Ekaterina Vl Bulinskaya

THE HITTING TIMES WITH TABOO FOR A RANDOM WALK ON AN INTEGER LATTICE

Keywords: random walks on integer lattices, hitting times, taboo probabilities, branching random walk. 2010 AMS classification: 60G50, 60J27, 60G17

For a symmetric, homogeneous and irreducible random walk on Z d , d ∈ N, having zero mean and a finite variance of jumps, we study the passage times (taking values in [0, ∞]) determined by the starting point x, the hitting state y and the taboo state z. We find the probability that these passages times are finite and analyze the tails of their cumulative distribution functions. In particular, it turns out that for the random walk on Z d , except for a simple (nearest neighbor) random walk on Z, the order of the tail decrease is specified by dimension d only. In contrast, for a simple random walk on Z, the asymptotic properties of hitting times with taboo essentially depend on the mutual location of the points x, y and z. These problems originated in our recent study of branching random walk on Z d with a single source of branching.

Introduction

A random walk is a classical model being the source of numerous interesting problems having elegant solutions. The monographs [START_REF] Lawler | Intersections of Random walks[END_REF], [START_REF] Madras | The Self-Avoiding Walk[END_REF] and [START_REF] Spitzer | Principles of Random Walk[END_REF] devoted to various properties of random walks have become the reference books for many researchers. The recent publications such as [START_REF] Chen | Random Walk Intersections: Large Deviations and Related Topics[END_REF] and [START_REF] Lawler | Random Walk: A Modern Introduction[END_REF] demonstrate its non-vanishing popularity. The reason is the abundance of applications of this simple probabilistic model (see, e.g., [START_REF] Berg | Random Walks in Biology[END_REF] and [START_REF] Weiss | Aspects and Applications of the Random Walk[END_REF]). Moreover, a lot of new complicated models were constructed on the basis of random walk (one can mention those investigated in [START_REF] Caravenna | Random Walk Models and Probabilistic Techniques for Inhomogeneous Polymer Chains[END_REF] and [START_REF] Révész | Random Walk in Random and Non-Random Environments[END_REF]). Thus, many assertions of the theory of random walks have been obtained as auxiliary (technical) lemmas needed for understanding other models of interest.

The present paper also provides results concerning random walk on integer lattice Z d (d ∈ N) which arose during the study of another model, namely, branching random walk (BRW) on Z d with a single source of branching (different modifications of the model were considered, e.g., in papers [START_REF] Albeverio | Asymptotics of branching symmetric random walk on the lattice with a single source[END_REF], [START_REF] Bulinskaya | Limit Distributions of Particle Numbers in Branching Random Walk[END_REF], [START_REF] Topchii | Catalytic Branching Random Walk in Z d with branching at the origin only[END_REF] and [START_REF] Yarovaya | Criteria of the exponential growth of particles numbers in models of branching random walks[END_REF]). Its main features are the following. If the starting point of the process is x ∈ Z d , x = 0, then the parent particle performs random walk on Z d until hitting the origin. If the starting point is 0, or the parent particle has just hit it, then after exponentially distributed time the particle may die producing a random number of offsprings. Otherwise, it leaves the origin (jumps randomly to a point x ′ ∈ Z d \{0}) and behaves afterwards as a memoryless particle starting at x ′ . At the birth moment the newborn particles are located at the origin. They and all their descendants evolve according to the scheme described above independently of each other as well as of their parents history.

In the framework of BRW a natural question arises: what is the limit distribution, as t → ∞, of the number of particles µ(t; y) at each point y ∈ Z d at time t? Evidently, µ(t; y) essentially depends on the trajectory of the parent particle. Indeed, if the parent particle hits the origin earlier than the time t then µ(t; y) can take any value in Z + . Otherwise, µ(t; y) = 1 or µ(t; y) = 0 if the parent particle is located at time t at point y or outside it, respectively. This is the motivation to introduce the notion of hitting time with taboo for a random walk on Z d . More precisely, for any x, y, z ∈ Z d such that y = z, let function H x,y,z (t), t ≥ 0, be (improper) cumulative distribution function (c.d.f.) of the first hitting time (or the first return time if x = y) of point y if the starting point of the random walk is x and the point z is a taboo state. In the next section we give a formal definition of this function. In the present work we find the limit value H x,y,z (∞) = lim t→∞ H x,y,z (t) and analyze the asymptotic behavior of H x,y,z (∞) -H x,y,z (t), as t → ∞, for a symmetric, homogeneous, irreducible random walk on Z d (d ∈ N) having zero mean and a finite variance of jumps.

It turns out that the most interesting case is d = 1 since there are two quite different kinds of asymptotic behavior of H x,y,z (t), t → ∞, depending on whether the random walk is simple (nearest neighbor) or not. It is also worth mentioning that, for the random walk on Z d , except for a simple random walk on Z, the value H x,y,z (∞) ∈ (0, 1) and the order of decrease of function H x,y,z (∞) -H x,y,z (t) is determined by dimension d only, regardless of x, y and z. In contrast, for a simple random walk on Z, the mutual location of points x, y and z determines the value H x,y,z (∞) ∈ [0, 1] as well as the order of decrease of H x,y,z (∞) -H x,y,z (t) as t → ∞.

Finally, we recall that the properties of hitting times (or, more generally, passage times including first entrance and last exit times) and taboo probabilities for a Markov chain with stationary transition probabilities were exposed in [START_REF] Chung | Markov Chains with Stationary Transition Probabilities[END_REF] and [START_REF] Syski | Passage Times for Markov Chains[END_REF], respectively. For the taboo probabilities see also [START_REF] Zubkov | Inequalities for Transition Probabilities with Taboos and Their Applications[END_REF] and references therein. The counterpart of the function H x,y,z (t) for a Markov chain with stationary transition probabilities has been used earlier (see, e.g., [START_REF] Bulinski | Theory of Stochastic Processes[END_REF], p. 202, and [START_REF] Karlin | A Second Course in Stochastic Processes[END_REF], p. 31). However, it was treated as an auxiliary tool and has not been studied per se.

Main results

We assume that all random variables (taking values in [-∞, ∞]) are defined on a probability space (Ω, F , P). We study a class of random walks on Z d more general than that proposed in [START_REF] Lawler | Random Walk: A Modern Introduction[END_REF], Ch.1, Sec.2.

Definition 1 For d ∈ N, let A = (a(x, y)) x,y∈Z d be a matrix such that elements a(x, y) ≥ 0 whenever x = y, a(x, x) < 0, y∈Z d a(x, y) = 0 and sup x∈Z d |a(x, x)| < ∞. Then a continuous time Markov chain S = {S(t), t ≥ 0} having the state space Z d , i.e. S(t) : Ω → Z d for each t ≥ 0, and generator A is called a random walk on Z d .

The existence of such a Markov chain is guarantied by Theorem 2 in [START_REF] Gikhman | The Theory of Stochastic Processes II[END_REF], Ch.3, Sec.2.

From here on we consider only a symmetric, homogeneous, irreducible random walk S having a finite variance of jumps. Symmetry and homogeneity mean that for x, y ∈ Z d one has a(x, y) = a(y, x) and a(x, y) = a(0, yx) =: a(yx), respectively. Random walk is called irreducible if for any x, y ∈ Z d there exists t > 0 such that P(S(t) = y|S(0) = x) > 0, i.e. all the points of the lattice Z d can be reached. Furthermore, a random walk is said to have a finite variance of jumps if x∈Z d x 2 a(0, x) < ∞ where 0 is the origin of Z d and • is a norm in this space.

The conditions imposed on elements a(x, y) allow us to use an explicit construction of the random walk on Z d with generator A (see, e.g., Theorem 1.2 in [START_REF] Brémaud | Markov Chains: Gibbs fields, Monte-Carlo simulation, and Queues[END_REF], Ch.9, Sec.1). According to this construction S is a regular jump process with right continuous trajectories. A jump of the process from state x ∈ Z d to state y ∈ Z d , x = y, occurs with probability a(x, y)/a(x, x) independently of the earlier evolution of S. Moreover, for transition times of the process τ (0) := 0 and τ (n) := inf{t ≥ τ (n-1) : S(t) = S(τ (n-1) )}, n ≥ 1, the following statement holds. Random variables {τ (n+1)τ (n) } ∞ n=0 are independent and each of them has exponential distribution with parameter a := -a(0, 0). In what follows we consider the version of the process S constructed in this way.

Note that for the sequence {τ (n) } ∞ n=1 of transition times of S one can consider embedded chain {S n , n ∈ Z + } defined by way of S n := S(τ (n) ). Mention in passing that {S n , n ∈ Z + } is a discrete-time homogeneous Markov chain having transition probabilities a(x, y)/a for x = y, x, y ∈ Z d (see, e.g., [START_REF] Brémaud | Markov Chains: Gibbs fields, Monte-Carlo simulation, and Queues[END_REF], Ch.8, Sec.4).

Recall some other facts concerning the introduced random walk on Z d . Let p(t; x, y), t ≥ 0, x, y ∈ Z d , be the transition probabilities of the random walk, that is,

p(t; x, y) := P(S(t + u) = y|S(u) = x) = P(S(t) = y|S(0) = x) for any u ≥ 0. Then for h → 0+ p(h; x, y) = a(x, y)h + o(h), if x = y, p(h; x, x) = 1 + a(x, x)h + o(h).
Due to [START_REF] Brémaud | Markov Chains: Gibbs fields, Monte-Carlo simulation, and Queues[END_REF], Ch.8, Sec.3, the transition probabilities of the process S satisfy the backward Kolmogorov equations dp(t; x, y) dt = z∈Z d a(x, z)p(t; z, y), p(0; x, y) = δ y (x).

