Asymptotic Estimates Of The Green Functions And Transition Probabilities For Markov Additive Processes

In this paper we shall derive asymptotic expansions of the Green function and the transition probabilities of Markov additive (MA) processes ( ξ n , S n ) whose ﬁrst component satisﬁes Doeblin’s condition and the second one takes valued in Z d . The derivation is based on a certain perturbation argument that has been used in previous works in the same context. In our asymptotic expansions, however, not only the principal term but also the second order term are expressed explicitly in terms of a few basic functions that are characteristics of the expansion. The second order term will be important for instance in computation of the harmonic measures of a half space for certain models. We introduce a certain aperiodicity condition, named Condition (AP), that seems a minimal one under which the Fourier analysis can be applied straightforwardly. In the case when Condition (AP) is violated the structure of MA processes will be clariﬁed and it will be shown that in a simple manner the process, if not degenerate, are transformed to another one that satisﬁes Condition (AP) so that from it we derive either directly or indirectly (depending on purpose) the asymptotic expansions for the original process. It in particular is shown that if the MA processes is irreducible as a Markov process, then the Green function is expanded quite similarly to that of a classical random walk on Z d .


Introduction
Let (T, T ) be a measurable space. Let (ξ n , S n ) (n = 0, 1, 2, . . .) be a Markov additive process (MA process in abbreviation) taking values in the product space T × Z d , namely it is a time homogeneous Markov process on the state space T × Z d whose one step transition law is such that the conditional distribution of (ξ n , S n − S n−1 ) given (ξ n−1 , S n−1 ) does not depend on the value of S n−1 .
Among various examples of MA processes those which motivated the present work are Markov processes moving in the Euclidian space whose transition laws are spatially periodic like the random walk on a periodic graph or the Brownian motion on a jungle-gym-like manifold. These processes are expected to be very similar to classical random walks or Brownian motions in various aspects but with different characteristic constants that must be determined. The results in this paper will serve as the base for verification of such similarity in fundamental objects of the processes like the hitting distribution to a set as well as for computation of the corresponding characteristic constants. (Another example is given in Remark 5.) From the definition of MA process it follows that the process ξ n is a Markov process on T and that with values of ξ n being given for all n, the increments Y n := S n − S n−1 constitute a conditionally independent sequence. Let p n T (ξ, dη) denote the n-step transition probability kernel of the ξ n -process and P ξ the probability law (on a measurable space (Ω, F) common to ξ) of the process (ξ n , S n ) starting at (ξ, 0). Set p T = p 1 T . We suppose that p T is ergodic, namely it admits a unique invariant probability measure, which we denote by µ, and that Doeblin's condition is satisfied. With the ergodic measure at hand the latter may be presented as follows: (H.1) there exist an integer ν ≥ 1 and a number ε > 0 such that If there is no cyclically moving subset for the process ξ n , this amounts to supposing that for some constant ρ < 1, (H.1 ′ ) |p n T (ξ, A) − µ(A)| ≤ const ρ n (n = 1, 2, 3, . . .) uniformly in A ∈ T and ξ ∈ T . (See Remark 2 in the next section for the other case. ) We also suppose that where E ξ represents the expectation under P ξ .
In this paper we shall derive asymptotic expansions of the Green function (d ≥ 2) and the transition probabilities of the process (ξ n , S n ) that satisfies (H.1) and (1). The derivation is based on a certain perturbation argument that has already been applied in previous works on (local) central limit theorems (cf. (6), (20), (17), (18) etc.) or on estimates of the Green functions (cf. (1), (9)) of MA processes. In our asymptotic expansions, however, not only the principal term but also the second order term are expressed explicitly in terms of a few basic functions that are characteristics of the expansion. The second order term will be important in computation of the harmonic measures for the walks restricted on a half space ( (16)) (of which we shall give a brief outline after the statement of a main result) or other regions ((4), (14), (19), (22)).
We introduce a certain aperiodicity condition, named Condition (AP), that seems a minimal one under which the Fourier analysis can be applied straightforwardly. In the case when Condition (AP) is violated the structure of MA processes will be studied in detail and it will be shown that in a simple manner the process, if not degenerate, can be transformed to another one that satisfies Condition (AP) so that from it we derive either directly or indirectly (depending on purpose) the asymptotic expansions for the original process. It in particular is shown that if (ξ n , S n ) is irreducible as a Markov process on T × Z d , then the Green function is expanded quite similarly to that of a classical random walk on Z d (Theorems 1 and 2). In the case when T is countable we also obtain a local limit theorem that is valid without assuming Condition (AP) nor irreducibility (Theorem 15). The results obtained here directly yield the corresponding ones for classical random walks on Z d , of which the estimates of transition probabilities valid uniformly for space-time variables outside parabolic regions seem not be in the existing literatures (Theorem 4, Corollary 6).
Other aspects of MA processes have been investigated under less restrictive settings than the present one by several authors: see (13) and (9) as well as references therein for renewal theorems or (21) for large deviations; a general theory of continuous time MA processes (construction, regularity property, Lévy-Ito-like decomposition etc.) is provided by (2). There are a lot of interesting examples of MA processes and several of them are found in (1). The term 'Markov additive process' is used by Cinlar (2). In some other works it is called by other names: a process defined on a Markov process, a random walk with internal states, a semi-Markov process or a Markov process with homogeneous second component etc.
The plan of the paper is as follows. In Section 1 we introduce basic notations and the aperiodicity condition (AP) and present the asymptotic expansions for the Green function and the transition probabilities. Several Remarks are made upon the results after that. The proofs of the asymptotic expansions are given for the Green function and for the transition probabilities in Sections 2 and 3, respectively, by assuming that the condition (AP) holds and ξ n makes no cyclic transition, whereas the case of its making cyclic transitions and that when (AP) is violated are dealt with in Sections 4 and 5, respectively. In Section 6 we compute derivatives at 0 of κ(θ) the principal eigenvalue of a transfer operator. In Section 7 we show that the characteristic functions of S n converge to zero geometrically fast away from the origin. The consequences of these two sections are taken for granted and applied in Sections 2 and 3. In the last section four Appendices are given: the first three provide certain standard proofs or statements of facts omitted in Sections 2 and 3; in the third we review a standard perturbation argument for the Fourier operator which is introduced in Section 2 and made use of in the proof of the main theorems.

