Abstract
Abstract. A complete representation of the Martin boundary of killed random walks on the quadrant N * × N * is obtained. It is proved that the corresponding full Martin compactification of the quadrant N * × N * is homeomorphic to the closure of the set {w = z/(1 + |z|) : z ∈ N * × N * } in R 2 . The method is based on a ratio limit theorem for local processes and large deviation techniques.
Introduction
The concept of Martin boundary was first introduced for Brownian motion by Martin [14] and next extended for countable discrete time Markov chains by Doob [7] and Hunt [9] . For a Markov chain (Z(t)) on a countable set E with Green's function G(z, z ′ ), the Martin compactification E M is the smallest compactification of the set E for which the Martin kernels K(z, ·) = G(z, ·)/G(z 0 , ·) extend continuously. See Rogers and Williams [17] for example. The Martin boundary for homogeneous random walks in Z d was obtained by Ney and Spitzer [15] . We identify the Martin boundary of a killed random walk (Z + (t)) on the positive quadrant N * × N * . Such a random walk has a sub-stochastic transition matrix (p(z, z ′ ) = µ(z ′ − z), z, z ′ ∈ N * × N * ) with some probability measure µ on Z 2 , it is identical to a homogeneous random walk (S(t)) on the 2-dimensional lattice Z 2 before first exits from the quadrant N * × N * and is killed upon the time τ= inf{n ≥ 0 : S(n) ∈ N * × N * }.
The random walk (Z + (t)) is therefore not homogeneous : transition probabilities on the boundary of the quadrant N * × N * are not the same as in the interior. For non-homogeneous Markov processes, the problem of Martin boundary identification is usually non-trivial and there are few examples where it was resolved : Cartier [3] described the Martin boundary of random walks on non-homogeneous trees, Doney [6] identified the Martin boundary of a homogeneous random walk (Z(n)) on Z killed on the negative half-line {z : z < 0}. Alili and Doney [1] identified the Martin boundary for space-time random walk S(n) = (Z(n), n) for a homogeneous random walk Z(n) on Z killed on the negative half-line {z : z < 0}. All these results were obtained by using a special linear structure of the processes. The Martin boundary of Brownian motion on a half-space was obtained in the book of Doob [7] by using an explicit form of the Green's function.
In Kurkova and Malyshev [13] the full Martin compactification is obtained by using methods of complex analysis for nearest neighbors random walks on a halfplane Z × N and in the quadrant Z 2 + = N × N. In a recent paper of Raschel [16] , the Martin boundary is obtained for nearest neighbor random walks in N × N with an absorption condition on the boundary also by using methods of complex analysis. Because of the use of the specific algebraic setting of elliptic curves, these methods seem to be difficult to apply when the jump sizes are more general.
The results of Kurkova and Malyshev [13] exhibit a formal similarity between the limiting behavior of the Martin kernel and the optimal large deviation trajectories obtained by Ignatyuk, Malyshev and Scherbakov [12] . A natural idea is then to study the Martin compactification by using large deviation methods. The large deviation approach was first proposed in the papers of Ignatiouk-Robert [11, 10] in order to identify the Martin boundary for partially homogeneous random walks on a half-space Z d−1 × N. The minimal harmonic functions were determined there by using the methods of Choquet-Deny theory (see Woess [19] ) and then the limiting behavior of the Martin kernel was obtained by using an explicit representation of the harmonic functions combined with the large deviation estimates of Green's function and the ratio limit theorem of Markov-additive processes. Unfortunately, the methods of Choquet-Deny theory and the ratio limit theorem are valid only for Markov-additive processes, i.e. when transition probabilities are invariant with respect to the translations on some directions. In the setting of the present paper, for a random walk in the quadrant N * × N * , such an invariance property cannot hold. Our paper is the first step towards a more ambitious program : to identify the Martin boundary for general partially homogeneous random walks in N n . The main idea of our method is the following : to study the asymptotic behavior of the Martin kernels K(z, z n ) for a sequence of points z n which tends to infinity with lim n z n /|z n | = q, one should consider a twisted random walk conditioned to go to infinity in the direction q. For a non-zero vector q ∈ R 2 + , such a twisted homogeneous random walk will visit at least one of the boundaries (−N) × Z or Z×(−N) only a finite number of times. If the corresponding boundary {0}×N [resp. N × {0}] is removed, the resulting process is then identical to the homogeneous random walk (S(t)) before the first time when it hits the set Z × (−N) [resp. (−N)× Z]. The limiting behavior of the Martin kernel of this process corresponding to the direction q is already known in such a setting. The limiting behavior of the Martin kernel of the original process (Z + (t)) should be essentially the same but with a correction given by a potential function. When both coordinates of q are positive, this idea is transformed into a rigorous proof with the aid of large deviation estimates and a generalization of a ratio limit theorem of the paper [11] . When one of the coordinates of q is zero, i.e. when the process is conditioned to go to infinity along one of the boundaries, our proof is much more complicated. In this case we combine large deviation techniques and the ratio limit theorem with delicate estimates obtained from Harnack's inequalities.
We assume that the probability measure µ on Z 2 satisfies the following conditions :
(H0) The homogeneous random walk S(t) = (S 1 (t), S 2 (t)) on Z 2 having transition probabilities p S (z, z ′ ) = µ(z ′ − z) is irreducible and has a non zero mean m= is finite everywhere on R d .
Under the above assumptions, the set D= {a ∈ R 2 : ϕ(a) ≤ 1}
is compact and strictly convex, the gradient ∇ϕ(a) exists everywhere on R 2 and does not vanish on the boundary ∂D = {a ∈ R 2 : ϕ(a) = 1}, the mapping (1.2) a → q(a)= ∇ϕ(a)/|∇ϕ(a)| determines a homeomorphism from ∂D to the unit 2-dimensional sphere S 2 = {q ∈ R 2 : |q| = 1} (see [8] ). We denote by q → a(q) the inverse mapping of (1.2) and we let a(q) = a(q/|q|) for a non-zero q ∈ R 2 . According to this notation, a(q) is the only point in ∂D where the vector q is normal to the convex set D. Throughout this paper we denote by N the set of all non-negative integers and we let N * = N \ {0}. The set of all non-negative real numbers is denoted by R + = [0, +∞[ and R * + =]0, +∞[ denotes the set of all strictly positive real numbers. It is convenient moreover to introduce the following notations :
is the first time when the random walk (S(t)) exits from the quadrant N * × N * .