As usual, δ y (x) is equal to 1 or 0 for x = y or x = y, respectively. The Fourier transform applied to (1) yields (see [START_REF] Topchii | Catalytic Branching Random Walk in Z d with branching at the origin only[END_REF] and [START_REF] Yarovaya | Branching Random Walks in Inhomogeneous Medium[END_REF], Ch.2, Sec.1)

p(t; x, y) = 1 (2π) d [-π,π] d e φ(θ)t+i(θ,y-x) dθ, x, y ∈ Z d , (2)
where (•, •) stands for the scalar product in Euclidean space R d and

φ(θ) := z∈Z d a(0, z) cos(z, θ), θ ∈ [-π, π] d .
Furthermore, by the method of steepest descent one derives from (2) the asymptotic relation

p(t; x, y) ∼ γ d t d/2 , x, y ∈ Z d , t → ∞. (3)
Here

γ d := 1 (2π) d/2 | det φ ′′ θθ (0)| and φ ′′ θθ (0) := ∂ 2 φ(θ) ∂θ i ∂θ j θ=0 i,j∈{1,...,d} .
A similar result for transition probabilities of a discrete time random walk on Z d was established in [START_REF] Spitzer | Principles of Random Walk[END_REF], Ch.2, Sec.7. Moreover, according to [START_REF] Yarovaya | Branching Random Walks in Inhomogeneous Medium[END_REF], Ch.2, Sec.1, equality (2) implies that

p(t; 0, 0) -p(t; x, y) ∼ γd (y -x) t d/2+1 , x, y ∈ Z d , t → ∞, (4)
where γd (z

) := 1 2(2π) d R d (υ, z) 2 e 1 2 (φ ′′ θθ (0)υ,υ) dυ, z ∈ Z d . Set G λ (x, y) := ∞ 0 e -λt p(t; x, y) dt, λ ≥ 0, x, y ∈ Z d , that is, G λ (x, y)
is the Laplace transform of the transition probability p(•; x, y). In view of (2) the introduced function can be represented in the form

G λ (x, y) = 1 (2π) d [-π,π] d e i(θ,y-x) λ -φ(θ) dθ. (5)
Taking into account (3) it is easily seen that for d ≥ 3 the Green's function G 0 (x, y) is finite for all x, y ∈ Z d . However, for d = 1 or d = 2, one has lim λ→0+ G λ (x, y) = ∞. This phenomenon corresponds to transience of the random walk for d ≥ 3 and its recurrence for d = 1 or d = 2 (for more details see, e.g., [START_REF] Lawler | Random Walk: A Modern Introduction[END_REF], Ch.4, Sec.1 and 2).

By virtue of (4) the function lim λ→0+ (G λ (0, 0) -G λ (x, y)) is finite for all d ∈ N and x, y ∈ Z d . Therefore we can define

ρ d (x) := a lim λ→0+ (G λ (0, 0) -G λ (0, x)), if x = 0, 1, if x = 0.
Equality (5) allows us to give another formula for

ρ d (x) if x = 0 ρ d (x) = a (2π) d [-π,π] d cos(x, θ) -1 φ(θ) dθ. (6)
Now we are able to introduce some basic notation and formulate our main results. Set τ := inf{t ≥ 0 : S(t) = S(0)}, i.e. the stopping time τ (with respect to the natural filtration of the process S) is the time of the first exit from the starting point of the random walk. Taking into account the explicit construction of the random walk one has τ = τ (1) almost surely (a.s.) and, consequently, G(t)

:= P(τ ≤ t|S(0) = x) = 1 -e -at , t ≥ 0. Let τ y,z := inf{t ≥ τ : S(t) = y, S(u) = z, τ ≤ u ≤ t} for y, z ∈ Z d , y = z. As usual, inf{t ∈ ∅} = ∞.
We call the stopping time τ y,z a hitting time of the state y with the taboo state z. Denote its (improper) c.d.f. given the starting point x by H x,y,z (t) := P(τ y,z ≤ t|S(0) = x), x, y, z ∈ Z d , y = z, t ≥ 0.

Note that the random variable τ - y,z := inf{t ≥ 0 : S(t + τ ) = y, S(u) = z, τ ≤ u ≤ t + τ }, y, z ∈ Z d , y = z, is not a stopping time w.r.t. the natural filtration of the process S. However, τ y,z can be also called a hitting time of the state y with the taboo state z, since in this case we count time not from the moment of the process start but from the moment of the first exit out of the starting point. In a similar way, we set H - x,y,z (t) := P(τ - y,z ≤ t|S(0) = x), x, y, z ∈ Z d , y = z, t ≥ 0.

Obviously, τ y,z = τ - y,z + τ a.s. Moreover, in view of strong Markov property of the random walk w.r.t. the stopping time τ (see, e.g., [START_REF] Brémaud | Markov Chains: Gibbs fields, Monte-Carlo simulation, and Queues[END_REF], Ch.8, Sec.4) the random variables τ - y,z and τ are independent. Hence, τ y,z has an absolutely continuous (improper) distribution as a sum of two independent random variables, one of them having a density. On the other hand, the explicit construction of S shows that the distribution of τ - y,z has an atom at zero when x = y, namely, H -

x,y,z (0) = a(x, y)/a. For x, y ∈ Z d such that y = 0, we specify the functions C d (x, y) by formulae

C 1 (x, y) := ρ 1 (y -x) + ρ 1 (x) -ρ 1 (y) 4aπγ 1 + aπy 2 γ 3 1 (ρ 1 (y -x) -ρ 1 (x) -ρ 1 (y)) ρ 2 1 (y) + 2aπxyγ 3 1 ρ 1 (y) , C 2 (x, y) := ρ 2 (y -x) + ρ 2 (x) -ρ 2 (y) 4aγ 2 , C d (x, y) := 2γ d (ρ d (y -x) + ρ d (x) -ρ d (y)) a(d -2)(G 0 (0, 0) + G 0 (0, y)) 2 , d ≥ 3.
The following theorems are the main results of the paper.

Theorem 1 Let x, y, z ∈ Z d be such that y = z. Then for the random walk S on Z d , except for a simple random walk on Z, one has

H x,y,z (∞) = ρ d (x -z) + ρ d (y -z) -ρ d (y -x) 2ρ d (y -z) ∈ (0, 1), d = 1 or d = 2, (7)
H x,y,z (∞) = G 0 (0, 0)ρ d (y -z) -G 0 (0, 0)ρ d (y -x) + G 0 (y, z)ρ d (x -z) ρ d (y -z)(G 0 (0, 0) + G 0 (y, z)) ∈ (0, 1), d ≥ 3, (8)
Moreover, as t → ∞,

H x,y,z (∞) -H x,y,z (t) ∼ C 1 (x -z, y -z) √ t for d = 1, (9)
H x,y,z (∞) -H x,y,z (t) ∼ C 2 (x -z, y -z) ln t for d = 2, (10)
H x,y,z (∞) -H x,y,z (t) ∼ C d (x -z, y -z) t d/2-1 for d ≥ 3 ( 11
)
where C d (•, •), d ∈ N, are positive functions defined above.

Theorem 2 Let S be a simple random walk on Z. If x, y, z ∈ Z and y = z then

1 -H x,y,z (t) ∼ √ 2 |y -x| √ a π √ t , x < y < z or z < y < x, (12)
1 -

1 2|y -z| -H y,y,z (t) ∼ 1 √ 2 a π √ t , x = y, (13)
xz yz -H x,y,z (t) = o(e -a ε t ), z < x < y or y < x < z,

1 2|y -z| -H z,y,z (t) = o(e -a ε t ), x = z, (14)
H x,y,z (t) ≡ 0, x < z < y or y < z < x, (15)
for some ε ∈ (0, 1).

Theorem 3 Theorems 1 and 2 hold true if function H x,y,z at the left-hand sides of (7)-( 16) is replaced by H - x,y,z , whereas the right-hand sides of these formulae remain intact. The proofs of Theorems 1 and 2 are given in Sections 4 and 5, respectively. They bear on application of the Laplace-Stieltjes and the Laplace transforms to the main equations involving H x,y,z and H x,z,y as well as functions H x,y , H z,y , H x,z and H y,z . Here H x,y (t), x, y ∈ Z d , t ≥ 0, is the (improper) c.d.f. of the hitting time of point y given the starting point x. After that we employ the Tauberian theorems techniques and, for d sufficiently large, we essentially use Faà di Bruno's formula for the n-th derivative of two functions superposition, n ≥ 1. An essential difficulty arising on this way is to prove the positivity of functions C d (x, y), x, y ∈ Z d , y = 0, under the conditions of Theorem 1. Theorem 3 proved in Section 6 can be viewed as a corollary of Theorems 1 and 2.

Auxiliary results

For a nonnegative function κ(t) and a nonnegative nondecreasing function χ(t), t ≥ 0, we set

κ(λ) := ∞ 0 e -λt κ(t) dt, χ(λ) := ∞ 0 e -λt dχ(t), λ > 0,
whenever the integrals exist. Recall a useful relation linking the Laplace-Stieltjes transform of the bounded function χ(t) and the Laplace transform of χ(∞)-χ(t) where χ(∞) := lim t→∞ χ(t), namely,

(χ(∞) -χ)(λ) = χ(∞) -χ(λ) λ , λ > 0. ( 17
)
This relation results from the formula of integration by parts. Introduce a stopping time τ y := inf{t ≥ τ : S(t) = y} called the hitting time of point y ∈ Z d . Denote its (conditional) c.d.f. given the starting point x by H x,y (t) := P(τ y ≤ t|S(0) = x), x, y ∈ Z d , t ≥ 0. The limit value H x,y (∞) := lim t→∞ H x,y (t) and the asymptotic behavior of function H x,y (∞) -H x,y (t), as t → ∞, were found in papers [START_REF] Bulinskaya | Catalytic Branching Random Walk on a Two-Dimensional Lattice[END_REF]- [START_REF] Bulinskaya | Limit Distributions Arising in Branching Random Walks on Integer Lattices[END_REF], [START_REF] Topchii | Catalytic Branching Random Walk in Z d with branching at the origin only[END_REF] and [START_REF] Vatutin | Catalytic branching random walk and queueing systems with random number of independent servers[END_REF] for y = 0 and x ∈ Z d . In Lemma 1 we reformulate these results in a more general way.