Main Results
Let (ξ n , S n ) be a MA process on T × Z d and p n T and P ξ the transition probability of the first component and the probability law of the process, respectively, as in Introduction. We suppose the conditions (H.1) and (1) to be true unless otherwise stated explicitly. To state the main results we introduce some further notation. Let p stand for the integral operator whose kernel is p T : pf = p T (·, dη)f (η). Define an R d -valued function h on T by and let c be a solution of (1 − p)c = h (µ-a.e.) that satisfies µ(c) := cdµ = 0, which, if it exists, is unique owing to the ergodicity of p T . We shall impose some moment condition on the variable Y 1 , which incidentally implies that h is bounded and that c exists and is bounded and given by c(ξ) = (1 − p) −1 h(ξ) := lim z↑1 ∞ n=0 z n p n h(ξ).
(2) (If (H.1 ′ ) is true, the last series is convergent without the convergence factors, hence c(ξ) = lim n→∞ E ξ [S n ]; see Remark 1 after Corollary 6 and Section 4 in the case when there are cyclically moving subsets; the convention that (1−p) −1 is defined by the Abel summation method as above will be adopted throughout the paper.) PutỸ n = Y n − c(ξ n−1 ) + c(ξ n ). Then Let Q be the matrix whose components are the second moments ofỸ 1 under the equilibrium P µ := µ(dξ)P ξ and denote its quadratic form by Q(θ): where θ ∈ R d , a d-dimensional column vector, y · θ stands for the usual inner product in R d and E µ denotes the expectation by P µ . Owing to (3) we have We also define functions h * and c * , dual objects of h and c, by and where p * denotes the conjugate operator of p in L 2 (µ): c * is a unique solution of (1 − p * )c * = h * such that µ(c * ) = 0 as before. Alternatively c * may be defined as a µ-integrable function such that µ(c for every bounded f . The transition probability p of the process (ξ n , S n ) may be expressed as We denote by p n the n-step transition probability which is defined by iteration for n ≥ 1 and p 0 ((ξ, x), · ) = δ (ξ,x) ( · ) as usual.
We call the process (ξ n , S n ) symmetric if p((ξ, x), (dη, y)) is symmetric relative to µ×the counting measure on Z d . From the expression of p given above we see that it is symmetric if and only if p T is symmetric relative to µ and q ξ,η (x) = q η,ξ (−x) for every x and almost every (ξ, η) with respect to µ(dξ)p(ξ, dη). If this is the case, h * = h and c * = c.
We introduce two fundamental conditions, one on irreducibility and the other on aperiodicity.
Irreducibility. A MA process (ξ n , S n ) is called irreducible if there exists a set T • ∈ T with µ(T • ) = 1 such that if ξ ∈ T • , x ∈ Z d and µ(A) > 0 (A ∈ T ), then for some n P ξ [ξ n ∈ A, S n = x] > 0. (6) Condition (AP). We say a MA process (ξ n , S n ) or simply a walk S n satisfies Condition (AP) or simply (AP) if there exists no proper subgroup H of the additive group Z d such that for every positive integer n, the conditional law P [S n − S 0 = · | σ{ξ 0 , ξ n }] on Z d is supported by H + a for some a = a(ξ 0 , ξ n ) ∈ Z d with P µ -probability one; namely it satisfies (AP) if no proper subgroup H of Z d fulfills the condition Here σ{X} denotes the σ-fields generated by a random variable X. (See Corollary 22 in Section 7 for an alternative expression of (AP).) Condition (AP) is stronger than the irreducibility and often so restrictive that many interesting MA processes do not satisfy it (some examples are given in 5.2 of Section 5). For the estimate of Green function we need only the irreducibility, while the local central limit theorem becomes somewhat complicated without (AP). In practice the results under the supposition of (AP) are enough, the problem in general cases of intrinsic interest often being reduced to them in a direct way as well as by means of a simple transformation of the processes (cf. Section 5).
We shall see in Section 5.4 that the matrix Q is positive definite if the process is irreducible.
We use the norm , then x coincides with the usual Euclidian length |x| The Green functions. We define the Green function (actually a measure kernel) G((ξ, x), (dη, y)) for d ≥ 3 by for some δ ∈ [0, 1) such that δ = 0 if d = 4, and for some nonnegative integer m. Then the Green function G admits the expansion (as |x| → ∞) with {x j } representing a certain homogeneous polynomial of x of degree j whose coefficients, depending on η as well as ξ, are L 1 (µ(dη))bounded uniformly in ξ; and T 1 + · · · + T m is understood to be zero if m = 0. Moreover T 1 is of the form where U (x) is a homogeneous polynomial of degree 3 (given by (30) in Section 2 ) that does not depend on variables ξ, η. If (ξ n , S n ) is symmetric (in the sense above), then U = 0 and c * = c.
In the two dimensional case we define Theorem 2. Let d = 2. Suppose that (ξ n , S n ) is irreducible and that for some δ ∈ (0, 1) and some integer m ≥ 0, sup ξ E ξ [|Y 1 | 2+m+δ ] < ∞ . Then G admits the following asymptotic expansion Here T k and R m are as in the preceding theorem (but with d = 2) and C ξ (dη) is a bounded signed measure on T (given by (31) in the next section). Moreover In Theorems 1 and 2, some explicit expressions are presented not only for the principal term but also for the second order term, i.e. T 1 , in the expansion of G. The second order term sometimes plays an important role in applications. In (16) a random walk on a periodic graph, say (V, E) whose transition law is spatially periodic according to the periodicity of the graph is studied and the results above are applied to compute the hitting distribution of a hyper plane where the second order term mentioned above is involved. Typically the vertex set of the graph is of the form V = {u + γ : u ∈ F, γ ∈ Γ} where F is a finite set of R d and Γ is a d-dimensional lattice spanned by d linearly independent vectors e 1 , . . . , e d ∈ R d . The random walk X n moves on V whose transition law is invariant under the natural action of Γ so that π(X n ) is a Markov chain on F , where π F : V → F denotes the projection, and is viewed as a MA process on T × R d . Suppose that there is a hyper plane M relative to which the reflection principle works so that ifv denotes the mirror symmetric point of v relative to M and u, v ∈ V \ M , then the Green function G + of the walk X n killed on M is given by if u, v are on the same side of V separated by M and G + (u, v) = 0 otherwise. Let V + be one of two parts of V separated by M and e a unit vector perpendicular to M . Then under suitable moment condition one can deduces from Theorems 1 and 2 that for u, v ∈ V + , and A is the matrix made of column vectors e 1 , . . . , e d . The asymptotic formula for the hitting distribution of M is readily obtained form this.
In the next result the same assumption as in Theorem 2 is supposed.
In the definition of G one may subtract (2πσ 2 n) −1 µ(dη) instead of p n ((ξ, 0), (dη, 0)); so defineG bỹ G((ξ, x), (dη, y)) = ∞ n=1 p n ((ξ, x), (dη, y)) − 1 2πσ 2 n µ(dη) , which only causes an alteration of the constant term C ξ (A) and an additional error term, r(x)µ(dη) say, of order O( x −4 ) in the expansion of Theorem 2: the constant term is given by where γ is Euler's constant. (r(x) appears only if ξ n makes a cyclic transition.) Corollary 3 is actually a corollary of Theorem 2 and Theorem 4 below. For a proof, see the end of the proof of Theorem 2 given in the next section.
We shall prove these results first under the supposition of (AP), and then reduce the general case to them . In the case when the condition (AP) does not hold, it is natural to consider the minimal subgroup H that satisfies (7) and the process, denoted by a n , that is the projection of S n on the quotient group Z d /H. If the process (ξ n , a n ), which is a Markov process on T × (Z d /H), is ergodic (namely it has a unique invariant probability measure), then the Green function is shown to well behave. This ergodicity of course follows from the irreducibility of (ξ n , S n ) (see Lemma 10). On the other hand, if it is not ergodic, the formulae in the theorems above must be suitably modified for obvious reason.
Local Central Limit Theorem. The method that is used in the proof of Theorems 1 and 2 also applies to the derivation of local central limit theorems just as in the case of classical random walks on Z d . We give an explicit form of the second order term as in the estimate of the Green function given above. The next order term is computable in principle though quite complicated (cf. Corollary 5).
In the expansion of the Green function there is no trace of cyclic transitions of ξ n (if any), which is reflected in the transition probability p n ((ξ, x), (A, y)) for obvious reason. In general there may be cyclically moving subsets of T so that the set T can be partitioned into a finite number of mutually disjoint subsets T 0 , . . . , T τ −1 (τ ≥ 1) such that for j, k, ℓ ∈ {0, . . . , τ − 1}, To state the next result it is convenient to introduce the probability measures µ j (j = 0, . . . , τ −1) on T which are defined by If τ = 1, we set µ 0 = µ.