S 2 += {q ∈ R 2 + : |q| = 1} and Γ += {a ∈ ∂D : q(a) ∈ S 2 + }. For a ∈ Γ + and z = (x 1 , x 2 ) ∈ N * × N * we set
Green's function of the process (Z(t)) :
Our paper is organized as follows : in Section 2, the main idea and the main steps of the proof of our result are sketched. Section 3 is devoted to the preliminary results. In Section 4 we prove that the functions h a with a ∈ Γ + defined by (1.3) are finite, harmonic for the Markov process (Z + (t)) and strictly positive. Section 5 is devoted to the large deviation results. It is shown that the family scaled processes Z ε + (t) = εZ + ([t/ε]) satisfies sample path large deviation principle. The logarithmic estimates of Green's function are obtained from the corresponding large deviation bounds. In Section 6 the large deviation estimates are used to decompose Green's function G + (z, z n ) into a main part corresponding to an optimal large deviation way to go from z to z n and the negligible part. In Section 7, we generalize the ratio limit theorem of Ignatiouk-Robert [11] . The decomposition into a main and a negligible parts of Green's function G + (z, z n ) and the ratio limit theorem are next combined in Section 8 in order to complete the proof of Theorem 1.
Local processes and renewal equations: a sketch of proofs
The main steps of our method can be summarized as follows :
-For a sequence (z n ) ∈ N * × N * with lim n z n /|z n | = q and lim n |z n | = +∞, Green's function G + (z, z n ) of the Markov process (Z + (t)) is represented in terms of a local random walk which is Markov-additive and has the same transition probabilities as the original random walk (Z + (t)) in a neighborhood of the point q|z n |. -Next, large deviation estimates are used to decompose G + (z, z n ) into a main part corresponding to an optimal large deviation way to go from z to z n and the negligible part. Such a decomposition allows to get the limit of the Martin kernel
from the limiting behavior and the uniform bounds of the Martin kernel of the corresponding local process. When both coordinates of the vector q = (q 1 , q 2 ) are non-zero, the local Markovadditive process is simply a homogeneous random walk (S(t)) on Z 2 having transition probabilities p(z, z ′ ) = µ(z ′ − z). This is the simplest case in our proof. Recall that our Markov process (Z + (t)) is identical to the homogeneous random walk (S(t)) on Z 2 before the time τ= inf{n ≥ 0 : S(n) / ∈ N * × N * } and is killed upon τ . The following renewal equation represents Green's function G + (z, z ′ ) of the Markov process (Z + (t)) in terms of Green's function G(z, z ′ ) of the random walk (S(t)) :
Ney and Spitzer [15] proved that for any q ∈ S 2 + and any sequence of points z n ∈ N * × N * with lim n |z n | = ∞ and lim n z n /|z n | = q,
for all z ∈ Z 2 (see also Section 7 in [11] for an alternative simple proof of this result). Using the renewal equation (2.1) one can get therefore the equality
if one can prove the exchange of limits
Relation (1.4) will follow finally from the relation (2.3) because the function h a is strictly positive on N * × N * (see Proposition 4.1 below). Relation (2.4) is therefore a key relation for our problem.
While the above idea seems quite simple, the proof of (2.4) is non-trivial because the convergence (2.2) is not uniform and the classical convergence theorems are here difficult to use. With our approach, we first decompose the right hand side of (2.1) into a main part
and the corresponding negligible part
by using the large deviation estimates of Green's function G(z, z ′ ) and
whenever the last limit exists, and consequently, (1.4) holds whenever lim n→∞ w∈Z 2 \(N * ×N * ): |w|<δ|zn|
To prove the last relation one needs the estimates (2.5) sup
for which the series
converge. These estimates are obtained in Section 7 with a suitable exponential function C(z) by using the ratio limit theorem applied to the random walk (S(t)).
The case when one of the coordinates of the vector q is equal to zero, , i.e. when the sequence (z n ) tends to infinity along one of the boundaries of the domain, is much more delicate to handle. First of all, we cannot use here the renewal equation (2.1) because the function exp(a(q) · z) − E z exp(a(q) · S(τ )), τ < ∞ is in this case identical to zero. If q = (1, 0), then contrary to the previous case, for n large enough, the point z n is far from the boundary {0} × R + but still near the boundary R + × {0} of the quadrant R + × R + . In this case, one should consider a Markovadditive process having the same statistical behavior as the process (Z + (t)) near the boundary N × {0} and far from the boundary {0} × N. This is a homogeneous random walk (Z 1 + (t)) on Z × N * having a sub-stochastic transition matrix
It is identical to the homogeneous random walk (S(t)) before the first time
when the random walk S(t) = (S 1 (t), S 2 (t)) exits from the half-plane Z × N * and killed upon the time τ 2 . Our Markov process (Z + (t)) is therefore identical to (Z 1 + (t)) before the time τ 1= inf{t ≥ 0 : S 1 (t) ≤ 0}.
Theorem 1 of [11] proves that for any sequence of points z n ∈ N * × N * with lim n |z n | = ∞ and lim n z n /|z n | = q = (1, 0),
with a strictly positive function h
Similarly to the previous case we decompose the right hand side of the renewal equation (2.6) into a main part and the corresponding negligible part by using the large deviation estimates of Green's functions G + (z, z ′ ) and G 1 + (z, z ′ ) and we show that (1.4) holds whenever there exists a function C 1 + (z) for which the series
converge and for δ > 0, (2.8) sup
The proof of (2.8) with a desired function C 1 + (z) is the most delicate part of our proof. With the ratio limit theorem we are able to estimate the quantities
* and w ∈ Z × {0} satisfying the inequality max{|z|, |w|} < δ|z n | with δ > 0 small enough. These estimates are combined next with Harnack's inequality in order to get (2.8) with a suitable exponential function C 1 + (z). The proof of (1.4) for a sequence z n ∈ N * × N * with lim n |z n | = ∞ and lim n z n /|z n | = q = (0, 1) is quite similar, it is sufficient to exchange the roles of the coordinates of the Markov process (Z + (t)). Instead of the local process (Z 1 + (t)) one should consider here the homogeneous random walk (Z
Preliminary results
For a given a ∈ D= {a ∈ R 2 : ϕ(a) ≤ 1}, let us consider a new twisted homogeneous random walk (S a (t)) on Z 2 having transition probabilities
The transition matrix of such a random walk is sub-stochastic because according to the definition of the set D,
Proposition 3.1. For every a ∈ D, the quantity E z exp(a · (S(τ ) − z)), τ < ∞ is equal to the probability that the twisted random walk (S a (t)) starting at the point z ever exits from the positive quadrant N * × N * .