Lemma 1 Let x, y ∈ Z d . Then H x,y (∞) = 1 for d = 1 and d = 2, (18)
H x,x (∞) = 1 - 1 a G 0 (0, 0) for d ≥ 3 and y = x, (19)
H x,y (∞) = G 0 (x, y) G 0 (0, 0) for d ≥ 3 and y = x. ( 20
)
Moreover, as t → ∞, one has

1 -H x,y (t) ∼ ρ 1 (y -x) a γ 1 π √ t for d = 1, (21)
1 -H x,y (t) ∼ ρ 2 (y -x) a γ 2 ln t for d = 2, (22)
H x,y (∞) -H x,y (t) ∼ 2γ d ρ d (y -x) a (d -2) G 2 0 (0, 0) t d/2-1 for d ≥ 3. ( 23
)
Proof. In view of definitions of τ and τ x as well as of conditional probability and a Markov chain we see that

p(t; x, x) = P(S(t) = x, τ > t|S(0) = x) + P(S(t) = x, τ ≤ t|S(0) = x) = P(τ > t|S(0) = x) + P(S(t) = x, τ x ≤ t|S(0) = x) = 1 -G(t) + t 0 P(S(t) = x|τ x = u, S(0) = x) dP(τ x ≤ u|S(0) = x) = 1 -G(t) + t 0 P(S(t) = x|S(u) = x) dH x,x (u).
Thus we deduce the following formula

p(t; x, x) = 1 -G(t) + t 0 p(t -u; x, x) d H x,x (u), x ∈ Z d , t ≥ 0. ( 24
)
Note that ( 5) implies the identity G λ (x, x) = G λ (0, 0) for any x ∈ Z d . Therefore, application of the Laplace-Stieltjes transform to (24) leads to

Ȟx,x (λ) = 1 - (1 -G)(λ) G λ (x, x) = 1 - (1 -G)(λ) G λ (0, 0) = Ȟ0,0 (λ). ( 25
)
In particular, by the Laplace-Stieltjes transform uniqueness (see, e.g., [START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF], Ch.13, Sec.1) it follows that

H x,x (t) = H 0,0 (t), t ≥ 0, x ∈ Z d . (26)
For different d, asymptotic properties of H 0,0 (t), t → ∞, were established in [START_REF] Bulinskaya | Catalytic Branching Random Walk on a Two-Dimensional Lattice[END_REF], [START_REF] Topchii | Catalytic Branching Random Walk in Z d with branching at the origin only[END_REF] and [START_REF] Vatutin | Catalytic branching random walk and queueing systems with random number of independent servers[END_REF]. More exactly, the limit value H - 0,0 (∞) := lim t→∞ H - 0,0 (t) and the asymptotic behavior of H - 0,0 (∞) -H - 0,0 (t), t → ∞, were found for the function H - 0,0 (t), t ≥ 0, related to H 0,0 by H 0,0 (t) = G * H - 0,0 (t) (as usual, * denotes the convolution). However, these results are easily extended to the case of function H 0,0 (t). Hence, due to (26) relations ( 18), ( 19) and ( 21)-( 23) are proved for x = y.

Let x = y. Combined definitions of τ y , conditional probability and a Markov chain imply

p(t; x, y) = P(S(t) = y, τ y ≤ t|S(0) = x) = t 0 P(S(t) = y|τ y = u, S(0) = x) dP(τ y ≤ u|S(0) = x) = t 0 P(S(t) = y|S(u) = y) dH x,y (u).
Consequently, for x = y and t ≥ 0 we get the formula

p(t; x, y) = t 0 p(t -u; y, y) d H x,y (u). ( 27
)
By virtue of (5) identities

G λ (x, y) = G λ (x -y, 0) = G λ (y -x, 0) and G λ (y, y) = G λ (0, 0) are valid.
Then application of the Laplace-Stieltjes transform to [START_REF] Vatutin | Catalytic branching random walk and queueing systems with random number of independent servers[END_REF] yields

Ȟx,y (λ) = G λ (x, y) G λ (y, y) = G λ (x -y, 0) G λ (0, 0) = G λ (y -x, 0) G λ (0, 0) = Ȟx-y,0 (λ) = Ȟy-x,0 (λ). ( 28
)
Now using the Laplace-Stieltjes transform uniqueness we obtain

H x,y (t) = H x-y,0 (t) = H y-x,0 (t), t ≥ 0. ( 29
)
The limit value H z,0 (∞), z ∈ Z d \{0}, and the asymptotic behavior of H z,0 (∞) -H z,0 (t), t → ∞, were found in [START_REF] Bulinskaya | Catalytic Branching Random Walk on Three-Dimensional Lattice[END_REF] and [START_REF] Bulinskaya | Limit Distributions Arising in Branching Random Walks on Integer Lattices[END_REF]. It follows that by ( 29) relations ( 18), ( 20) and ( 21)-( 23) are established for x = y. The proof of Lemma 1 is completed.

The next lemma provides a linear integral equation involving functions H x,y,z (t) and H x,z,y (t) which is the basis of our theorems proofs.

Lemma 2 Let x, y, z ∈ Z d and y = z. Then the following equation holds

H x,y (t) = H x,y,z (t) + t 0 H z,y (t -u) dH x,z,y (u), t ≥ 0. ( 30
)
Proof. Firstly note that the random variables τ y and τ y,z coincide a.s. on event {τ y,z < ∞} in view of their definition, that is τ y I(τ y,z < ∞) = τ y,z I(τ y,z < ∞) a.s. (here I(B) stands for the indicator of a set B). Consequently, {τ y ≤ t, τ y,z ≤ t} = {τ y,z ≤ t} for each t ≥ 0. Moreover, one can see that τ y I(τ z,y < ∞) ≥ τ z,y I(τ z,y < ∞) a.s. Therefore, for every nonnegative t, one has {τ y ≤ t, τ y,z > t} = {τ y ≤ t, τ y,z = ∞} = {τ y ≤ t, τ z,y < ∞} = {τ y ≤ t, τ z,y ≤ t}. These relations allow us to write

H x,y (t) = P(τ y ≤ t, τ y,z ≤ t|S(0) = x) + P(τ y ≤ t, τ y,z > t|S(0) = x) = P(τ y,z ≤ t|S(0) = x) + P(τ y ≤ t, τ z,y ≤ t|S(0) = x) = H x,y,z (t) + t 0 P(τ y ≤ t|τ z,y = u, S(0) = x) dH x,z,y (u).
Using definitions of τ y and τ z,y as well as that of a Markov chain it can be verified that P(τ y ≤ t|τ z,y = u, S(0) = x) = P(τ y ≤ t -u|S(0) = z). The previous reasoning supplemented with this equality entails the desired statement. Lemma 2 is proved.

The obtained equation has a natural interpretation. Namely, the term at the left-hand side of (30) accounts for the random walk trajectories starting at point x and hitting point y until time t. Every path of such kind belongs to one of two types. The trajectories of the first type do not pass point z before hitting y. They are taken into account by the first summand at the right-hand side of [START_REF] Yarovaya | Criteria of the exponential growth of particles numbers in models of branching random walks[END_REF]. As to a trajectory of the second type, it hits point z at time u, 0 ≤ u ≤ t, before reaching y so that the part of the trajectory after hitting z is a path starting at z and reaching y until time tu. The second summand at the right-hand side of ( 30) is responsible for the trajectories of the second type.

The next statement is a Lemma 2 corollary which provides the Laplace-Stieltjes transform of a solution of equation [START_REF] Yarovaya | Criteria of the exponential growth of particles numbers in models of branching random walks[END_REF].

Corollary 1 If x, y, z ∈ Z d and y = z then for any λ > 0 Ȟx,y,z (λ) = Ȟx,y (λ) -Ȟx,z (λ) Ȟz,y (λ) 1 -Ȟz,y (λ) Ȟy,z (λ) . (31)
Proof. Due to Lemma 2 we have a system of two linear integral equations in the functions H x,y,z (t) and H x,z,y (t)

H x,y (t) = H x,y,z (t) + H x,z,y * H z,y (t), H x,z (t) = H x,z,y (t) + H x,y,z * H y,z (t).
Application of the Laplace-Stieltjes transform to each equation of the system leads to a new system of algebraic equations in Ȟx,y,z (λ) and Ȟx,z,y (λ)

Ȟx,y (λ) = Ȟx,y,z (λ) + Ȟx,z,y (λ) Ȟz,y (λ), Ȟx,z (λ) = Ȟx,z,y (λ) + Ȟx,y,z (λ) Ȟy,z (λ).

Solving this system we obtain [START_REF] Zubkov | Inequalities for Transition Probabilities with Taboos and Their Applications[END_REF]. The proof is complete.

The next proposition clarifies the invariance of functions H x,y,z (•) indexed by x, y, z ∈ Z d , z = y, under shifts and reflections of space Z d . Corollary 2 Let r, x, y, z ∈ Z d and y = z. Then

H x+r,y+r,z+r (t) = H x,y,z (t), H -x,-y,-z (t) = H x,y,z (t), t ≥ 0. ( 32
)
Proof. The desired assertion is a direct consequence of formulae ( 26), ( 29) and [START_REF] Zubkov | Inequalities for Transition Probabilities with Taboos and Their Applications[END_REF].

For proving Theorem 1 we also need the following lemma. Its derivation is based on considering of all the possible jumps of the random walk after latency period τ .

Lemma 3 For x, y, z ∈ Z d such that y = z, one has H - x,y,z (∞) -H - x,y,z (t) = r∈Z d ,r =x,r =y,r =z a(x, r) a (H r,y,z (∞) -H r,y,z (t)), t ≥ 0. (33)
Proof. In view of definitions of random variables τ and τ - y,z one has

H - x,y,z (∞) -H - x,y,z (t) = P(t < τ - y,z < ∞|S(0) = x) = r∈Z d ,r / ∈{x,y,z} P(t < τ - y,z < ∞, S(τ ) = r|S(0) = x) = r∈Z d ,r / ∈{x,y,z} P(S(τ ) = r|S(0) = x)P(t < τ - y,z < ∞|S(τ ) = r, S(0) = x).
Here P(t < τ - y,z < ∞|S(τ ) = r, S(0) = x) = P(t < τ y,z < ∞|S(0) = r) by the identity τ y,z = τ + τ - y,z a.s. and due to strong Markov property of the random walk w.r.t. the stopping time τ (see, e.g., [START_REF] Brémaud | Markov Chains: Gibbs fields, Monte-Carlo simulation, and Queues[END_REF], Ch.8, Sec.4). Moreover, according to the explicit construction of S the relation P(S(τ ) = r|S(0) = x) = a(x, r)/a is also true. Combining the obtained equalities with the previous argument one comes to (33). Lemma 3 is proved.