Theorem 4.
Let τ and µ j be as above. Suppose that Condition (AP) holds and as n + |x| → ∞. Here a ∧ b stands for the minimum of a and b; P n,k (x) is a polynomial of x such that P n,2 ≡ 0 and if k ≥ 3 where P A j (y) is a polynomial (depending on ξ, ℓ as well as A but determined independently of m) of degree at most 3j and being odd or even according as j is odd or even. The first polynomial P A 1 is given by where H (defined via (40) (valid also for the case τ ≥ 2)) is a linear combination of Hermite polynomials (in d-variables) of degree three with coefficient independent of ξ, A; in particular H is identically zero if the process is symmetric.
The error estimate in Theorem 4 is fine: the expansions of the Green functions as in Theorems 1 and 2 can be obtained from Theorem 4 (except for d = 4) in view of the inequality if d = 2, δ > 0 or d = 3, δ ≥ 0; and a similar one with C/|x| k+δ on the right side for d ≥ 5 (we need to multiply the right side by log |x| for d = 4).
Corollary 5. Suppose that (ξ n , S n ) satisfies (AP) and as n → ∞ uniformly for |x| ≤ C √ n and ξ ∈ T (with C being any positive constant). Here a A (ξ) is a certain function on T .
In the special case when the ξ n process degenerates into a constant we have the result for a classical random walk on Z d which is an extension of that of Spitzer (22) . Its n-step transition probability p n (x, y) is given in the form p n (y − x).

Corollary 6.
Suppose that the random walk is strongly aperiodic in the sense of (22), p 1 (x)x = 0 and p 1 (x)|x| k+δ ] < ∞ for some k ≥ 2 and δ ∈ [0, 1). Then as n + |x| → ∞. Here P n,k (x) is a polynomial of x as described in Theorem 4 except that P A j therein is independent of A (κ(θ) in (40) is nothing but the characteristic function of p 1 ).
Remarks for Theorems 1 to 4.
Remark 1 If there are cyclically moving subsets of T for the process (ξ n ) as in (12), the relation (H.1 ′ ) does not hold any more. However, under Doeblin's condition and ergodicity, it holds that for j, k, ℓ ∈ {0, . . . , τ − 1}, uniformly in ξ ∈ T j , A ∈ T . (For proof see Doob (3), Section V.5 (especially pages 205-206, 207). The function c, defined as a solution of (1 − p)c = h, is obtained by taking the Abel sum as in (2) of possibly divergent series p n h; c * may be similarly given. (In the special case when every T i consists of one point, the process is a deterministic cyclic motion on a finite set and For the general case see Section 4.) Remark 2 One might consider a class of MA processes which satisfy Condition (AP) and wish to prove that the estimates as given above hold uniformly for the processes in the class (cf. (10), (26)). For expecting such uniform estimates to be true it is reasonable to suppose that the following bounds hold uniformly for the class: where Π is the principal part of p made of eigenprojections corresponding to eigenvalues of modulus unity (see Section 6). (For (iii) see (83) of Section 7.) Unfortunately it is not fully clear whether the uniformity of these bounds is sufficient since it does not seem to provide appropriate bounds for derivatives of κ(θ), M θ (f ) (cf. Section 2 for the notation).