Proof. Indeed, let
be the first time when the twisted random walk (S a (t)) exits from the quadrant N * × N * . Then for any t ∈ N,
and consequently,
The set Γ + = {a ∈ ∂D : q(a) ∈ S 2 + } endowed with a topology induced by the usual topology of R 2 is homeomorphic to a segment with the end points in a(1, 0) and a(0, 1). The points a(1, 0) and a(0, 1) are said to be critical. We denote by
the first time when the process S(t) = (S 1 (t), S 2 (t)) exits from the half-plane N * ×Z and we let
Remark that according to our notations
Proposition 3.2. Every non-critical point of Γ + has a neighborhood where the
Proof. For a ∈ D, by Proposition 3.1, the quantity
is equal to the probability that the twisted sub-stochastic homogeneous random walk (S a (t)) starting at z ever hits the set on the event {τ = τ 1 < +∞} and S 2 (τ ) ≤ 0 on the event {τ = τ 2 < +∞}, from this it follows that
Proof. The proof of this proposition use essentially the same arguments as the proof of Proposition 3.2. For a ∈ D,
is equal to the probability that the twisted sub-stochastic homogeneous random walk (S a (t)) starting at z hits the set (−N) × Z before hitting the set Z × (−N). This proves that the functions a → E z (exp(a·S(τ )), τ = τ 1 < τ 2 ) are finite on D for all z ∈ N * ×N * . Moreover, let us consider the points a(0, 1) = (a Then the set Ω= {a = (a 1 , a 2 ) ∈ R 2 : a 1 > max{a
} is an open neighborhood of the point a(1, 0) and for any a = (a 1 , a 2 ) ∈ Ω \ D there is a pointâ = (â 1 ,â 2 ) on the boundary of the set D withâ 2 = a 2 andâ 1 < a 1 (see Figure 2 ). Since on the event {τ = τ 1 < τ 2 }, S 1 (τ ) ≤ 0 we conclude that for any z ∈ N * × N * ,
The functions a → E z (exp(a · S(τ )), τ = τ 1 < τ 2 ) are therefore finite on Ω for all z ∈ N * × N * . Finally, for δ > 0 small enough, a = a(1, 0) + (0, δ) ∈ Ω and hence, the last inequality proves also (3.2). Proposition 3.3 is therefore proved.
A straightforward consequence of Proposition 3.3 is the following statement.
Proof. Indeed, on the event τ = τ 1 < τ 2 , for any δ > 0,
and consequently
Since by Proposition 3.3, the right hand side of the last relation is finite for all z ∈ N * × N * and δ > 0 small enough , we conclude that the function (3.3) is finite on N * × N * .
With the same arguments one gets similar results for the critical point a(0, 1). It is sufficient to exchange the roles of the first and the second coordinates of (S(t)).
To show that the functions (1.3) are well defined we will need moreover the following statement.
Lemma 3.1. For a random walk (ξ(t)) on Z having zero mean and transition probabilities
This lemma was proved in the core of the proof of Lemma 5.3 in the paper [11] . Corollary 3.1 combined with Lemma 3.1 implies the following proposition.
Proof. To prove this proposition let us first notice that
where for a = a(1, 0), by Corollary 3.1,
To prove that the function
Next, we consider a twisted random walk (S a (t)) on Z 2 with transition probabilities
) is a random walk on Z having a mean
and satisfying the conditions of Lemma 3.1. This lemma applied with ξ(t) = S a 2 (t) and
2 )|) we conclude that (3.4) holds and consequently, the function
is finite on N * × N * .
Harmonic functions.
The main result of this section is the following proposition. Before to prove this proposition we consider the following lemmas.
Lemma 4.1. For a ∈ Γ + , the function
Proof. Indeed, suppose that a ∈ Γ + . Then under the hypotheses (H0) and (H3), the twisted random walk
and a finite variance. If q(a) = (0, 1), the first coordinate S a 1 (t) of S a (t) is therefore a recurrent random walk on Z. In this case, the first time when S a 1 (t) becomes negative or zero is almost surely finite for any starting point S a (0) = z ∈ N * × N * and consequently, the twisted random walk (S a (t)) almost surely exits from the quadrant N * × N * . By Proposition 3.1 from this it follows that
The same arguments but with a recurrent random walk (S 2 (t)) prove this equality when q(a) = (1, 0).
Suppose now that q(a) ∈ {(1, 0), (0, 1)}. Then by the strong law of large numbers, S a (t)/t → m(a) almost surely as t → ∞ for any initial state S a (0) = z. From this it follows that for any S a (0) = z and ε > 0 there is an almost surely finite positive random variable N ε such that for all t ≥ N z,ε ,
Since q(a) ∈ {(1, 0), (0, 1)}, the both coordinates of the mean vector m(a) are positive and non-zero and consequently, there exist N > 0 andε > 0 for which the set z ∈ Z 2 : |z − m(a)t| <εt for some t ≥ N is included to the quadrant N * × N * . For the initial state S a (0) = 0, from this it follows that almost surely
The minimums min are therefore almost surely finite and consequently, for someẑ = (
The last inequality combined with Proposition 3.1 shows that
To complete our proof it is now sufficient to notice that under the hypotheses (H2), for any z ∈ N * × N * , the probability that the random walk (S a (t)) starting at z hits the pointẑ before the first exit from the quadrant N * × N * is non-zero and consequently, for some t = t(z,ẑ) ∈ N,
Lemma 4.1 is therefore proved.
Lemma 4.2. The functions
are well defined and non-negative on N * × N * .