The next result which is known as Faà di Bruno's formula plays an important role while establishing Theorem 1 for d ≥ 3.

Theorem 4 (see [START_REF] Tacács | Combinatorial Methods in the Theory of Stochastic Processes[END_REF], Appendix, Subsec.9) Let functions W (u) and V (u) have n-th derivatives. Then for the n-th derivative of function U(u) = W (V (u)) the following formula is valid

U (n) (u) = n 1 ,n 2 ,n 3 ,...∈Z + : n 1 +2n 2 +3n 3 +...=n W (n 1 +n 2 +n 3 +...) (V ) n! n 1 ! n 2 ! n 3 ! . . . V ′ 1! n 1 V ′′ 2! n 2 V ′′′ 3! n 3 . . .. (34)
The last auxiliary result will be useful for proving Theorem 2.

Lemma 4 For each x ∈ N the following equality holds true

π -π 1 -cos xθ 1 -cos θ dθ = 2πx. (35) Proof. For each n ∈ N one has π -π cos nθ -cos (n + 1)θ 1 -cos θ dθ - π -π cos (n -1)θ -cos nθ 1 -cos θ dθ = π -π 2 cos nθ -cos (n + 1)θ -cos (n -1)θ 1 -cos θ dθ = π -π 2 cos nθ(1 -cos θ) 1 -cos θ dθ = π -π 2 cos nθ dθ = 0.
Whence for all n ∈ N the values of the following integrals are the same and equal, for instance, the value of the integral for n = 1

π -π cos (n -1)θ -cos nθ 1 -cos θ dθ = π -π 1 -cos θ 1 -cos θ dθ = 2π.
Hence, for each x ∈ N we obtain

π -π 1 -cos xθ 1 -cos θ = x n=1 π -π cos (n -1)θ -cos nθ 1 -cos θ dθ = x n=1 2π = 2πx.
Thus, equality (35) is established and the proof of Lemma 4 is complete.

Proof of Theorem 1

Firstly, in view of Corollary 2 it suffices to prove Theorem 1 for z = 0. Secondly, we have to consider separately the cases d = 1, d = 2 and d ≥ 3, since the demonstration of the theorem essentially depends on dimension d. Before passing to the non-simple random walk on Z let us write down some formulae which have the same form for all d ∈ N. Let x, y ∈ Z d \{0} and x = y. Using ( 17), ( 25), ( 28) and [START_REF] Zubkov | Inequalities for Transition Probabilities with Taboos and Their Applications[END_REF] as well as recalling that

∞ 0 e -λt (1 -G(t)) dt = (λ + a) -1 , λ ≥ 0, we deduce the following relations (H x,y,0 (∞) -H x,y,0 )(λ) = H x,y,0 (∞) λ - G λ (x, y)G λ (0, 0) -G λ (0, x)G λ (0, y) (G 2 λ (0, 0) -G 2 λ (0, y))λ , ( 36
) (H 0,y,0 (∞) -H 0,y,0 )(λ) = H 0,y,0 (∞) λ - G λ (0, y) (G 2 λ (0, 0) -G 2 λ (0, y))(λ + a)λ , ( 37
) (H y,y,0 (∞) -H y,y,0 )(λ) = H y,y,0 (∞) λ - 1 λ + G λ (0, 0) (G 2 λ (0, 0) -G 2 λ (0, y))(λ + a)λ . ( 38
)
4.1 The case d = 1

In this subsection we focus on a non-simple random walk on Z although some arguments are valid for a simple random walk on Z d as well. At the beginning we find the limit value H x,y,0 (∞) for x, y ∈ Z, y = 0, due to the formula H x,y,0 (∞) = lim λ→0+ Ȟx,y,0 (λ). To this end we write the following asymptotic representation of the function Ȟ0,r (λ), r ∈ Z,

Ȟ0,r (λ) = 1 - ρ 1 (r) √ λ a γ 1 √ π + o( √ λ), λ → 0+, (39)
which results, in view of [START_REF] Lawler | Intersections of Random walks[END_REF], from application of Tauberian theorem (Theorem 4 in [START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF], Ch.13, Sec.5) to relation [START_REF] Shiriaev | [END_REF]. Then substituting (39) in [START_REF] Zubkov | Inequalities for Transition Probabilities with Taboos and Their Applications[END_REF] and taking into account [START_REF] Yarovaya | Branching Random Walks in Inhomogeneous Medium[END_REF] we calculate lim λ→0+ Ȟx,y,0 (λ) and, consequently, we get formula for H x,y,0 (∞) which is true for a simple random walk on Z as well. However, to complete the derivation of ( 7) for d = 1 we have to show that H x,y,0 (∞) ∈ (0, 1). Note that by virtue of ( 18) equality ( 30) implies the equivalences

H x,y,0 (∞) = 0 ⇔ H x,0,y (∞) = 1 and H x,y,0 (∞) = 1 ⇔ H x,0,y (∞) = 0.
Hence, to verify that H x,y,0 (∞) ∈ (0, 1) for all x, y ∈ Z, y = 0, it suffices to check that H x,y,0 (∞) > 0 for all x, y ∈ Z, y = 0. In its turn the latter holds true if, for instance, relation ( 9) is satisfied with C 1 (x, y) > 0 for all x, y ∈ Z, y = 0. Thus, the initial problem of validating that H x,y,0 (∞) ∈ (0, 1) is reduced to proving [START_REF] Bulinskaya | Limit Distributions of Particle Numbers in Branching Random Walk[END_REF] and showing positivity of the function C 1 (•, •). The rest part of the subsection is devoted to establishing the last two claims. The demonstration of ( 9) is based on formulae (36), ( 37) and ( 38) in which we substitute asymptotic decompositions of functions G λ (0, 0) and G λ (0, 0) -G λ (0, r), as λ → 0+, for r = x, r = y and r = yx. Due to Tauberian theorem (Theorem 4 in [START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF], Ch.13, Sec.5) the asymptotic behavior of G λ (0, 0)

for d = 1 is encoded in relation (3), namely, G λ (0, 0) = γ 1 √ π √ λ + o 1 √ λ , λ → 0 + . (40)
Furthermore, using the same Tauberian theorem we infer that

G λ (0, 0) -G λ (0, r) = a -1 ρ 1 (r) -2 √ π γ1 (r) √ λ + o( √ λ), λ → 0+, r ∈ Z\{0}. (41)
Indeed, a -1 ρ 1 (r) = ∞ 0 (p(t; 0, 0)p(t; 0, r)) dt < ∞ on account of (4) and

G λ (0, 0) -G λ (0, r) -a -1 ρ 1 (r) = ∞ 0 e -λt (p(t; 0, 0) -p(t; 0, r)) dt -a -1 ρ 1 (r) = λ ∞ 0 e -λt t 0 (p(u; 0, 0) -p(u; 0, r)) du dt -λ ∞ 0 e -λt a -1 ρ 1 (r) dt = -λ ∞ 0 e -λt ∞ t (p(u; 0, 0) -p(u; 0, r)) du dt. ( 42
)
Here ∞ t (p(u; 0, 0)p(u; 0, r)) du ∼ 2γ 1 (r) t -1/2 (as t → ∞) by formula (4) and Theorem 31 in [START_REF] Vatutin | Harris Branching Processes[END_REF]. Now we substitute relation (41) and the formula for H x,y,0 (∞) appearing in ( 7) into (36), (37) and (38). After collecting terms we substitute (40) in the obtained formulae. Omitting tiresome calculations we deduce that for x, y ∈ Z, y = 0, and λ → 0+

(H x,y,0 (∞) -H x,y,0 )(λ) ∼ ρ 1 (x) + ρ 1 (y -x) -ρ 1 (y) 4 a √ π γ 1 √ λ + a √ π (ρ 1 (y)γ 1 (x) + ρ 1 (y -x)γ 1 (y) -ρ 1 (x)γ 1 (y) -ρ 1 (y)γ 1 (y -x)) ρ 2 1 (y) √ λ .
This can be rewritten in the form

(H x,y,0 (∞) -H x,y,0 )(λ) ∼ C 1 (x, y) √ π √ λ , λ → 0+, (43)
given that γ1 (r) = πr 2 γ 3 1 , r ∈ Z. The latter identity is valid by virtue of definitions of γ 1 and γ1 (r) since γ 1 = 1/ -2πφ ′′ (0) and

γ1 (r) = r 2 4π +∞ -∞ υ 2 e φ ′′ (0) υ 2 /2 dυ = r 2 -4πφ ′′ (0) -φ ′′ (0) +∞ -∞ u 2 e -u 2 /2 du = r 2 -2φ ′′ (0) -2πφ ′′ (0)
.

We took into account that the variables change u = υ -φ ′′ (0) reduces the integration to writing

+∞ -∞ u 2 e -u 2 /2 du = √ 2π.
Note that relation ( 43) is valid for a simple random walk on Z as well. However, in this case the function C 1 (x, y) takes nonnegative values (not strictly positive values).

If C 1 (x, y) > 0 then application of Tauberian theorem (Theorem 4 in [START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF], Ch.13, Sec.5) to relation (43) leads to the desired statement [START_REF] Bulinskaya | Limit Distributions of Particle Numbers in Branching Random Walk[END_REF]. Thus, to complete the proof of Theorem 1 for d = 1 we only need to verify the positivity of function C 1 (x, y), x, y ∈ Z, y = 0.