Remark 3
For the local central limit theorem the zero mean condition (1) is not essential. With the mean vector b = E µ [Y 1 ] we have only to replace x by x − nb on the right sides of (13) and (15) to have the corresponding formulas. The same proofs as in the case of mean zero go through if h and h * are defined by Remark 4 We have supposed the Doeblin's condition (H.1) to hold, which amounts to supposing the uniform bound of the exponentially fast convergence (H.1 ′′ ). We may replace it by the L p (µ) (p ≥ 1) bound under some auxiliary condition on p T (which is satisfied eg. if T is a countable set), although the estimates stated in Theorems above must be generally not uniform relative to ξ 0 = ξ any more. (Cf. (1) for a such extension.)

Remark 5
The case when the distributions of S n is not supported by any lattice can be dealt with in a similar way under some reasonable conditions. If the asymptotic formulae are for measure kernels and understood in a weak sense (cf. the last section of (25)), it suffices to suppose, in place of Condition (AP), that for some positive integer n • , If it is for the density, we need suppose a more restrictive one, eg, the condition that for some integers n • , k ≥ 1 which in particular implies that if 2j ≥ k, the conditional distribution of S jn • with ξ 0 , ξ n • being fixed has a square integrable density. It follows from each of (16) and (17) that for every ε > 0 with positive P µ -probability, so that the estimates as given in Section 7 can be verified in a similar way as therein. It may be worth pointing out that to have an expansion analogous to the right side of (8) the first several terms P 1 , . . . , P ℓ must be discarded from the Green function G since they may possibly behave very badly even when they possess densities.
Let (M n , u n ) (n ≥ 1) be an i.i.d. sequence of pais of random d × d-matrices M n and random d-vectors u n and define ξ 0 = I (unit matrix), ξ n = M 1 · · · M n , S 0 = 0 and recursively S n+1 = S n + ξ n u n+1 . The pair (ξ n , S n ) is then a Markov additive process starting from (I, 0). If M n are taken from the special orthogonal group SO(d), the conditions (16) as well as (H.1) is satisfied under mild restrictions on the distribution of (M 1 , u 1 ) (cf. (1)). This model is s closely related to the random difference equation Y n = M n Y n−1 + u n : in fact, given Y 0 , Y n has the same distribution as S n + ξ n Y 0 . (Cf. (15), (1) for more general cases of M n . ) Other expressions of Q. In some of previous works the covariance matrix Q is expressed in apparently different forms, which we here exhibit. Define Let R be another symmetric matrix defined through the quadratic form: Then R(θ) = 2µ((c * · θ)(h · θ)) = 2µ((h * · θ)(c · θ)) and In fact from the identity E · [Y 1 ] − c + pc = 0, we deduce the equality which, on integrating by µ, yields (18).
, is the central limit theorem variance for the sequence h(ξ n ) · θ under the stationary process measure P µ , in particular σ 2 ≤ (det Q • ) 1/2d . The last inequality is not necessarily true in the asymmetric case (see Example 6.1 of (16)).
Let m(ξ, η) be the first moment of the conditional law of Y 1 given (ξ 0 , Then h(ξ) = p T (ξ, dη)m(ξ, η), and we infer that Let Q m denote the central limit theorem variance for the sequence m(ξ n , ξ n+1 ).
and, on performing summation over n, These are valid only for |θ| ≤ δ • . We may extend M ξ and κ to arbitrary functions that are sufficiently smooth and |κ| < 1 for |θ| > δ • and define R ξ (θ) as the remainder. Set ∆ = [−π, π) d and choose these extensions of M ξ and κ so that they vanish in a neighborhood of the boundary ∂∆. It holds that In the rest of this section we suppose that the process (ξ n , S n ) satisfies (AP). (The general case will be treated in Section 5.) In this proof we further suppose that Then for each m both κ(θ) and M ξ (θ) are m-times differentiable functions of θ and there exists a positive constant r = r m < 1 such that Here (and later on without exception) the gradient operator ∇ acts on a function of θ. The proof of this estimate is postponed to Section 7 (Lemma 20). From it together with (21) it follows that the second term on the right side of (25) approaches zero as |x| → ∞ faster than |x| −m for every m.
We shall derive in Section 6 the following identities (Proposition 18 and Lemma 20). Here the somewhat abusing notation (θ · ∇) 2 κ(0) stands for k,j θ k θ j ∇ k ∇ j κ(0), which may be expressed by another one Tr(θ 2 ∇ 2 κ(0)), where θ 2 is understood to be a d × d-matrix whose (k, j) entry is θ k θ j and similarly for ∇ 2 . We infer from where {θ k } denotes a homogeneous polynomial of θ ∈ R d of degree k. On the other hand by where B f (θ) is a real function given by Let d ≥ 3. Then, following the usual manner of evaluation of Fourier integrals (cf. Appendix B), we deduce from (25) together with what is remarked right after it that for ξ, η ∈ T , where U (x) is a homogeneous polynomial of degree 3 and given by In the case when the process (ξ n , Y n ) is symmetric, it is clear that c * = c; the equality U = 0 follows from Proposition 18 (iii) of Section 6.
Let d = 2. The proof may proceed as in (25) or (5). We outline it in below to identify the second order term. The proof is based on the Fourier inversion formula and make the decomposition Then one deduces that the first integral on the right side equals where B = {θ : Q(θ) ≤ σ 2 } (⊂ ∆) and γ is Euler's constant (cf. (22), (5)). (Here the last integral is not absolutely convergent and needs to be defined as a principal value in a suitable sense (cf. Lemma 2 of (5) ).) If we define By (28) and (23) we have in a neighborhood of the origin. With this we estimate the sum of the last two integrals in (32) as in the case d ≥ 3.
Proof of Corollary 3. This proof amounts to computing the sum In view of Theorem 4 (see (14)) and Theorem 2 it suffices to show that as |x| → ∞, as well as, for dealing with the case when ξ n is cyclic of period τ (≥ 2), that These estimates will be shown in Section 8 (Appendix C).
Remark 6 Consider the case when ξ n makes a cyclic transition and/or S n does not satisfy (AP). As in (22) (p.310) we modify the process (ξ n , S n ) by adding δ ∈ (0, 1) of the probability that it does not move at each step and multiplying by 1 − δ the original probabilities: for the new law P ′ . This transforms the cyclically moving ξ n into noncyclic one and many processes that do not satisfy (AP) into those satisfying (AP), but not all as it always does in the case of random walks on Z d (see Section 5 for a counter example as well as relevant matters). Denoting by p ′ , κ ′ etc. the corresponding objects for the modified process, we have ) and both µ θ and e θ remain the same. If d ≥ 3, we have hence the required estimate of G follows from that of G ′ straightforwardly, provided that the modified process satisfies (AP) for all sufficiently small δ > 0.