Proof. Indeed, Proposition 3.4 proves that the function (4.1) is well defined. To prove that this function is non-negative on N * × N * let us notice that by dominated convergence theorem from Proposition 3.4 it follows that
Moreover, the function z = (x 1 , x 2 ) → x 2 exp(a(1, 0)·z) is harmonic for the random walk S(t) because according to the definition of the point a(1, 0), for any z = (x 1 , x 2 ),
Hence, the sequence S 2 (n) exp(a(1, 0) · S(n)) is a martingale relative to the natural filtration of (S(n)) and by the stopping-time theorem, for a = a(1, 0) and any
where the last relation holds because on the event {τ > n}, S 2 (n) > 0. The last inequality combined with (4.3) shows that the function (4.1) is non-negative on N * × N * . To prove that the function (4.2) is also well defined and non-negative on N * ×N * , it is sufficient to exchange the roles of the first and the second coordinates of
Proof of Proposition 4.1. Suppose first that a ∈ {a(1, 0), a(0, 1)}. Then by Lemma 4.1, the function
is finite and strictly positive on N * ×N * . Furthermore, for the homogeneous random walk (S(t)) on Z 2 , the exponential function z → exp(a · z) is harmonic and the function
satisfies the equality
Since Z + (t) is killed upon to the first time τ when S(t) exits from N * × N * and is identical to S(t) for t ≤ τ , we conclude that the function h a is harmonic for the random walk (Z + (t)). When a ∈ {a(1, 0), a(0, 1)}, the function h a is therefore harmonic for (Z + (t)) and strictly positive on N * × N * . Consider now the case when a = a(1, 0) = (a 
By Lemma 4.2, the function h a is therefore well defined and non-negative on N * ×N * . To prove that this function is harmonic for the Markov process (Z + (t)) it is sufficient to notice that
Finally, let us show that the function h a is strictly positive. For this we first notice that by Harnack's inequality,
because the function h a is non-negative and harmonic for the Markov process (Z + (t)). Moreover,
because according to the assumption (H2), the Markov process (Z + (t)) is irreducible on N * × N * . These relations show that the function h a is either strictly positive everywhere on N * × N * or identically zero on N * × N * . To prove that the function h a is strictly positive on N * × N * it is therefore sufficient to show that h a (z) > 0 for some z ∈ N * × N * . We get this inequality for z = (x 1 , x 2 ) with x 2 = 1 and x 1 > 0 large enough by using the following relations :
Moreover, from Proposition 3.3 it follows that for a = a(1, 0) = (a ′ 1 , a ′ 2 ) and any δ > 0 small enough there is a pointâ = (â 1 ,â 2 ) ∈ ∂D withâ 1 < a
Since the right hand side of the last inequality tends to zero as x 1 → ∞, from this it follows that h a (z) > 0 for z = (x 1 , x 2 ) ∈ N * × N * with x 2 = 1 and x 1 > 0 large enough. Proposition 4.1 is therefore proved.
Large deviation results
In this section we obtain large deviation results for the family of scaled process and we deduce from them the logarithmic asymptotics of Green's function. To get the large deviation results for scaled processes (εZ + ([t/ε]) we need to show that the original non-scaled process (Z + (t)) satisfies the following communication condition. 
there is a sequence of points z 0 , z 1 , . . . , z n ∈ E 0 with z 0 = z, z n = z ′ and n ≤ C|z ′ − z| such that Proof. Indeed, under the hypotheses (H2), the random walk (Z + (t)) is irreducible on N * × N * . Hence, for every unit vector e ∈ {(1, 0), (0, 1)} there is n e ∈ N such that P (1,1) (Z + (n e ) = (1, 1) + e) > 0 and consequently, there are u 
. According to this definition, a good rate function is lower semi-continuous. 2) Let (Z(t)) be a Markov process on
2 ) with a rate function
, and
P [z/ε] denotes here the distribution of the Markov process (Z(t)) corresponding to the initial state Z(0) = [z/ε] where [z/ε] is the nearest lattice point to z/ε in E. For t ∈ N and ε > 0, we denote by [t/ε] the integer part of t/ε.
Under the hypotheses (H3), the jump generating function ϕ of (1.1) is finite everywhere on R 2 and hence, by Mogulskii's theorem (see [4] ), the family of scaled random walks S ε (t) = εS([t/ε]) satisfies sample path large deviation principle in
2 ) with a good rate function
The convex conjugate (log ϕ) * of the function log ϕ is defined by
Under the hypotheses (H4), (log ϕ) 
respectively.
For the random walk (Z + (t)) killed upon the first exit from the quadrant N * ×N * , we obtain the following statement. with the good rate function
Proof. The proof of this proposition is quite similar to the proof of Proposition 4.1 of the paper [11] . We recall here the main steps of this proof. For any c ≥ 0 and for any compact set V ⊂ R 2 , the set To get the lower large deviation bound
, it is sufficient to show that for any absolutely continuous function φ :
If φ(t) belongs to the interior of the set R + × R + for every t ∈ [0, T ], then for δ ′ > 0 small enough, on the event
the process (Z(t)) does not exit from the set N * × N * and consequently, Z ε + (t) = S ε (t) for all t ≤ T . In this case, the above inequality follows therefore from sample path large deviation principle for the family of scaled process S ε (t) = εS([t/ε]). To get (5.7) for an arbitrary absolutely continuous function φ : [0, T ] → R + ×R + , we use the communication condition. Recall that by Proposition 5.1, our random walk (Z + (t)) satisfies the following communication condition on N * × N * : there exist θ > 0 and C > 0 such that for any z = z ′ , z, z ′ ∈ N * × N * there is a sequence of points z 0 , z 1 , . . . , z n ∈ N * × N * with z 0 = z, z n = z ′ and n ≤ C|z ′ − z| such that
From this it follows that the mapping
2 ) (see the proof of Proposition 4.1 of the paper [11] ). Letting φ n (t) = φ(t)+(1/n, 1/n) for all t ∈ [0, T ] and n ∈ N * we obtain therefore a sequence of absolutely continuous functions φ n : [0, T ] → R + × R + for which the inequality
is already verified and
Since according to the definition (5.6) of the rate function I + [0,T ] , for every n ∈ N, 
follows from the corresponding upper large deviation bound of Mogulskii's theorem, for the scaled random walk S ε (t) = εS(t/ε) with the closed set F += {φ ∈ F : φ(t) ∈ R + × R + , ∀t ∈ [0, T ]} because
and according to the definition of the rate functions I Proposition 5.4. For any q ∈ R 2 + , z ∈ N * × N * and any sequences ε n > 0 and z n ∈ N * × N * with lim n ε n = 0 and lim n ε n z n = q, the following relations hold 
it is sufficient to notice that for any z ∈ N * × N * and n ∈ N,
where G a(q) (z, z ′ ) denotes Green's function of the twisted random walk (S a(q) (t)) on Z 2 with transition probabilities 
is quite similar to the proof of Proposition 4.2 of Ignatiouk-Robert [11] . We first notice that for any r > 0, T > 0 and 
Letting moreover T = 1/|∇ϕ(a)| with a = a(q) we obtain 
Next, we use the communication condition of Proposition 5.1. By Proposition 5.1, for any z = z ′ , z, z ′ ∈ N * ×N * there is a sequence of points w 0 , w 1 , . . . , w k ∈ N * ×N * with w 0 = z, w k = z ′ and k ≤ C|z ′ − z| such that
From this it follows that for any
Using the inequality G + (z, z n ) ≥ G + (z, z ′ ) P z ′ (Z + (t) = z n ) for z ′ = z n , we get therefore
Moreover, using the inequality Card{z ∈ Z 2 : |z − a| < R} ≤ (2R + 1) 2 with R = r/ε n and a = q/ε n we obtain
for all those n ∈ N for which |q − ε n z n | < r and consequently, for any r > 0,
Letting at the last inequality r → 0 and using (5.13) we get (5.12). The equality (5.8) is therefore verified. The proof of (5.9) and (5.10) is quite similar.