According to Corollary 2 we may assume that y > 0. Then there exist three possible relative positions of points x, y and 0, namely, x ≥ y, 0 ≤ x < y and x < 0. Let us consider at first the case x ≥ y. To check strict positivity of C 1 (x, y) for such x and y we search for a lower estimate of it. To this end note that ρ 1 (x) + ρ 1 (yx)ρ 1 (y) ≥ 0 in view of formula for H y,x,0 (∞) appearing in [START_REF] Bulinskaya | Catalytic Branching Random Walk on Three-Dimensional Lattice[END_REF] and the evident inequality H y,x,0 (∞) ≤ 1. Obviously, C 1 (0, y) ≥ 0 and, consequently, ρ 2 1 (y) -4a 2 π 2 y 2 γ 4 1 ≥ 0. Combining these inequalities we see that for x ≥ y

C 1 (x, y) = (ρ 1 (x) + ρ 1 (y -x) -ρ 1 (y))(ρ 2 1 (y) -4a 2 π 2 y 2 γ 4 1 ) 4 a π ρ 2 1 (y) γ 1 + 2ρ 1 (y -x)aπy 2 γ 3 1 ρ 2 1 (y) + 2aπγ 3 1 y(x -y) ρ 1 (y) ≥ 2ρ 1 (y -x)aπy 2 γ 3 1 ρ 2 1 (y) + 2aπγ 3 1 y(x -y) ρ 1 (y) > 0.
Before passing to the case 0 ≤ x < y we derive one more useful relation. Recall that τ y,0 ≥ τ - y,0 a.s. because τ y,0 = τ - y,0 + τ a.s. Since the random variable τ is exponentially distributed, P(τ y,0 < ∞) = P(τ - y,0 < ∞). Therefore P(t < τ y,0 < ∞) ≥ P(t < τ - y,0 < ∞) and hence H x,y,0 (∞) -H x,y,0 (t) ≥ H - x,y,0 (∞) -H - x,y,0 (t). Due to Lemma 3 and the monotonicity property of the Laplace transform it follows that

(H x,y,0 (∞) -H x,y,0 )(λ) ≥ r∈Z,r =x,r =y,r =0 a(x, r) a (H r,y,0 (∞) -H r,y,0 )(λ), λ > 0. ( 44
)
Now turn to the case 0 ≤ x < y. We show the positivity of C 1 (x, y) successively for x = y -1, x = y -2, . . ., x = 0. If x = y -1 then there exists a point r > y such that a(y -1, r) > 0 as otherwise the random walk is simple. Therefore, by virtue of (43) and of the just established inequality C 1 (r, y) > 0, 0 < y < r, there is a summand of the order 1/ √ λ at the right-hand side of (44). Consequently, by (43) the left-hand side of (44) is also of the order 1/ √ λ, λ → 0+, and the relation C 1 (y -1, y) > 0 with y > 0 is proved. Recall that the random walk under consideration is irreducible, that is, the greatest common divisor of all r ∈ Z\{0} such that a(r) > 0 equals 1 (see [START_REF] Borovkov | Probability Theory[END_REF], Ch.12, Sec.3). This implies for x = y -2 that there exists a point r > y or r = y -1 such that a(y -2, r) > 0. Since for such r we have just shown the positivity of C 1 (r, y), in view of (43) it means that the left-hand side of (44) has the order of growth 1/ √ λ, as λ → 0. So we conclude that C 1 (y -2, y) > 0 with y > 0. Considering successively the cases x = y -3, x = y -4, . . ., x = 0, we prove according to the described scheme that C 1 (x, y) > 0 for all x ∈ [0, y), x ∈ Z.

The case x < 0 < y is also treated by indirect methods, since the immediate estimation of C 1 (x, y) for such x and y is rather difficult. We consider in succession x = -1, x = -2, . . .. At first we assume that for the random walk S there exist at least three points r 1 > 0, r 2 > 0 and r 3 > 0 such that a(r 1 ) > 0, a(r 2 ) > 0 and a(r 3 ) > 0. If x = -1 then at the right-hand side of (44) there is at least one summand indexed by r where r > 0, r = y and a(-1, r) > 0. Similarly to the previous discussion we infer that the left-hand side of (44) has the order of growth 1/ √ λ, as λ → 0+, and C 1 (-1, y) > 0. If x = -2 then at the right-hand side of (44) there exists at least one summand indexed by r = -1 or r > 0, such that r = y and a(-2, r) > 0. Since for such r the positivity of C 1 (r, y) has been already proved, in view of ( 43) and (44) we deduce that C 1 (-2, y) > 0. The positivity of C 1 (x, y) for x = -3, x = -4, . . . is verified in the same manner.

We continue to deal with the case x < 0 < y assuming that there exist exactly two points r 1 > 0 and r 2 > 0 such that a(r 1 ) > 0 and a(r 2 ) > 0. Let x = -1. If there is a point r > 0, r = y, such that a(-1, r) > 0 (i.e. r 1 = r+1 or r 2 = r+1) then by virtue of ( 43) and (44) as well as of the verified inequality C 1 (r, y) > 0 we get the desired relation C 1 (-1, y) > 0. Otherwise, one has r 1 = 1 and r 2 = y + 1. If y > 1 then a(-2, y -1) = a(y + 1) > 0 and, consequently, C 1 (-2, y) > 0 in view of (43), (44) and the established earlier estimate C 1 (y -1, y) > 0. Therefore, if r 1 = 1, r 2 = y + 1 and y > 1 we obtain C 1 (-1, y) > 0 due to (43) and (44) as well as the relation a(-1, -2) = a(1) > 0 and the just proved inequality C 1 (-2, y) > 0. However, for r 1 = 1, r 2 = y + 1 and y = 1, the previous arguments do not work and we have to employ the following facts concerning the embedded chain {S n , n ∈ Z + }.

For the embedded chain {S n , n ∈ Z + } introduce a passage time T := min{n > 0 : S n ≥ -1}. According to [START_REF] Afanasiev | Random Walks and Branching Processes[END_REF], Theorems 15.1 and 15.2,

P(T < ∞|S(0) = -1) = 1, P(T > n|S(0) = -1) ∼ c √ n , n → ∞, ( 45
)
where c is some positive constant.

For brevity we write P -1 (•) := P(•|S(0) = -1). Since we assume that r 1 = 1 and r 2 = 2, the process {S n , n ∈ Z + } can perform jumps to the nearest-neighbor points or to the next nearest points. Hence,

P -1 (T = n, S n = -1) = P -1 (T = n, S n-1 = -2, S n = -1) + P -1 (T = n, S n-1 = -3, S n = -1) = P -1 (S k < -1, 0 < k < n -1, S n-1 = -2, S n = -1) + P -1 (S k < -1, 0 < k < n -1, S n-1 = -3, S n = -1) = a(-2, -1) a P -1 (S k < -1, 0 < k < n -1, S n-1 = -2) + a(-3, -1) a P -1 (S k < -1, 0 < k < n -1, S n-1 = -3) ≥ min{a(1), a(2)} a P -1 (T = n). (46)
Let us return to a lower estimate of H -1,1,0 (∞) -H -1,1,0 (t) = P -1 (t < τ 1,0 < ∞), as t → ∞. Denote by N = {N(t), t ≥ 0} the Poisson process constructed by means of the random sequence {τ (n+1)τ (n) } ∞ n=0 , i.e. N is the Poisson process with intensity a. Considering all the possible jumps of S, taking into account (46) and invoking the explicit construction of S we derive that

P -1 (t < τ 1,0 < ∞) = ∞ n=1 P -1 (τ 1,0 = τ (n) , t < τ 1,0 < ∞) = ∞ n=1 P -1 (S k = 0, S k = 1, 0 < k < n, S n = 1, t < τ (n) < ∞) = ∞ n=1 P -1 (S k ≤ -1, 0 < k < n -1, S n-1 = -1, S n = 1, N(t) < n) = ∞ n=1 P -1 (S k ≤ -1, 0 < k < n -1, S n-1 = -1, S n = 1)P(N(t) < n) ≥ a(-1, 1) a ∞ n=1 P -1 (S k < -1, 0 < k < n -1, S n-1 = -1)P(N(t) < n) = a(2) a ∞ n=1 P -1 (T = n -1, S n-1 = -1)P(N(t) < n) ≥ a(2) min{a(1), a(2)} a 2 ∞ n=1 P -1 (T = n -1) n-1 k=0 e -at (at) k k! = a(2) min{a(1), a(2)} a 2 ∞ k=0 P -1 (T ≥ k) e -at (at) k k! . (47)
Thus, finding a lower estimate of P -1 (t < τ 1,0 < ∞) is reduced to establishing a lower estimate of the last series. It is not difficult to check that (N(t)at)/ √ at Law -→ ξ with ξ ∼ N (0, 1), as t → ∞. Consequently, P(N(t) ∈ (atb √ at, at + b √ at)) → P(ξ ∈ (-b, b)) > 0, as t → ∞, for each b > 0. Hence, in view of (45)

∞ k=0 P -1 (T ≥ k) e -at (at) k k! ≥ P(T ≥ [at + b √ at]) P(N(t) ∈ (at -b √ at, at + b √ at)) ∼ c √ at P(ξ ∈ (-b, b)), t → ∞. (48)
Combining ( 43), ( 47) and (48) as well as taking into account the monotonicity property of the Laplace transform we conclude that C 1 (-1, 1) > 0. Thus, we have shown that C 1 (-1, y) > 0 for each y > 0. Let us pass to the starting point x = -2. Recall that we assume that there exist exactly two points r 1 > 0 and r 2 > 0 such that a(r 1 ) > 0 and a(r 2 ) > 0. If there is point r > 0, r = y, or r = -1 such that a(-2, r) > 0 (that is, r 1 = r + 2 or r 2 = r + 2) then due to (43) and (44) as well as the verified inequality C 1 (r, y) > 0 we get the desired estimate C 1 (-2, y) > 0. Otherwise, one has r 1 = 2 and r 2 = y + 2. In particular, a(-4, -2) > 0 and a(-4, y -2) > 0. Since the random walk is irreducible, one can guarantee that y -2 = 0. Consequently, C 1 (-4, y) > 0 in view of (43), (44) and the earlier established inequality C 1 (y -2, y) > 0 with y > 0 and y = 2. In its turn C 1 (-2, y) > 0 by virtue of ( 43), (44) and the just proved estimate C 1 (-4, y) > 0. Therefore, the positivity of C 1 (x, y) for x = -2 and y > 0 is demonstrated. For x = -3, x = -4, . . ., verification of C 1 (x, y) positivity, y > 0, is implemented similarly to the case x = -2. Consequently, Theorem 1 for d = 1 is proved.