Proof of Local Central Limit Theorem.
By means of the expression (23) together with (28) one can derive an asymptotic expansion of the transition probability p n (α, dβ) in a usual manner. Here we first review the derivation of the expansion in the case when |x|/ √ n is bounded above mainly for identification of the second order term and then discuss it in the case when |x|/ √ n is bounded off zero. It is supposed that τ = 1, namely there are no cyclically moving sets for ξ n process (see section 4 in the case τ ≥ 2).
The case |x|/ √ n < C. We consider mostly the case k = 5, namely E ξ |Y 1 | 5+δ < ∞. Owing to (26) and (27) we see (Remember that M ξ (θ) = e θ (ξ)µ θ (f ).) By the second relation we have Therefore on using (22) where H • (y) is an odd polynomial of degree three defined by is a linear combination of Hermite polynomials of degree three. For the verification one divides the range of integration √ n∆ into three parts according as use Condition (AP) to estimate the last part with the help of Proposition 20 of Section 7.
Since the Fourier transform of a function of the form {θ j }e − 1 2 Q(θ) is a Gaussian density times a polynomial of degree j (hence odd or even according as j is odd or even), the formula (39) is nothing but the formula stated in the theorem in the case when |x|/ √ n is bounded above. The constant term of the transform being equal to the integral over R d of the function, the constant a A (ξ) in the formula (15) is given by The contribution of P (θ, n) with the term −a f (ξ)/2n subtracted is bounded by This verifies Corollary 5.
The case |x|/ √ n > C. Put and ω = x/|x|, ∇ ω = ω · ∇. The proof is based on the following identities: as well as the relation (22): . The method of using these identities is an extension of that found in Spitzer (22) in which k = 2. The arguments given below are mostly the same as for the classical random walk on Z d , but for the case k ≥ 3 they seem not be in existing literatures.
Proof of Lemma 7. We can expand ∇ m ω κ(θ) for m < k into a Taylor series up to the order k − m with the error estimate of o(|θ| k−m+δ ) to see that and for each ν = 0, 1, 2, . . ., as n → ∞ uniformly for |θ| ≤ n 1/6 .
For the rest of the proof we suppose that k ≥ 3 (the argument is easily adapted to the case k = 2). Noticing that we observe that I n − II n is of the form like that required for D n . As for the terms of I n , since |∇ j κ| ≤ C j (j = 1, 2, ..., k), from each factor (∇ m ω )κ(θ/ √ n) with m > 2 there arises a factor 1/ √ n m−2 , so that I (2) n is also of the same form as above (but without the term of order 1/ √ n ). These together with the smoothness of M concludes that D n is given as in (44) , so that its degree is k + 3j. ) It remains to show (46). But this follows from the the facts that among the factors in (48) κ and (∇ ω ) α j κ with α j < k are differentiable and . This completes the proof of the lemma.
We resume the proof of Theorem 4. First we prove it in the case δ = 0. Recalling the basic relations (42), (43), and (22) it is routine (as indicated in the case |x|/ √ n < C) to deduce from Lemma 7 that uniformly for |x|/ √ n > C, whereP k,n (y) is the polynomial appearing in the Fourier transform of e − 1 2 Q(θ) P k,n (θ). Since both the formula (15) of Theorem 4 and the above one are valid uniformly for C < |x|/ √ n < C −1 with arbitrary 0 < C < 1, the polynomial following e − x 2 /2nσ 2 (as its multiple ) on the right side of the latter must agree with that of the former within the indicated error estimate. This yields the required formula in the case δ = 0.
In the case δ > 0. Let R n,k be the function introduced in Lemma 7. It suffices to prove that for r := |x| > C √ n, and For the first estimate we are to apply (46). To this end we set η n,r = (π √ n/r)ω, where ω = x/r, Then, by virtue of the factor e − 1 2 Q(θ) in the integrand, the relation η n,r · x/ √ n = π, and the fact that the volume of the symmetric difference of n 1/6 ∆ and n 1/6 ∆ − η n,r is O(r −1 √ n · n (d−1)/6 ), we see that Since R n,k (θ + η n,r ) = o (1 + |θ|) k+δ / √ n k+δ provided that r > C √ n, the first term on the right side is o(|η n,r |/ √ n k+δ ) = o(1/ √ n k r δ ). On using (46) the second term also is o(1/ √ n k r δ ).