Principal part of renewal equations
For δ > 0 and a sequence of points z n ∈ N * × N * with lim n |z n | = +∞ and lim n z n /|z n | = q we define the sequence of functions Ξ q δ (z, z n ) by letting
if both coordinates of the vector q are non-zero. For q = (1, 0) we put
and for q = (0, 1) we let
Recall that (S(t)) is a homogeneous random walk on Z 2 having transition probabilities p S (z, z ′ ) = µ(z ′ −z) and Green's function G(z, z ′ ). Our random walk (Z + (t)) on N * × N * is identical to (S(t)) for t < τ = inf{n ≥ 0 : S(n) ∈ N * × N * } and killed upon the time τ . WE denote by (Z 1 + (t)) a random walk on Z × N * having a substochastic transition matrix (
It is identical to the homogeneous random walk (S(t)) for t < τ 2 = inf{n ≥ 0 : S(n) ∈ Z × N * } and killed upon the time τ 2 . Green's function of the Markov process (Z
The random walk (Z 2 + (t)) is identical to (S(t)) for t < τ 1 = inf{n ≥ 0 : S(n) ∈ N * × Z} and killed upon the time τ 1 . Comparison of (6.1), (6.2) and (6.3) with the renewal equations (2.1) and (2.6) shows that
The main result of this section proves that for any z ∈ N * × N * and δ > 0, the quantity Ξ q δ (z, z n ) represents the principal part of right hand side of the renewal equations (2.1) and (2.6) for z ′ = z n when lim n |z n | = +∞ and lim n z n /|z n | = q. This is a subject of the following proposition.
Proposition 6.1. Under the hypotheses (H0)-(H3),
for any z ∈ N * ×N * , δ > 0 and any sequence of points z n ∈ N * ×N * with lim n |z n | = +∞ and lim n z n /|z n | = q.
To prove this proposition we need the following lemma.
Lemma 6.1. Under the hypotheses (H0) and (H3), for any q ∈ S
2 + and δ > 0, there is a small ε > 0 such that (6.5) inf
Proof. Recall that under the hypotheses (H0) and (H3), the set
is compact and strictly convex (see [8] ), and according to the definition (1.2) of the mapping q → a(q), the point a(q) is the only point on the boundary of the set D where the vector q is normal to D. For any non-zero vector q ∈ R 2 , the point a(q) is therefore the only point in D where the linear function a → a · q achieves its maximum over a ∈ D. Hence, for any w ∈ R 2 ,
where the inequality holds with the equality if and only if a(w) = a(q) = a(q − w).
Since the mapping w → a(w) from the unit sphere S 2 to ∂D = {a ∈ R 2 : ϕ(a) = 1} is one to one, from this it follows that
Moreover, the set D = {a ∈ R 2 : ϕ(a) ≤ 1} being compact, the function w → a(w) · w + a(q − w) · (q − w) is convex, finite and therefore continuous on R 2 . From this it follows that for all R > 0 and δ > 0, ε(R, δ)= inf
and consequently, for 0 < ε < ε(R, δ)/R, we obtain inf
To get (6.5) it is now sufficient to show that for any ε > 0 small enough, there is R > 0 such that inf
Here, we use the following estimates : for any w ∈ R 2 and q ∈ S
Remark that the function λ(w)= a(w)·w+a(−w)·(−w) is continuous and positively homogeneous : (6.9) λ(w) = |w|λ(w/|w|).
Moreover, the same arguments as in the proof of the inequality (6.6) show that λ(w) > a(w) · w + a(w) · (−w) = 0 whenever a(w) = a(−w), and consequently λ(w) > 0 for all w = 0. Hence, letting ε 0= 1 2 min
λ(w) > 0 and c= 2 max a∈D |a| and using (6.9) at the right hand side of (6.8) we get
for all 0 < ε < ε 0 and w ∈ R 2 with |w| > c/ε 0 . The inequality (6.7) holds therefore for R = c/ε 0 and 0 < ε < ε 0 , and the inequality (6.5) is satisfied for 0 < ε < min{ε 0 , ε(c/ε 0 , δ)}.
Proof of Proposition 6.1. Let a sequence of points z n ∈ N * × N * be such that lim n |z n | = +∞ and lim n z n /|z n | = q. Then by Proposition 5.4,
and hence, to get (6.4) it is sufficient to show that (6.10) lim sup
Moreover, since the quantities Ξ q δ (z, z n ) − G + (z, z n ) are decreasing with respect to δ > 0, it is sufficient to prove this relation for small δ > 0. For this the following estimates are used : for any δ > 0, z ∈ Z 2 and n ∈ N,
when both coordinates of the vector q ∈ S 2 + are non-zero,
when q = (1, 0) and similarly,
for q = (0, 1). These estimates show that for any q = (x, y) ∈ S 2 + , δ > 0, z ∈ Z 2 and n ∈ N,
Remark furthermore that for all a, a ′ ∈ ∂D and z, w,
where G a (z, z ′ ) denotes Green's function of the twisted homogeneous random walk (S a (t)) on Z 2 with transition probabilities
). Using (6.12) with a = a(w/|z n |) = a(w) and a ′ = a(q − w/|z n |) at the right hand side of (6.11) and letting λ ε (q, w) = a(w) · w + a(q − w) · (q − w) − ε|w| and c= 2 max
exp −|z n |λ ε q, w/|z n | − ε|w| + c|z| + c z n − |z n |q for any ε > 0. Since lim n z n /|z n | = q and the series w∈Z 2 exp(−ε|w|) converge for every ε > 0, from the last inequality it follows that lim sup
To complete the proof of (6.10) it is now sufficient to notice that by Lemma 6.1, for any δ > 0, there is a small ε > 0 for which inf w∈R 2 :inf θ>0 |w−θq|≥κδ λ ε (q, w) > a(q) · q.