The case d = 2

As in the previous subsection one can find the limit value H x,y,0 (∞) for x, y ∈ Z 2 , y = 0, by the formula H x,y,0 (∞) = lim λ→0+ Ȟx,y,0 (λ). For this purpose we write the following asymptotic representation of the function Ȟ0,r (λ

), r ∈ Z 2 , Ȟ0,r (λ) = 1 + ρ 2 (r) a γ 2 ln λ + o 1 ln λ , λ → 0+, ( 49
)
which results from application of Tauberian theorem (Theorem 4 in [START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF], Ch.13, Sec.5) to relation [START_REF] Spitzer | Principles of Random Walk[END_REF] and from employing [START_REF] Lawler | Intersections of Random walks[END_REF]. Substituting (49) in ( 31) and taking into account [START_REF] Yarovaya | Branching Random Walks in Inhomogeneous Medium[END_REF] we find lim λ→0+ Ȟx,y,0 (λ) and, thus, H x,y,0 (∞) is obtained. However, to complete the derivation of ( 7) for d = 2 one has to show that H x,y,0 (∞) ∈ (0, 1). Similarly to Subsection 4.1, equalities ( 18) and ( 30) entail that H x,y,0 (∞) = 0 ⇔ H x,0,y (∞) = 1 and H x,y,0 (∞) = 1 ⇔ H x,0,y (∞) = 0. Therefore, for establishing that H x,y,0 (∞) ∈ (0, 1) for all x, y ∈ Z 2 , y = 0, it suffices to verify the inequality H x,y,0 (∞) > 0 for all x, y ∈ Z 2 , y = 0. In its turn the latter holds true if, for instance, relation ( 10) is satisfied with C 2 (x, y) > 0 for all x, y ∈ Z 2 , y = 0. Hence, the initial problem of checking that H x,y,0 (∞) ∈ (0, 1) is reduced to proving of [START_REF] Bulinski | Theory of Stochastic Processes[END_REF] and showing the positivity of function C 2 (•, •). The rest part of this subsection is devoted to validating the last two claims. The proof of (10) bears on formulae (36)-( 38) in which we substitute asymptotic decompositions of functions G λ (0, 0) and G λ (0, 0) -G λ (0, r), as λ → 0+, for r = x, r = y and r = yx. Due to Tauberian theorem (Theorem 2 in [START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF], Ch.13, Sec.5) the asymptotic behavior of G λ (0, 0) for d = 2 is implied by (3), namely,

G λ (0, 0) = -γ 2 ln λ + o(ln λ), λ → 0 + . ( 50
)
Using the same Tauberian theorem and relation (42), valid for all d ∈ N, we deduce that

G λ (0, 0) -G λ (0, r) = a -1 ρ 2 (r) + γ2 (r) λ ln λ + o(λ ln λ), λ → 0+, r ∈ Z 2 \{0}. (51)
Indeed, in view of ( 4) one has a -1 ρ 2 (r) = ∞ 0 (p(t; 0, 0)p(t; 0, r)) dt < ∞ and by formula (4) and Theorem 31 in [START_REF] Vatutin | Harris Branching Processes[END_REF] one infers that ∞ t (p(u; 0, 0)p(u; 0, r)) du ∼ γ2 (r)/t, as t → ∞. Now we substitute relation (51) and the formula for H x,y,0 (∞) appearing in ( 7) into (36)-(38). After collecting terms we substitute (50) in the obtained formulae. Omitting tedious calculations we establish that

(H x,y,0 (∞) -H x,y,0 )(λ) ∼ ρ 2 (x) + ρ 2 (y -x) -ρ 2 (y) -4 a γ 2 λ ln λ , λ → 0+, x, y ∈ Z 2 , y = 0.
If C 2 (x, y) > 0 then application of Tauberian theorem (Theorem 4 in [START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF], Ch.13, Sec.5) to the last asymptotic relation gives [START_REF] Bulinski | Theory of Stochastic Processes[END_REF]. Thus, to complete the proof of Theorem 1 for d = 2 we only have to verify the positivity of C 2 (x, y) for all x, y ∈ Z 2 , y = 0. To prove the claim we employ the method of ad absurdum. Assume that there exist x and y such that x ∈ Z 2 , y ∈ Z 2 \{0} and C 2 (x, y) = 0. Then the definition of C 2 (x, y) implies that

ρ 2 (x) + ρ 2 (y -x) -ρ 2 (y) = 0. ( 52
)
In view of ( 7) the last equality is equivalent to the relation H y,x,0 (∞) = 1. Therefore, formulae ( 18) and [START_REF] Yarovaya | Criteria of the exponential growth of particles numbers in models of branching random walks[END_REF] entail that H y,0,x (t) = 0 and H y,x,0 (t) = H y,x (t) for t ≥ 0. Hence, by virtue of ( 22) one has C 2 (y, x) > 0 and C 2 (y, x) = ρ 2 (yx)/(a γ 2 ). Taking into account the definition of C 2 (y, x) the last equality can be rewritten as ρ 2 (y)-ρ 2 (x) = 3ρ 2 (y -x). This relation combined with assumption (52) leads to contradiction. Hence, C 2 (x, y) > 0 for all x, y ∈ Z 2 , y = 0. The proof of Theorem 1 for d = 2 is complete.

The case d ≥ 3

In this subsection we assume that d ≥ 3. Basing on the formula H x,y,0 (∞) = lim λ→0+ Ȟx,y,0 (λ),

x, y ∈ Z d , y = 0, we find the limit value H x,y,0 (∞) with the help of ( 19), ( 20) and [START_REF] Zubkov | Inequalities for Transition Probabilities with Taboos and Their Applications[END_REF]. To prove (8) completely we have to check that H x,y,0 (∞) ∈ (0, 1). Obviously, in view of ( 19), ( 20) and ( 30) the relation H x,y,0 (∞) ≤ H x,y (∞) < 1 holds true. For verifying inequality H x,y,0 (∞) > 0 it suffices to establish [START_REF] Caravenna | Random Walk Models and Probabilistic Techniques for Inhomogeneous Polymer Chains[END_REF] and show the positivity of C d (x, y). So the rest part of the subsection is devoted to validating these two assertions. Further on we suppose that x and y are fixed points within Z d \{0}. Formulae (36) -( 38) can be rewritten in the following form

(H x,y,0 (∞) -H x,y,0 )(λ) = ρ d (y)(ρ d (y -x) + ρ d (x) -ρ d (y)) a 2 (G 0 (0, 0) + G 0 (0, y)) J 1 (λ; 0, y) + J 2 (λ; x, y) + J 3 (λ; x, y) - G 0 (0, 0)(ρ d (y -x) + ρ d (x) -ρ d (y)) a(G 0 (0, 0) + G 0 (0, y)) J 4 (λ; y) - a(G 0 (0, 0)G 0 (x, y) -G 0 (0, x)G 0 (0, y)) ρ d (y)(G 0 (0, 0) + G 0 (0, y)) J 5 (λ; y), (53)
(H 0,y,0 (∞) -H 0,y,0 )(λ) = ρ d (y) a 2 (G 0 (0, 0) + G 0 (0, y)) J 1 (λ; y; y) -G 0 (0, y) a(G 0 (0, 0) + G 0 (0, y)) J 4 (λ; y)

- G 0 (0, y) ρ d (y)(G 0 (0, 0) + G 0 (0, y)) J 5 (λ; y) + 1 a J 6 (λ; y, y), (54)
(H y,y,0 (∞) -H y,y,0 )(λ) = ρ d (y) a 2 (G 0 (0, 0) + G 0 (0, y)) J 1 (λ; 0, y) - G 0 (0, 0) a(G 0 (0, 0) + G 0 (0, y)) J 4 (λ; y) + G 0 (0, 0) ρ d (y)(G 0 (0, 0) + G 0 (0, y)) J 5 (λ; y) - 1 a J 6 (λ; 0, y) (55)
where for r ∈ Z d and λ > 0

J 1 (λ; r, y) := K d (λ; r) G 2 λ (0, 0) -G 2 λ (0, y) , J 2 (λ; r, y) := (K d (λ; y -r) -K d (λ; y))G λ (0, 0) G 2 λ (0, 0) -G 2 λ (0, y) , J 3 (λ; r, y) := (K d (λ; 0) -K d (λ; r))G λ (0, y) G 2 λ (0, 0) -G 2 λ (0, y) , J 4 (λ; y) := K d (λ; 0) -K d (λ; y) G 2 λ (0, 0) -G 2 λ (0, y) , J 5 (λ; y) := K d (λ; 0) -K d (λ; y) G λ (0, 0) -G λ (0, y) , J 6 (λ; r, y) := G λ (0, r) (λ + a)(G 2 λ (0, 0) -G 2 λ (0, y))
and

K d (λ; r) := G 0 (0, r) -G λ (0, r) λ .
Let us briefly describe the further scheme of the proof of [START_REF] Caravenna | Random Walk Models and Probabilistic Techniques for Inhomogeneous Polymer Chains[END_REF]. At first we differentiate [d/2] -1 times (symbol [•] stands for the integer part of a number) the both sides of (53), (54) and (55) w.r.t. variable λ at point λ > 0. Then we retrieve the asymptotic behavior of function J

([d/2]-1) 1
(λ; r, y), as λ → 0+, for each r ∈ Z d . Moreover, we verify that the functions J

([d/2]-1) 2 (λ; r, y), J ([d/2]-1) 3 (λ; r, y), J ([d/2]-1) 4 (λ; y), J ([d/2-1]) 5 (λ; y) and J ([d/2]-1) 6 (λ; r, y) are o(J ([d/2]-1) 1
(λ; r, y)), as λ → 0+, for each r. Afterwards one can employ Corollary 43 in [START_REF] Vatutin | Harris Branching Processes[END_REF] and, thus, find the asymptotic behavior of the functions H x,y,0 (∞) -H x,y,0 (t), H 0,y,0 (∞) -H 0,y,0 (t) and H y,y,0 (∞) -H y,y,0 (t), as t → ∞.

