asymptotic expansions.
Mixed power-exponential moment generating functions of first hitting times are defined by Alternatively, this can be written as n r e ρn g (ε) ij (n), ρ ∈ R, r = 0, 1, . . . , i, j ∈ X.
Differentiating both sides of relation (60) gives the following for ε ≤ min{ε 1 , ε 2 } and ρ < δ, where Let us rewrite relations (60), (61) and (62) in matrix notation. For each j = 0, we define column vectors and N × N matrices j P (ε) (ρ, r) = j p (ε) ik (ρ, r) where the elements are given by With these definitions we have Using Eqs. 60-67, we get for r = 0, and, for r = 1, 2, . . . , where Relations (68), (69) and (70) allows us to calculate mixed power-exponential moments of first hitting times for a fixed (sufficiently small) value of ε. In order to construct asymptotic expansions for these moments, we will use properties of asymptotic matrix expansions, which will be presented now.
Let A(ε) be an m × n matrix valued function. Suppose that A(ε) on the interval 0 < ε ≤ ε 0 can be represented as where A 0 , . . . , A k are m×n matrices with real-valued elements and o(ε k ) is an m×n matrix where all elements are of order o(ε k ). Then we say that A(ε) has an expansion of order k.
The following lemma collects some properties for asymptotic matrix expansions that will be used. These properties are known, but we give a short proof in order to make the paper more self-contained. k, and let B(ε) be a p × q matrix valued function which has an expansion of order l.

Lemma 5 Let A(ε) be an m × n matrix valued function which has an expansion of order
(i) If c is a real-valued constant, then C(ε) = cA(ε) has an expansion of order k and the coefficients are given by (ii) If m = p and n = q, then C(ε) = A(ε) + B(ε) has an expansion of order k ∧ l and the coefficients are given by (iii) If n = p, then C(ε) = A(ε)B(ε) has an expansion of order k ∧ l and the coefficients are given by (iv) If m = n and det(I − A 0 ) = 0, then the inverse matrix C(ε) = (I − A(ε)) −1 exists for sufficiently small ε and has an expansion of order k where the coefficients are given by Proof Parts (i), (ii), and (iii) are consequences of elementary algebraic relations. For the proof of part (iv) we first note that since (I − A(ε)) → (I − A 0 ) as ε → 0, and det(I − A 0 ) = 0, it follows that det(I − A(ε)) = 0 for sufficiently small ε. Thus, the matrix I − A(ε) has an inverse for sufficiently small ε. Furthermore, the elements of this inverse matrix are rational functions of the elements of A(ε). From this it follows that (I − A(ε)) −1 → (I − A 0 ) −1 , so we have the representation where C 0 = (I − A 0 ) −1 and M 0 (ε) → 0 as ε → 0. Now assume that k = 1. Then, using Eq. 71, Rewriting this relation and dividing by ε > 0, we get Letting ε tend to zero it follows that M 0 (ε)/ε → C 0 A 1 C 0 as ε → 0. From this and relation (71) we get the representation This proves part (iv) for k = 1. For a general k we can prove the result by induction using the same technique as above.
We will now use the results above to show how mixed power-exponential moments of first hitting times can be expanded in power series with respect to the perturbation parameter and how the coefficients can be calculated explicitly.
Let us introduce the following perturbation condition which is assumed to hold for some ρ < δ, where δ is the number from Lemma 4: Notice that condition D is a more general variant of condition D formulated in Section 2. Indeed, condition D is obtained by putting ρ = ρ (0) in condition D .
To prepare for the next result, note that it follows from condition D that the vectors p (ε) j (ρ, r) and matrices j P (ε) (ρ, r), defined by relations (65) and (66), respectively, have asymptotic expansions where the vector coefficients p j [ρ, r, n] are given by p j [ρ, r, n] = p 1j [ρ, r, n] p 2j [ρ, r, n] · · · p Nj [ρ, r, n] T , and the coefficients j P[ρ, r, n] = j p ik [ρ, r, n] are N × N matrices where the elements are given by j p ik [ρ, r, n] = p ik [ρ, r, n] i = 1, . . . , N, k = j, 0 i = 1, . . . , N, k = j. The following theorem is an essential tool for the proof of the main result of the present paper.
Theorem 2 Assume that conditions A, B, C, and D hold and fix some j = 0. Then:

exists for sufficiently small ε and has the expansion
(iii) For r = 1, . . . , k, we have the expansions , where the coefficients can be calculated recursively by the formulas Proof First note that under conditions A, B, and C, it follows from part (iii) of Lemma 4 that (0) j (ρ) < ∞, for all ρ ≤ δ. Thus, by applying Lemma 1 we see that the inverse matrix (I − j P (0) (ρ)) −1 exists for all ρ ≤ δ. Using this and condition D , part (i) now follows from part (iv) of Lemma 5.
For the proof of part (ii) notice that it follows from Eq. 68 and part (i) that for sufficiently small ε we have It follows from Eq. 73, part (i), condition D , and part (iii) of Lemma 5 that (ε) j (ρ) has an expansion of order k with coefficients given by Eq. 72. This proves part (ii). Now we consider (69) and (70) for r = 1: where It follows from Eq. 74 and part (i) that for sufficiently small ε, It follows from Eq. 75, part (ii), condition D , and parts (ii)-(iii) of Lemma 5 that It now follows from Eqs. 76,77,78,part (i), and part (iii) of Lemma 5 that This proves part (iii) for r = 1.
We prove the general result by induction. Let us assume that part (iii) holds for r = 1, . . . , u − 1, for some u ≤ k. Equations 69 and 70 give where It follows from Eq. 79 and part (i) that for sufficiently small ε, It follows from Eq. 80, part (ii), condition D , parts (i)-(iii) of Lemma 5, and the induction hypothesis that It now follows from Eqs. 81,82,83,part (i), and part (iii) of Lemma 5 that (ε) j (ρ, u) has an expansion of order k − u with coefficients given by This concludes the proof of Theorem 2.