Uniform ratio limit theorem for Markov-additive processes
In this section we improve the ratio limit theorem of the paper [11] . This result is next applied to get the desirable estimates (2.5) and (2.8) for the local processes (S(t)) and (Z 1 + (t)).
Uniform ratio limit theorem for general Markov-additive processes.

Recall that a Markov chain Z(t) = (A(t), M (t)) on a countable set
The first component A(t) of Z(t) = (A(t), M (t)) is said to be an additive part of the process Z(t), and the second component M (t) is its Markovian part. The Markovian part M (t) is a Markov chain on E with transition probabilities
The assumptions we need on the Markov-additive process (Z(t) = (A(t), M (t))) are the following :
l for some l ∈ N and the function
is finite in a neighborhood of zero in R d+l .
Our Markov-additive process (Z(t)) is not necessarily stochastic : its transition matrix can be strictly sub-stochastic in some points z = (x, y) ∈ Z d × E. The following property of Markov-additive processes is essential in our analysis. G(z, z ′ ) denotes here Green's function of the Markov process (Z(t)).
Proposition 7.1. Let the Markov-additive processes Z(t) = (A(t), M (t)) be transient and satisfy the conditions (A1) and (A2). Suppose moreover that for given
holds with some n > 0. Then for any 0 < σ < 1 and r > 0 there are C > 0 and θ > 0 such that
Proof. In a particular case, for n = 1, this statement was proved in the core of the proof of Proposition 3.2 of the paper [11] by using the method of Bernoully part decomposition due to Foley and McDonald [5] . When n > 1, for Green's functioñ
of the included Markov chainZ(t) = Z(nt), this result proves that for any r > 0 and 0 < σ < 1 there areC > 0 andθ > 0 such that
where
When n > 1, the inequality (7.3) holds therefore for r > 0 small enough with θ =θ and
To complete the proof of this proposition it is now sufficient to notice that the right hand side of (7.3) is increasing with respect to r > 0. Hence, if the inequality (7.3) holds with some C > 0 and θ > 0 for a small r > 0, then it is also satisfied for large r > 0 with the same constants C and θ.
The following statement is an immediate consequence of Proposition 7.1.
Proposition 7.2. Let a sequence of points z n ∈ Z d × E be such that lim n |z n | = ∞ and
Then under the hypotheses of Proposition 7.1,
Proof. Indeed, by Proposition 7.1, for any r > 0 and 0 < σ < 1 there are C > 0 and θ > 0 such that
Moreover, (7.4) shows that
Using the last relation at the right hand side of (7.6) we obtain
and letting next σ → 0 we get
The prove the inequality
it is now sufficient to exchange the roles of w and w ′ . The equalities (7.5) are therefore verified.
Suppose now that the Markov-additive process (Z(t)) satisfies the following more restrictive condition : (A1') There are θ > 0 and C > 0 such that for any
If the condition (A1') is satisfied then there is a bounded function n 0 : E → N * such that for any z = (x, y) ∈ Z d × E,
and hence, there is k ∈ N * (for instance, k = n! with n = max y n 0 (y)) such that
We denote byk the greatest common divisor of the set of all integers k > 0 for which
The following statement is a refined version of the ratio limit theorem obtained in [11] .
Proposition 7.3. Let a Markov-additive process Z(t) = (A(t), M (t)) be transient and satisfy the hypotheses (A1') and (A2). Suppose moreover that a sequence of points
Proof. Indeed, let K be the set of all integers for which the inequality (7.7) holds.
To prove this proposition we first notice that because of the assumption (A1') for any w ∈ Z × {0} there are ε > 0 and a bounded function n : E → N * such that
Using Markov property, we get therefore
By Proposition 7.2, from this it follows that
and also
for all w ∈ Z d × {0} and k ∈ K. To get the last inequality it is sufficient to replace w by −w in (7.8) . Consider now the subgroup K of Z generated by K. Since (7.8) and (7.9) are satisfied for all k ∈ K we conclude that (7.8) holds also for any k ∈ K and in particular for k =k becausek ∈ K (see Lemma A.1 of Seneta [18] ).
7.2. Applications to local processes. According to the above definition, our homogeneous random walk (S(t)) on Z 2 is Markov-additive : its additive part is the process S(t) itself and the Markovian part is empty. The quantityk is here the period of the random walk (S(t)). Proposition 7.3 applied for the process (S(t)) and the estimates (5.9) yield the following statement.
Proposition 7.4. For any sequence of points z n ∈ Z 2 with lim n |z n | = +∞ and
Proof. Indeed, for any a ∈ ∂D, the twisted homogeneous random walk (S a (t)) defined by (3.1) satisfies the conditions (A1') and (A2). The condition (A2) is satisfied because of the assumption (H3), and the communication condition (A1') is satisfied because the random walk (S a (t)) is irreducible (see the proof of Lemma 4.1 in [11] for more details). Moreover, Green's function G a (z, z ′ ) of the twisted random walk (S a (t)) satisfies the equality
Hence, for any sequence of points z n ∈ Z 2 with lim n |z n | = +∞ and lim n z n /|z n | = q ∈ S 2 , using (5.9) we get
and consequently, by Proposition 7.3,
The last relations combined with (7.11) prove (7.10).
We need the following consequence of this proposition.
Corollary 7.1. Let a sequence of points z n ∈ Z 2 be such that lim n |z n | = +∞ and lim n z n /|z n | = q ∈ S 2 . Then for any σ > 0 there are C ′ > 0, C ′′ > 0, δ > 0 and N > 0 such that
for all n ≥ N and z ∈ Z 2 with |z| < δ|z n |.