Before studying the limiting behavior of J

([d/2]-1) 1
(λ; r, y) we have to investigate the asymptotic properties of the functions

K d (λ; r), (G λ (0, 0) -G λ (0, y)) -1 , (G 2 λ (0, 0) -G 2 λ (0, y)) -1 , G λ (0, r)/(G 2 λ (0, 0) -G 2 λ (0, y)), G λ (0, r)/(λ + a)
and their derivatives up to the order

[d/2] -1 for each r ∈ Z d .
Integrating by parts it is easy to check the following equality

K d (λ; r) = ∞ 0 e -λt ∞ t p(u; 0, r) du dt, r ∈ Z d , λ > 0. (56)
Then for each n ∈ N function K d (λ; r) has the n-th derivative w.r.t. λ > 0 which can be calculated by the formula

K (n) d (λ; r) = (-1) n ∞ 0 e -λt t n ∞ t p(u; 0, r) du dt (57)
for each r ∈ Z d . For convenience we set K

∞ t p(u; 0, r) du ∼ 2γ d (d -2)t d/2-1 , t → ∞, r ∈ Z d . (58)
Then in view of (57) one has, as λ → 0+,

(-1) n K (n) d (λ; r) → ∞ 0 t n ∞ t p(u; 0, r) du dt < ∞, 0 ≤ n < d -1 2 -1. (59)
Moreover, using Tauberian theorem (Theorem 2 in [START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF], Ch.13, Sec.5) we observe that, as λ → 0+,

(-1) d/2-2 K (d/2-2) d (λ; r) ∼ 2γ d (d -2) ln 1 λ only for even d. (60)
Note that for uneven d the numbers [(d -1)/2] -1 and [d/2] -1 coincide whereas for even

d one has [(d -1)/2] -1 = d/2 -2.
Employing relations (57) and (58) as well as Tauberian theorem (Theorem 4 in [START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF], Ch.13, Sec.5) we infer that, as λ → 0+,

(-1) [d/2]-1 K ([d/2]-1) d (λ; r) ∼ 2γ d √ π (d-2) √ λ , if d is uneven, 2γ d (d-2)λ , if d is even. ( 61
)
We stress that the right-hand sides of ( 60) and (61) do not depend on point r ∈ Z d . Now we investigate the asymptotic properties of function G λ (0, 0) -G λ (0, y) and its derivatives up to the order [d/2] -1, as λ → 0+. Recall that for λ ≥ 0 one has the identity G λ (0, 0) -G λ (0, y) = ∞ 0 e -λt (p(t; 0, 0)p(t; 0, y)) dt where the asymptotic behavior of the function p(t; 0, 0)p(t; 0, y) (as t → ∞) is given by (4). Then by virtue of the counterpart of (57) for the function G λ (0, 0) -G λ (0, y) we get that, as λ → 0+,

(-1) n (G λ (0, 0)-G λ (0, y)) (n) → ∞ 0 t n (p(t; 0, 0) -p(t; 0, y)) dt < ∞, 0 ≤ n ≤ d 2 -1. (62)
Using Theorem 4 we are ready to estimate the asymptotic growth of the n-th derivative, 0 ≤ n ≤ [d/2] -1, of the function (G λ (0, 0) -G λ (0, y)) -1 , as λ → 0+. For this purpose we consider function V -1 as the external function W (V ) and we substitute G λ (0, 0) -G λ (0, y) instead of internal function V (•) in Theorem 4. Since W (n) (V ) = (V -1 )

(n) = (-1) n n! V -n-1 for every n ∈ Z + and G λ (0, 0) -G λ (0, y) → a -1 ρ d (y) > 0, as λ → 0+, then in view of Theorem 4 and relation (62) we obtain the estimate we are interested in

1 G λ (0, 0) -G λ (0, y) (n) = O(1), λ → 0+, 0 ≤ n ≤ d 2 -1. ( 63
)
Let us turn to the study of the asymptotic behavior of the function G λ (0, r) and its n-th derivatives, 0 < n ≤ [d/2] -1, as λ → 0+, for each r ∈ Z d . In view of the counterpart of (57) for G λ (0, r) as well as of relation [START_REF] Berg | Random Walks in Biology[END_REF] we see that, as λ → 0+,

(-1) n G (n) λ (0, r) → ∞ 0 t n p(t; 0, r)dt < ∞, 0 ≤ n ≤ d -1 2 -1. (64)
G λ (0, r) λ + a (n) = n k=0 C k n G (k) λ (0, r) 1 λ + a (n-k) = n k=0 C k n G (k) λ (0, r) (-1) n-k (n -k)! (λ + a) n-k+1
where n ∈ Z + and, hence, for each r ∈ Z d we obtain the desired estimates

G λ (0, r) λ + a (n) = O(1), λ → 0+, 0 ≤ n ≤ d -1 2 -1, (66)
G λ (0, r) λ + a

Repeating the argument used for deriving relations (64) and ( 65) we find the asymptotic behavior of the function G λ (0, 0) + G λ (0, y) and its derivatives up to the order [d/2] -1, as λ → 0+, (-1) n (G λ (0, 0)+G λ (0, y)

) (n) → ∞ 0 t n (p(t; 0, 0) + p(t; 0, y))dt < ∞, 0 ≤ n ≤ d -1 2 -1, (68) (-1) d/2-1 (G λ (0, 0) + G λ (0, y)) (d/2-1) ∼ 2γ d ln 1 λ only for even d. (69)
To investigate the asymptotic properties of the function (G λ (0, 0) + G λ (0, y)) -1 and its derivatives of order n, 0 < n ≤ [d/2] -1, as λ → 0+, we employ Theorem 4 once again. We take V -1 as the external function W (V ) and consider the function

G λ (0, 0) + G λ (0, y) as V (•) in Theorem 4. Since W (n) (V ) = (V -1 )
(n) = (-1) n n! V -n-1 for all n ∈ Z + and G λ (0, 0) + G λ (0, y) → G 0 (0, 0) + G 0 (0, y) > 0, as λ → 0+, due to Theorem 4 as well as relations ( 68) and (69) we get the estimates

1 G λ (0, 0) + G λ (0, y) (n) = O(1), λ → 0+, 0 ≤ n ≤ d -1 2 -1, ( 70
) 1 G λ (0, 0) + G λ (0, y) (d/2-1) = O ln 1 λ , λ → 0+, only for even d. (71)
To establish asymptotic behavior of the function (G 2 λ (0, 0) -G 2 λ (0, y)) -1 and its derivatives up to the order [d/2] -1, as λ → 0+, it suffices to apply the Leibniz formula along with estimates (63), ( 70) and (71). Indeed,

1 G 2 λ (0, 0) -G 2 λ (0, y) (n) = n k=0 C k n 1 G λ (0, 0) -G λ (0, y) (k) 1 G λ (0, 0) + G λ (0, y) (n-k)
for n ∈ Z + , and, consequently,

1 G 2 λ (0, 0) -G 2 λ (0, y) (n) = O(1), λ → 0+, 0 ≤ n ≤ d -1 2 -1, ( 72
) 1 G 2 λ (0, 0) -G 2 λ (0, y) (d/2-1) = O ln 1 λ , λ → 0+, only for even d. (73)
Let us investigate the asymptotic properties of the function G λ (0, r)/(G 2 λ (0, 0) -G 2 λ (0, y)) and its n-th derivatives, 0 ≤ n ≤ [d/2] -1, for each r ∈ Z d , as λ → 0+. The Leibniz formula implies that for all λ > 0 and

n ∈ Z + G λ (0, r) G 2 λ (0, 0) -G 2 λ (0, y) (n) = n k=0 C k n G (k) λ (0, r) 1 G 2 λ (0, 0) -G 2 λ (0, y) (n-k)
.

Taking into account (64), (65), ( 72) and (73), for each r ∈ Z d , we infer that (λ; r, y) for each r ∈ Z d , as λ → 0+. By the definition of J 1 (λ; r, y) after applying the Leibniz formula to its ([d/2] -1)-th derivative we obtain

G λ (0, r) G 2 λ (0, 0) -G 2 λ (0, y) (n) = O(1), λ → 0+, 0 ≤ n ≤ d -1 2 -1, (74)
G λ (0, r) G 2 λ (0, 0) -G 2 λ (0, y) (d/2-1) = O ln 1 λ , λ → 0+,
J ([d/2]-1) 1 (λ; r, y) = [d/2]-1 k=0 C k [d/2]-1 1 G 2 λ (0, 0) -G 2 λ (0, y) (k) K ([d/2]-1-k) d (λ; r).
Analyzing every summand in the last sum with the help of relations ( 59)-( 61) and ( 72), (73) we conclude that the first summand gives the main contribution to the asymptotic behavior of J

([d/2]-1) 1
(λ; x, y), λ → 0+. Indeed, the first summand has the order of growth 1/ √ λ for uneven d and has the order 1/λ for even d, respectively. On the other hand, the rest summands are O(1) for uneven d and are O(ln 2 λ) for even d, respectively. Therefore, for each r ∈ Z d , as λ → 0+, one gets

(-1) [ d 2 ]-1 J ([ d 2 ]-1) 1 (λ; r, y) ∼ 2γ d √ π (d-2)(G 2 0 (0,0)-G 2 0 (0,y)) √ λ , if d is uneven, 2γ d (d-2)(G 2 0 (0,0)-G 2 0 (0,y))λ , if d is even. (76)
To estimate the asymptotic behavior of the functions J