Cyclic Transitions of ξ n
We here advance formal analytical procedures for dealing with the case when the process ξ n cyclically moves, although what modification is to be done is intuitively clear.

The Case When (AP) Is Violated.
In this section we consider the case when Condition (AP) is violated, in other words, there exists a proper subgroup H for which the condition (7) holds. Throughout this section we denote by H the minimum of such subgroups. (The minimum exists since the class of H satisfying (7) is closed under intersection.) The arguments in this section are mostly algebraic and apply without the condition (H.1) except for the matters that obviously require (H.1) in this paper.
We divide this section into four parts. In the first one we introduce a new MA process, denoted by (ξ n ,Ŝ n ), which is obtained from (ξ n , S n ) by a simple transformation and prove that Condition (AP) is satisfied for it. In the next part we present several examples, which exhibit certain possibilities about ergodicity of the process (ξ n ). In the third we see that the degenerate case where the dimension of H is less than d may be reduced to a non-degenerate case. In the last part the non-degenerate case is considered. It is shown that if (ξ n ) is ergodic then the expansions of the Green function in Section 1 are valid without any modification (this will complete the proof of the results of Section 1); if it is non-ergodic, the expansions are still valid except for a constant factor and for a suitable restriction on the combination of initial and terminal points (depending on an ergodic component). Also as an asymptotic form of transition probability we present a fairly clear picture in the case when T is countable.

5.1.
Pick up a representative, a ∈ Z d say, of each coset in the quotient group Z d /H and let K be the set of such a's, so that each x ∈ Z d is uniquely represented as x = y + a with y ∈ H, a ∈ K. According to this representation of x we define π K by π K (x) = a. IfT = T × K, this gives rise to the mapping T × Z d →T × H which maps (ξ, x) ∈ T × Z d to (ξ,ŷ) wherê ξ = (ξ, π K (x)) andx = x − π K (x); and accordingly the new process, (ξ n ,Ŝ n ), taking on values inT × H, is induced from (ξ n , S n ): S n = S n − a n andξ n = (ξ n , a n ) where a n = π K (S n ).
Clearly (ξ n ,Ŝ n ) is a MA process onT × H.
Proposition 8. For every invariant measureμ, Condition (AP) holds for the process (ξ n ,Ŝ n ) that is regarded as a MA process onT × H.

5.2.
Even when S n is aperiodic in the sense that for every proper subgroup H ′ of Z d , µ({ξ ∈ T : ∃a ∈ Z d , P ξ [Y 1 ∈ H ′ + a] = 1}) < 1, there are various cases of the processξ n : it can be cyclic, non-ergodic, or non-cyclic and ergodic as exhibited in the examples given below. Ifξ n is not ergodic orτ > τ , the formulas of Theorems 1, 2 and 4 must be suitably modified.

Examples.
In these examples T is a quotient group Z/kZ ∼ = {0, . . . , k − 1} with k = 2 or 3; (ξ n ) is noncyclic and S n is aperiodic in the sense stated above except for the example (5); but Condition (AP) is not satisfied.

5.3.
In all the examples above we have ♯K < ∞, which, however, is not generally true. Given a Markov process ξ n on T satisfying (H.1), we take a measurable function ϕ : T → Z d and an initial random variable S 0 and define S n by which is clearly MA and satisfies (7) with H = {0} (moreover for a suitable ϕ the walk S n may be irreducible in usual sense if T is large enough). Clearly ♯K = ∞ and Q = 0. The converse is also true. (7) is satisfied with H = {0}), then S n is given in the form (59) with P µ -probability one.
Proof. Let H = {0}. Then there exist Z d -valued measurable functions ϕ n (ξ, η) (for n ≥ 1) such that with P µ -probability one, We divide the rest of proof into two steps.
In general H may be isomorphic to Z m : so that ♯K = ∞ and Q = 0 whereas Q is not positive definite. In such a case, lettingH be the largest subgroup of Z d such that H ⊂H ∼ = Z m , we can find another subgroup H • ∼ = Z d−m so that Z d =H + H • and K = K ′ + H • (direct sum) where K ′ =H/H (the quotient group); this induces the decomposition with ϕ a function on T taking on values in the lattice H • and a ′ n a K ′ -valued process, such that ifξ n = (ξ n , a ′ n ), then (ξ n ,Ŝ n ) is a MA process on (T × K ′ ) × H which satisfies (AP); clearly ♯K ′ < ∞. The proof is immediate from Proposition 9. (6)), then ♯K < ∞ andξ n is ergodic, and vice versa.

Example. Let d = 2 and H
Proof. If ♯K = ∞, (ξ n , S n ) cannot be irreducible owing to the decomposition (62). Ifξ n is not ergodic, (ξ n , S n ) cannot be irreducible; thus the first half of the lemma. The converse follows from Proposition 8.

5.4.
In what follows we suppose that ♯K < ∞, which is satisfied under the irreducibility. By virtue of (56) lim n −1 E µ |S n −Ŝ n | 2 = 0; hence Q =Q, in particular Q is positive definite according to Proposition 8. Applications of Theorems proven under (AP) to the process (Ŝ n ,ξ n ) yield the expansions of Green functions and transition probabilities of it, from which we can derive those for (S n , ξ n ). In this subsection we obtain such results in a rather direct way.
Without essential loss of generality we also suppose that S n is irreducible in the sense that for every proper sub-group

5.4.1.
The Green function in the case whenξ n is ergodic.
In view of Lemma 10 the following lemma completes the proof of the results of Section 1.