Proof. Indeed, the equalities (7.10) show that for any σ > 0 there are δ > 0 and N > 0 such that
for any unit vector e ∈ Z 2 and all n ≥ N , u ∈ Z 2 . Remark that for any z ∈ Z 2 there are unit vectors e 1 ∈ {(−1, 0), (1, 0)} and e 2 ∈ {(0, −1), (0, 1)}, non-negative integers n 1 , n 2 ∈ N and real numbers r 1 , r 2 ∈ [0, 1[ such that
If |z| < δ|z n | then letting u 0 =kr 1 e 1 +kr 2 e 2 and
The inequalities (7.13) applied with u = u k for each k = 0, . . . , n 1 + n 2 − 1 prove therefore that (7.14) and similarly,
for all n ≥ N . Remark finally that for any u ∈ Z 2 ,
where P u (S(t) = 0 for some t > 0) > 0 and P 0 (S(t) = u for some t > 0) > 0 because by assumption (H0), our random walk (S(t)) is irreducible. Using this relations together with (7.14) and (7.15) we conclude that (7.12) holds with To apply Proposition 7.3 in this case we need to identify the greatest common divisor of the set of all integers k > 0 for which (7.16) inf
This is a subject of the following lemma.
Lemma 7.1. The greatest common divisor of the set of all integers k > 0 for which (7.16) holds is equal to the periodk of the random walk (S(t)).
Proof. Indeed, if P 0 (S(k) = 0) > 0 for some k ∈ N * then there is a sequence of points u 0 , u 1 , . . . , u k ∈ Z 2 with u 0 = u k = 0 such that
Moreover, without any restriction of generality one can assume that for some l ∈ {1, . . . , k}, the second coordinate of the vectors e 1 = u 1 − u 0 , . . . , e l = u l − u l−1 is positive and the second coordinate of the vectors e l+1 = u l+1 −u l , . . . , e k = u k −u k−1 is negative or zero. Then (0, 1) + u i ∈ Z × N * for all i = 0, . . . , k and
Since according to the definition of the process (Z 1 + (t)), inf
we conclude that (7.16) holds if and only if P 0 (S(k) = 0) > 0 and consequently, the greatest common divisor of the set of all integers k > 0 for which (7.16) holds is equal to the periodk of the random walk (S(t)). Lemma 7.1 is therefore proved. for all w ∈ Z × {0}.
Proof. The proof of this proposition is quite similar to the proof of Proposition 7.4. Proposition 7.3 is applied here for the twisted random walk (Z a,1
with a = a(q), which is identical to (S a (t)) for
and killed upon the time τ a 2 . Lemma 4.1 of [11] proves that such a random walk satisfies the communication condition (A1'). The condition (A2) is satisfied here because by assumption (H3), for any a
The greatest common divisor of the set of all integers k > 0 for which
is clearly the same as for the original process (Z 1 + (t)). By Lemma 7.1, this is the periodk of the random walk (S(t)). Finally, Green's function G a,1
the twisted random walk (Z a,1
Using this relation together with (5.10), we conclude that for any sequence of points z n ∈ N * × N * with lim n |z n | = +∞ and lim n z n /|z n | = q = (1, 0),
and consequently, by Proposition 7.3, for any w ∈ Z × {0},
The last relation combined with (7.18) proves (7.17) .
From Proposition 7.5, using the same arguments as in the proof of Corollary 7.1 we get the following statement.
Corollary 7.2. Let a sequence of points z n ∈ N * × N * be such that lim n |z n | = +∞ and lim n z n /|z n | = q = (1, 0). Then for any σ > 0 there are
for all n ≥ N , z ∈ Z × N * and w ∈ Z × {0} with max{|z|, |w|} < δ|z n |.
To get the estimates (2.8) with a suitable function C 1 + (z) we need the following stronger statement. Proposition 7.6. Let a sequence z n ∈ N * × N * be such that lim n |z n | = +∞ and lim n z n /|z n | = q = (1, 0). Then for any σ > 0 there are C > 0, δ > 0 and N > 0 such that
The proof of this proposition uses Corollary 7.2 and the following results.
Lemma 7.2. Let (ξ(t)) be a discrete time Markov chain on Z and T be an almost surely finite stopping time relative to the natural filtration of (ξ(t)). Denote by T n the first time when the process (ξ(t)) hits the half-line {k ∈ Z : k ≤ −n} :
Proof. Indeed, for any N ∈ N, on the event {T = N }, the minimum
is almost surely finite. Under the hypotheses of our lemma, from this it follows that
and hence,
Remark finally that for any N ∈ N, the sequence P k (T = N, ξ N > −n) is increasing with respect to n and because of (7.19) , by dominated convergence theorem,
as n → −∞. Using monotone convergence theorem we conclude therefore that
as n → ∞ and consequently, also P k (T n > T ) → 1 as n → ∞.
Lemma 7.3. Let (ξ(t)) be an irreducible homogeneous random walk on Z with a non-negative mean and a finite variance. Denote by T 0 the first time when the process (ξ(t)) exits from the set N * :
and let T = inf{t ≥ 0 : ξ(t) = ξ(0) + 1} be the first time when the process (ξ(t)) hits the point ξ(0)+1. Then P n (T < T 0 ) → 1 as n → ∞.
Proof. Indeed, let if (ξ(t)) is an irreducible homogeneous random walk on Z with a non-negative mean and a finite variance then T = inf{t ≥ 0 : ξ(t) = ξ(0) + 1} is an almost surely finite stopping time relative to the natural filtration of (ξ(t)) and for T n = {t ≥ 0 : ξ(t) ≤ −n},
Lemma 7.3 is therefore a consequence of Lemma 7.2.
For the homogeneous random walk (S(t)), the above lemma implies the following statement.
Lemma 7.4. Letτ = inf{t ≥ 0 : S 2 (t) = S 2 (0) + 1}. Then for a = a (1, 0) ,
Proof. Indeed, consider the twisted random walk (S a (t)) on Z 2 having transition probabilities p a (z, z ′ ) = exp(a · (z ′ − z)) with a = a(1, 0). Then the same arguments as in the proof of Proposition 3.1 show that
is a homogeneous random walk on Z with zero mean
and a finite variance because according to the assumption (H3), the jump generating function 
is increasing and tends to 1 as k → ∞. 