Solidarity Property of Periodicity
In this section we show that the periodicity of the distribution of first return time satisfies a solidarity property.
It will be shown that the function g (ε) ii (n) have the same period for all states i = 0. In the proof of this result we will use the convolution operator. For two real-valued functions f (n), n = 0, 1, . . . , and g(n), n = 0, 1, . . . , the convolution is defined by Furthermore, for a function f (n), n = 0, 1, . . . , the k-fold convolution f ( * k) (n) is defined recursively by f ( * 0) (n) = χ(n = 0) and Let us introduce the following notation, j }, n = 0, 1, . . . , i, j, k ∈ X. In the proof of the following lemma we adopt a technique that is used in the proof of a similar result for continuous time semi-Markov processes given in Ç inlar (1974).

Lemma 6
If we for some ε ≥ 0 have g (ε) ij > 0 for all i, j = 0, then d i = d j for all i, j = 0.
Proof Choose i, j = 0 arbitrarily. The conclusion is trivial if i = j so let us assume that i = j .
By using the regenerative property of the semi-Markov process we can for all n = 0, 1, . . . , write down the following relations, Iterating Eq. 85 and using Eq. 84 we get Similarly, by using Eqs. 86 and 87 we get Since g (ε) ii (n) has period d i , it has all its mass concentrated on the set d i N = {d i , 2d i , . . .}. It follows from Eq. 88 with m = 0 that the functions j g (ε) ii (n), i g (ε) ij * j g (ε) ji (n) and i g (ε) ij * i g (ε) jj * j g (ε) ji (n) are all concentrated on the set d i N. Since i g (ε) ij * j g (ε) ji (n) is not identically equal to zero, it also follows from Eq. 88 that i g (ε) jj (n) concentrates on d i N. It can now be concluded that all functions on the right hand side of Eq. 89, except for possibly the last one, are concentrated on d i N. Using this, and that g (ε) jj (n) is the limit of the right hand side of Eq. 89 as m → ∞, we have for any n / ∈ d i N, ji * ( j g (ε) ii ) ( * (m+1)) * g (ε) ij (n ) = 0.
This means that g (ε) jj (n) is concentrated on the set d i N and we can conclude that d j ≥ d i . By using analogous arguments as above, Eqs. 88 and 89 can also be used to show that d i ≥ d j .
In conclusion, d i = d j .

Exponential Expansions for Perturbed Regenerative Processes
This section presents exponential asymptotic expansions for perturbed discrete time regenerative processes. The results in this section are obtained by applying a corresponding result for discrete time renewal equations given in Silvestrov and Petersson (2013).
Notice that f (ε) (n) are possibly improper distributions with defect that is, the defect is given by the stopping probability in one regeneration period. Moment generating functions for first regeneration times are defined by We will assume that the distributions of first regeneration times satisfy the following conditions: where the limiting distribution f (0) (n) is non-periodic and not concentrated at zero.
R 2 : There exists δ > 0 such that: The solution of the following characteristic equation plays a crucial role for the asymptotic behavior of the probabilities P (ε) (n, A), Our first lemma gives some basic properties for the solution of Eq. 90. The proof can be found in Silvestrov and Petersson (2013).
The root ρ (ε) of the characteristic equation is only given as the solution of a non-linear equation. In order to give a more detailed description of the asymptotic behavior of ρ (ε) as ε → 0 we can construct an asymptotic expansion. This requires some perturbation conditions on the following mixed power-exponential moment generating functions, n r e ρn f (ε) (n), ρ ∈ R, r = 0, 1, . . .
In order to apply the theory of perturbed renewal equations, we also need the following condition: Furthermore, we define 0 = {A ∈ : q (ε) (n, A) → q (0) (n, A) as ε → 0, n = 0, 1, . . .} and Our next theorem shows how we can construct an asymptotic expansion for the root of the characteristic equation based on the coefficients given in condition R 3 and how this yields exponential asymptotic expansions for the probabilities P (ε) (n, A), A ∈ 0 . This result is proved in Silvestrov and Petersson (2013) for a general renewal equation under slightly different conditions. In the following proof we show that the conditions in the present paper are sufficient in order to apply this result to prove Theorem 3.
Theorem 3 Assume that conditions R 1 , R 2 , and R 3 hold.
Let us finally show that (β) holds for any A ∈ 0 . It follows from Lemma 7 that for every β > 0, we have ρ (ε) ≤ ρ (0) + β for sufficiently small ε. Let us choose β such that 0 < β < γ , where γ is the value from condition

Proof of the Main Result
In this section, we give the proof of the main result of the present paper. The proof is based on the results presented in Sections 4-8.
Assume that the initial distribution of the semi-Markov process ξ (ε) (n) is concentrated at some state i = 0. Then ξ (ε) (n) is a regenerative process with regeneration times being successive return times to state i. If state 0 is an absorbing state, these regeneration times are possibly improper random variables. In Section 8 it was assumed that the regeneration times were proper random variables. However, the probabilities P (ε) ij (n), i, j = 0, do not depend on the transition probabilities from state 0. This means that we can modify these transition probabilities without affecting the probabilities P (ε) ij (n), i, j = 0. For example, if we take Q (ε) ij (n) = χ(n = 1)/(N + 1), then return times to any fixed initial state i = 0 can serve as proper regeneration times. We can apply the results of Section 8 to this modified process and then it follows that the results also hold for the process where 0 is an absorbing state.