Lemma 11.
Suppose thatξ n is ergodic. Then the expansions of the Green functions in Theorems 1 and 2 and Corollary 3 hold true.
The expansions in Theorems 1 and 2 and Corollary 3 are derived from the estimates of E ξ [e iSn·θ f (ξ n )] in a neighborhood |θ| < ε. In the proof of Theorem 11 we shall see that even in the case when (AP) is violated for the walk S n the computation based only on such estimates leads to correct results, provided thatξ n is ergodic.
In view of Proposition 8 we can apply the results of Sections 2 through 4 to the· process with a Fourier domain∆ in place of ∆. If a ∈ K, b = π K (x) and x =x + b, since |∆| = (2π) d /♯K. (For the present purpose we may put a = 0 but this proof will apply to the nonergodic case.) Owing to the relationŜ n + b = S n = S n − S 0 + a (P ξ -a.s.) and the additivity property of the walk S n , we can rewrite the right side above as ClearlypT satisfies Doeblin's condition, so that the distribution of (ξ n , a n ) converges toμ geometrically fast. Supposeτ = 1. Then, we can discard the event a n = b − a (mod K) and the factor ♯K simultaneously up to an error of order o(ρ n 1 ) (with 0 < ρ 1 < 1), which results in In carrying out the Fourier integration we use this expression on the ε-neighborhood of θ = 0 and (64) on the rest to follow the computation of Section 2. The caseτ > 1 can be dealt with as before (see Section 4). This proves Lemma 11.
The next lemma, though not used in this paper, is sometimes useful to translate results for (ξ n , S n ) to those for (ξ n ,Ŝ n ) and vice versa.

5.4.2.
The Green function in the case whenξ n is non-ergodic.
Letξ n be not ergodic. ThenT is decomposed into more than one ergodic components. We regard K as the quotient group Z d /H.

Lemma 13.
Let m be the number of ergodic components ofξ n . Then m ≤ ♯K and there exist a subgroup K ′ of K and a decomposition T = a ∈K/K ′ T ( a ) such that m = ♯(K/K ′ ) and the class of setsT a ∈ K/K ′ makes the ergodic decomposition ofT = T × K, where a ∈ K/K ′ (a ∈ K) is identified with a coset a + K ′ (⊂ K); the corresponding invariant measures are given bŷ respectively.

Proof. Pick up an ergodic component E ⊂T and set
T (a) = {ξ ∈ T : (ξ, a) ∈ E} (a ∈ K), so that E = a T (a) × {a}. Since (ξ n , a n ) is a MA process onT , namely the distribution of the increment a n − a n−1 is determined by the value of ξ n−1 independently of a n−1 , for every a ∈ T According to Lemma 13 the formulas in Section 1 is modified as follows.
Let G ( a • ) denote the Green function for the process restricted on an irreducible component is an ergodic component for (ξ n ) described in Lemma 13. Then a version of the formula corresponding to (8) is stated as follows: if and w = y − x, then (no change except for the factor ♯(K/K ′ ) and for the restriction on the combinations of the initial point (ξ, a) and the terminal set A × {b}) .
The modification for two dimensional case is similar.
For the proof we may proceed as for Theorem 11 except that we divide by ♯K ′ instead of ♯K when we discard the event a n = b − a (mod H) in the formula (65), which gives rise to the factor ♯(K/K ′ ) to the right side of (66).
We have an analogue of Lemma 12 also in the case whenξ n is nonergodic: it may read that