Proof. Indeed, for any x ∈ Z, the sequence k → P (0,k) S(τ ) = (x, k + 1),τ < τ 2 is increasing because for any k ∈ N * ,
, and S 2 (t) > 0 for all t ≤τ ) ≥ P (0,k+1) (S(τ ) = (x, k + 2), and S 2 (t) > 1 for all t ≤τ ) = P (0,k) (S(τ ) = (x, k + 1), and S 2 (t) > 0 for all t ≤τ )
By monotone convergence theorem from this it follows that
as k → ∞, σ → 0 and N → ∞. Moreover, using again monotone convergence theorem we get
Since by Lemma 7.4, the right hand side of the last relation is equal to 1 we conclude that
as k → ∞, σ → 0 and N → ∞, and consequently, for any ε > 0 there are N ε > 0, k ε > 0 and σ ε > 0 such that for all N ≥ N ε , k ≥ k ε and 0 < σ ≤ σ ε ,
Lemma 7.5 is therefore proved.
Consider now an increasing sequence of stopping timesτ k defined as follows :
Then from Lemma 7.5 using strong Markov property we obtain the following statement. Lemma 7.6. Let a = a(1, 0). Then for any ε > 0 there are C ε > 0, N ε > 0 and σ ε > 0 such that for all N ≥ N ε , 0 < σ ≤ σ ε and k ≥ 1,
Proof. Indeed, by strong Markov property, the left hand side of the above inequality is greater than
and hence, for any ε > 0 with the same quantities N ε > 0, σ ε > 0 and k ε > 0 as in Lemma 7.5, the inequality (7.20) holds for all N ≥ N ε , 0 < σ ≤ σ ε and k ≥ 1 with
Proof of Proposition 7.6. Let a sequence of points z n ∈ N * × N * be such that lim n |z n | = +∞ and lim n z n /|z n | = q = (1, 0). To simplify the notations, we denote throughout the proof of Proposition 7.6 a(1, 0) = a.
Then by Corollary 7.2, for any σ > 0 there are C ′ > 0, C ′′ > 0, δ σ > 0 and n σ > 0 such that
for all those n ≥ n σ , x ∈ Z and k ∈ N * for which max{|x|, k} < δ σ |z n |.
Furthermore, recall that the process (Z 1 + (t)) is identical to the homogeneous random walk (S(t)) on Z 2 before the first time when the second coordinate S 2 (t) of S(t) becomes zero or negative and is killed upon the time τ 2= inf{t ≥ 0 : S 2 (t) ≤ 0}. Hence, for any n ∈ N and k ∈ N * 
and consequently, for all n ≥ n σ , N > 0 and k ≥ 1 satisfying the inequalities 0 < N (k − 1) < δ σ |z n | and 1 < k < δ σ |z n |, using the first inequality of (7.21) we get and hence, using Lemma 7.6 we conclude that for any ε > 0, there are C ε > 0, N ε > 0 and σ ε > 0 such that
whenever 0 < σ < σ ε , n ≥ n σ , 1 ≤ k < δ σ |z n | and δ σ |z n | > (1 − k)N ε .
Since |z n | → +∞, this proves that for any ε > 0 there areĈ ε > 0,δ ε > 0 and n ε > 0 such that for all n ≥n ε and k ∈ N * . To complete the proof of our proposition we combine now the estimates (7.22) with (7.21) . From now on ε > 0 and σ > 0 are arbitrary and independent from each other. For n ≥ max{n σ ,n ε } and z = (x, k) ∈ Z × N * satisfying the inequalities |x| ≤ δ σ |z n | and k ≤δ ε |z n |, the second inequality of (7.21) together with (7.22) imply that When ε = δ, the last inequality proves Proposition 7.6 with δ = min{δ ε , δ σ } > 0, N = max{n σ ,n ε } > 0 and C = C ′′Ĉ ε /P z0 (Z 1 + (t) = (0, 1) for some t > 0).
Proof of Theorem 1
Let a sequence of point z n ∈ N * × N * be such that lim n |z n | = +∞ and lim n z n /|z n | = q ∈ S and by Proposition 3.2, the series (8.3) with the above exponential function C(w) converge for an arbitrary C > 0 if σ > 0 is small enough. When both coordinates of lim n z n /|z n | = q are non-zero, the equality (8.1) is therefore proved.
Suppose now that lim n z n /|z n | = q = (1, 0). Here, to get (8.1) it is sufficient to show that for some δ > 0 and C 0 > 0, The proof of this equality uses the same arguments as above but with the help of Proposition 3.3, Proposition 7.6 and the results of the paper [11] . Theorem 1 of [11] proves the point-wise convergence Moreover, remark that the right hand side of (8.6) is zero when z ∈ Z × (−N). Hence, the function h 1 a(q),+ can be extended to Z 2 by letting h 1 a(q),+ (z) = 0 for z ∈ Z × (−N) and consequently, h 1 a(q),+ (S(τ )) = 0 whenever τ = τ 2 < ∞. Since clearly, {τ = τ 1 < τ 2 < ∞} ∪ {τ = τ 2 < ∞} = {τ < ∞}, the right hand side of the above relation is therefore equal to Finally, for any z = (x 1 , x 2 ) ∈ N * × N * , using strong Markov property we obtain h 1 a(q),+ (z) − E z h 1 a(q),+ (S(τ )), τ < ∞ = x 2 exp(a(q) · z) − E z S 2 (τ 2 ) exp(a(q) · S(τ 2 )), τ 2 < ∞ − E z S 2 (τ ) exp(a(q) · S(τ )), τ < ∞ + E z E S(τ ) S 2 (τ 2 ) exp(a(q) · S(τ 2 )), τ 2 < ∞ , τ < ∞ = x 2 exp(a(q) · z) − E z S 2 (τ ) exp(a(q) · S(τ )), τ < ∞ = h a(q) (z) and hence, the left hand side of (8.5) is equal to h a(q) (z)/h 1 a(q),+ (z 0 ). This proves that (8.5) holds with C 0 = 1/h 1 a(q),+ (z 0 ) > 0 and consequently, for q = (1, 0), the equality (8.1) is also proved.
The proof of (8.1) for q = (0, 1) uses exactly the same arguments, it is sufficient to exchange the roles of the first and the second coordinates.

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