Local central limit theorems.
Under our assumption that ♯K < ∞ we have the following result in place of the decompositions (59) and (62).
Remark 7. (i) In the case τ > 1 the two conditions (71) and (72) are not exclusive of each other: if τ −1 e • ∈ Z d , then on suitably modifying ϕ and regarding τ −1 e • as an element of K, which we rewrite as e • , the latter is reduced to the former. On recalling that P µ [a 0 = 0] = 1 these formulas actually give expressions for the increments a n − a 0 when S 0 is not necessarily 0.
(ii) The condition (72) is equivalently expressed as A recipe for finding e • may be found from (71) and (74) as well as in the proof of Lemma 14. In the case when p τ T (ξ, ξ) > 0 for some ξ = ξ • in particular, it is given by e • = a τ a.s.(P ξ • ) where a τ is necessarily nonrandom under the premise.
In the rest of this section we suppose that T is countable. This supposition is used only through Lemma 14. It is recalled that our only hypothesis here is ♯K < ∞; the irreducibility may fail to hold.
Theorem 15. Let ϕ and e • be as in Lemma 14. Suppose that T is countable. Then, without assuming Condition (AP) the local central limit theorems in Section 1 remain true if the right sides of the formulas (13) and (15) are multiplied by the function according as (71) or (72) holds.
or on multiplying f (ξ n ) and taking expectation, Recalling what is noticed at the beginning of this proof we apply this identity with f such that f (η) = 0 whenever x = ϕ(η) − ϕ(ξ) + ne • (mod H), so that it reduces to p n θ+u f = e ix·u p n θ f . Hence we have only to sum up the contributions of the integrals on neighborhoods of u ∈ K ♭ to see that the same computation as in Section 3 leads to the desired result owing to the identity ♯K ♭ = ♯K.
From the relation (75) and Proposition 20 we obtain the following corollary.
If (s, τ ) = 1, the ergodic decomposition may be finer than that given in Proposition 17 depending on how the cyclically moving sets T j is related to ϕ. For instance, if K ′′ is a subgroup of K such that K = K ′ + K ′′ (direct sum) and the cyclically moving subsets T j are of the form In particular, there can be s distinct ergodic components even in the case K ′ = K.
Remark 8. In the case ♯T < ∞ a local central limit theorem for MA processes as given in Theorem 15 (but up to the principal order term) is obtained by Krámli and D. Szász (18) under the condition that the covariance matrix Q is positive definite and (ξ n ) makes no cyclic transition. Their approach is somewhat different from ours. Keilson and Wishart (17), studying a central limit theorem for MA processes on T × R with ♯T < ∞, show among others that Q = 0 if and only if it is degenerate in the sense that the walk is represented as in (59). Our proof is applicable to MA processes on T × R d . 6 Derivatives of κ(θ) and M ξ (θ) at 0 In this section we compute the derivatives of the principal eigenvalue κ(θ) based on the perturbation method of which we shall review in Appendix D. Let p T and p be a probability kernel on T and the bounded operator on L ∞ (µ) associated with it as defined in Section 1. We suppose that the basic assumptions mentioned in Introduction ( i.e., (H.1) and (1)) hold.
Let p θ be the operator with the kernel defined by (20): . Denote its principal eigenvalue by κ(θ) (for |θ| small enough); let e θ and µ θ be the corresponding eigenfunction and its dual object which are normalized so that µ(e θ ) = µ θ (e θ ) = 1 as in Section 2.
Proposition 20. Let m be a non-negative integer. Suppose that sup ξ E ξ [|Y 1 | m ] < ∞ and Condition (AP) holds for S n . Then, for each ε > 0 there exists a positive constant r < 1 such that if f (ξ) is bounded, In the proof of Proposition 20 given below we need to express Condition (AP) of the walk S n in terms of the characteristic functions To this end we first prove a preliminary result that we formulate in a general setting.
Lemma 21. Let X λ , λ ∈ Λ be a family of random variables taking on values in Z d and ν(dλ) a probability measure on Λ and suppose that P [X λ = x] is λ-measurable. Denote by F λ the support of the law of X λ : The following two conditions are equivalent.
Proof. For a nonempty set F ⊂ Z d , taking any x ∈ F , we denote by [F ] the smallest subgroup including F − x. Clearly [F ] does not depend on x. Now suppose that the equality in (i) of the lemma holds true for a θ ∈ ∆ \ {0} and let H be the set of all x such that x · θ ∈ 2πZ. Then for ν-almost all λ, E e iX λ ·θ = 1, or equivalently, x · θ ∈ 2πZ for all x ∈ [F λ ], so that for ν-almost all λ, [F λ ] ⊂ H. Since H is a proper subgroup we have (ii). The converse is obvious.
Proof. It suffices to show that Condition (AP) is violated if for every n • ≥ 1 there exists a proper subgroup H for which the probability in (82) equals unity. In view of the preceding lemma this follows from the inequality (1 ≤ k < m) since it shows that if the probability in (82) equals unity for n • = m, then it does for every n • ≤ m and since if this condition (with the same m) is satisfied by two subgroups, so is by the intersection of them.
Studying a Markov chain on Z d with a transition law having a certain periodicity structure Takenami (24) introduces a condition analogous to that in the corollary 22 and proves it to be satisfied by the Markov chain under a certain circumstance. Babillot (1) and Givarc'h (6) call a MA process aperiodic if the condition (82) holds with n • = 1 for each proper subgroup H.
By the same argument we verify the following as n → ∞. Here o(|η| δ r n ) is uniform for |η| < 1, so that the infinite sum over n ≥ 0 of the supremums on the left side is o(|η| δ ).
For estimation of Fourier integral of the error term ε k (θ)/Q 2 (θ) we repeat integration by parts k − 2, k − 1 or k times according as d = 2, d = 3 or d ≥ 4. To complete the proof of Theorems 1 and 2 it now is sufficient to prove the next lemma.
B. Our evaluation of Fourier integrals on the torus ∆ made in Section 3 is based on the following two formulae (i) and (ii) as well as the results of the preceding section: the former two are used to dispose of the integral on a neighborhood of origin and the latter ones are on the rest.
(i) Let D be a d-dimensional bounded domain containing the origin and having piece-wise smooth boundary ∂D. Let g be a function on R d of the form {θ k }/|θ| s+k with k a non-negative integer and s a real number such that s < d. Then for every integer n satisfying n ≥ d − s, D g(θ)e ix·θ dθ = g ∧ (x) + n l=1 B D,l (x) (i|x|) l − 1 (i|x|) n D c (−ω · ∇) n g(θ)e ix·θ dθ (x ∈ R d ).
Here ω = x/|x| and B D,l denotes the boundary integral ∂D (−ω · ∇) n g(θ)e ix·θ ω · dS; if n = d − s the last integral is not absolutely convergent and must be understood to be the principal value; g ∧ denotes the Fourier transform R d g(θ)e ix·θ dθ in the sense of Schwartz distribution on the punctured space R \ {0}, namely g ∧ (x) (x = 0) is identified by the relation gϕ ∧ dx = g ∧ ϕdx to hold for every smooth function ϕ that vanishes outside a compact set of R d \ {0}. Proof is standard (see eg. Lemma 2.1 of (25)).
(ii) If ϕ k is a homogeneous harmonic polynomial of degree k, then for s ∈ R, C. Here is given proofs of (33) and (34). For the first one it suffices to prove that When r is large, the first term on the right side may be written as where k = ⌊r 2 ⌋ ( the largest integer that does not exceed r 2 ) and similarly for the last term. By elementary computation we see that for a > 0 (log t)e −t dt + log a, and r = 1 2 min{1 − ρ, |ω − 1|}, then the spectrum Σ θ is divided by the circle C r = {|1 − z| = r} into two parts in such a manner that the part outside C r is contained in one of the open disks |z| < (1 + ρ)/2, |z − ω j | < r, j = 1, . . . , τ − 1, and the part inside continuously moves to 1 as θ approaches the origin; and that the latter consists of a single eigenvalue, κ(θ) say, which is simple (cf. (12): either p.34 or p.212). Let Π θ 0 denote the projection operator corresponding to this eigenspace. Then (p θ − Π θ 0 ) n L ∞ (µ) ≤ C[(1 + ρ)/2] n and Here e θ is an eigenfunction for the eigenvalue κ(θ) and µ θ is a dual object: these may be defined (if δ • is small enough) by with Ξ(θ) = µΠ θ 0 1, and the product e θ ⊗µ θ stands for the operator given by the complex measure kernel e θ (ξ)µ θ (dη). They are normalized so that µ(e θ ) = µ θ (e θ ) = 1.