([d/2]-1) 2
(λ; r, y) and J

([d/2]-1) 3
(λ; r, y), as λ → 0+, we use the Leibniz formula once again, namely,

J ([d/2]-1) 2 (λ; r, y) = [d/2]-1 k=0 C k [d/2]-1 G λ (0, 0) G 2 λ (0, 0) -G 2 λ (0, y) (k) (K d (λ; y -r) -K d (λ; y)) ([d/2]-1-k) , J ([d/2]-1) 3 (λ; r, y) = [d/2]-1 k=0 C k [d/2]-1 G λ (0, y) G 2 λ (0, 0) -G 2 λ (0, y) (k) (K d (λ; 0) -K d (λ; r)) ([d/2]-1-k) .
By virtue of (59)-( 61), (74) and (75) at the right-hand sides of the last equalities there are only the first summands which grow as o(1/ √ λ) for uneven d and as o(1/λ) for even d, respectively, as λ → 0+. The rest summands are O(1) for uneven d and are o(ln 2 λ) for even d. Thus, on account of (76) for each r ∈ Z d , as λ → 0+, we come to the desired estimates

J ([d/2]-1) 2 (λ; r, y) = o J ([d/2]-1) 1 (λ; 0, y) , J ([d/2]-1) 3 (λ; r, y) = o J ([d/2]-1) 1 (λ; 0, y) . ( 77
)
Let us study the asymptotic growth of the function J

([d/2]-1) 4
(λ; y) as λ → 0+. The definition of J 4 (λ; y) and the Leibniz formula entail

J ([d/2]-1) 4 (λ; y) = [d/2]-1 k=0 C k [d/2]-1 1 G 2 λ (0, 0) -G 2 λ (0, y) (k) (K d (λ; 0) -K d (λ; y)) ([d/2]-1-k) .
Due to relations (59)-( 61 (λ; r, y) , λ → 0 + .

We turn to estimating of the asymptotic behavior of the function J ([d/2]-1) 5

(λ; y), as λ → 0+. The definition of J 5 (λ; y) and the Leibniz formula imply that

J ([d/2]-1) 5 (λ; y) = [d/2]-1 k=0 C k [d/2]-1 1 G λ (0, 0) -G λ (0, y) (k) (K d (λ; 0) -K d (λ; y)) ([d/2]-1-k) .
On account of relations (59)-( 61) and ( 63) we see that the first summand at the last sum is o(1/ √ λ) for uneven d and is o(1/λ) for even d, respectively. All the other summands have the order of growth O(1) for uneven d and o(ln λ) for even d, respectively, as λ → 0+. By (76) it follows that for each r ∈ Z

d J ([d/2]-1) 5 (λ; y) = o J ([d/2]-1) 1 (λ; r, y) , λ → 0 + . ( 79
)
The last step is the revealing the asymptotic properties, as λ → 0+, of the function J ([d/2]-1) 6

(λ; r, y). By the definition of J 6 (λ; r, y) together with the Leibniz formula one can write

J ([d/2]-1) 6 (λ; r, y) = [d/2]-1 k=0 C k [d/2]-1 1 G 2 λ (0, 0) -G 2 λ (0, y) (k) G λ (0, r) λ + a ([d/2]-1-k)
.

Recalling relations (66), ( 67), ( 72) and ( 73), we deduce that if d is uneven then all the summands at the last sum are O(1), as λ → 0+. However, if d is even then the first and the last summands have the order of growth O(ln λ) whereas the others have the order of growth O(1), as λ → 0+.

In any case in view of (76) the following formula holds true (λ; r, y) , λ → 0 + .

Combining results (53) -( 55) and ( 76) -(80) we conclude that

∞ 0 e -λt t [d/2]-1 (H x,y,0 (∞) -H x,y,0 (t)) dt ∼ 2γ d (ρ d (y -x) + ρ d (x) -ρ d (y)) a(d -2)(G 0 (0, 0) + G 0 (0, y)) 2 × √ π/ √ λ, if d is uneven, 1/λ, if d is even, ( 81
) ∞ 0 e -λt t [ d 2 ]-1 (H 0,y,0 (∞) -H 0,y,0 (t)) dt ∼ ∞ 0 e -λt t [ d 2 ]-1 (H y,y,0 (∞) -H y,y,0 (t)) dt ∼ 2γ d a(d -2)(G 0 (0, 0) + G 0 (0, y)) 2 √ π/ √ λ, if d is uneven, 1/λ, if d is even, (82)
as λ → 0+. If C d (x, y) > 0 for all x, y ∈ Z d \{0}, x = y (it is easily seen that C d (0, y) and C d (y, y) are always positive for d ≥ 3) then application of Corollary 43 in [START_REF] Vatutin | Harris Branching Processes[END_REF] to the last asymptotic relations implies [START_REF] Caravenna | Random Walk Models and Probabilistic Techniques for Inhomogeneous Polymer Chains[END_REF].

Thus, to complete the proof of Theorem 1 for d ≥ 3 we only have to verify the positivity of C d (x, y) for x, y ∈ Z d \{0}, x = y. The main idea of the demonstration is the same as that exploited for proving the positivity of C 2 (•, •) in Subsection 4.2. Namely, we assume the contrary, that is, for some d ∈ N, x ∈ Z d \{0, y} and y ∈ Z d \{0} the relation C d (x, y) = 0 is valid. This is equivalent to the following relations

G 0 (0, 0) + G 0 (0, y) = G 0 (x, y) + G 0 (0, x) ⇔ G 0 (0, 0) -G 0 (0, x) = G 0 (x, y) -G 0 (0, y). (83)
Note that the formula for H y,x,0 (∞) appearing in [START_REF] Bulinskaya | Limit Distributions Arising in Branching Random Walks on Integer Lattices[END_REF] and the definition of function ρ d (•) allow us to write

H y,x,0 (∞) = G 0 (0, 0)(G 0 (x, y) -G 0 (0, y)) + G 0 (0, y)(G 0 (0, 0) -G 0 (0, x)) G 2 0 (0, 0) -G 2 0 (0, x)
.

Combining (83) and the last equality we see that H y,x,0 (∞) = (G 0 (0, 0) -G 0 (0, x))(G 0 (0, 0) + G 0 (0, y)) G 2 0 (0, 0) -G 2 0 (0, x) = G 0 (0, 0) + G 0 (0, y) G 0 (0, 0) + G 0 (0, x) = G 0 (x, y) + G 0 (0, x) G 0 (0, 0) + G 0 (0, x) .

However, by formula [START_REF] Yarovaya | Criteria of the exponential growth of particles numbers in models of branching random walks[END_REF] one has H y,x,0 (∞) ≤ H y,x (∞) and due to [START_REF] Révész | Random Walk in Random and Non-Random Environments[END_REF] it follows that G 0 (x, y) + G 0 (0, x) G 0 (0, 0) + G 0 (0, x) ≤ G 0 (x, y) G 0 (0, 0) ⇔ G 0 (0, 0) ≤ G 0 (x, y).

The obtained contradiction completes the proof of ( 11) and the proof of Theorem 1 as well.

Proof of Theorem 2

Firstly note that in view of Corollary 2 it suffices to establish Theorem 2 for z = 0. Recall that for a simple random walk on Z the transition intensities a(r, r + 1) = a(r, r -1) = a/2 and a(r, r + k) = 0 for all r ∈ Z and k ∈ Z such that |k| > 1. By definition of the function φ(θ) and by that of the constant γ 1 one gets φ(θ) = a(cos θ -1), θ ∈ [-π, π], and γ 1 = 1/ √ 2aπ. Hence, equality [START_REF] Bulinskaya | Catalytic Branching Random Walk on a Two-Dimensional Lattice[END_REF] and Lemma 4 imply that ρ 1 (r) = |r|, r ∈ Z\{0}. Observe that the formula for H x,y,0 (∞) in ( 7) is valid for a simple random walk on Z as well. Therefore, we infer that H x,y,0 (∞) = 0 for y < 0 < x and x < 0 < y, H x,y,0 (∞) = x/y for 0 < x < y and y < x < 0, H x,y,0 (∞) = 1 for 0 < y < x and x < y < 0, H 0,y,0 (∞) = 1/(2|y|) and H y,y,0 (∞) = 1-1/(2|y|). Thus, for y < 0 < x and x < 0 < y one has H x,y,0 (•) ≡ 0, that is, relation ( 16) is proved. For 0 < y < x and x < y < 0 due to [START_REF] Yarovaya | Criteria of the exponential growth of particles numbers in models of branching random walks[END_REF] we conclude that H x,y,0 (∞) = 1 = H x,y (∞) and H x,0,y (∞) = 0. This is equivalent to relations H x,0,y (•) ≡ 0 and H x,y,0 (•) ≡ H x,y (•). Since the asymptotic behavior of the function H x,y (∞) -H x,y (t) is established in Lemma 1, relation [START_REF] Chen | Random Walk Intersections: Large Deviations and Related Topics[END_REF] is also proved. Moreover, formula (43) is true for a simple random walk on Z and, consequently, applying Tauberian theorem (Theorem 4 in [START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF], Ch.13, Sec.5) we find the asymptotic behavior of H y,y,0 (∞) -H y,y,0 (t), as t → ∞, with C 1 (y, y) = 1/(2 √ aπ) > 0. Thus, relation ( 13) is proved. So we have to investigate only the asymptotic properties of H x,y,0 (∞) -H x,y,0 (t) when

x ∈ [0, y), x ∈ Z, that is, to verify relations ( 14) and [START_REF] Gikhman | The Theory of Stochastic Processes II[END_REF].

To this end recall a well-known result (the gambler ruin problem) for embedded chain {S n , n ∈ Z + }. Namely, P(0 < S k < y, 0 < k ≤ n|S(0) = x) ≤ (1ε 0 ) n for some ε 0 ∈ (0, 1) and x ∈ [0, y], x ∈ Z (see, e.g., [START_REF] Shiriaev | [END_REF], Ch.1, Sec.9). Bearing on this result we derive that H x,y,0 (∞) -H x,y,0 (t) = (1ε 0 ) n (at) n e -at n! = e -a ε 0 t .

It follows that relations ( 14) and ( 15) are valid with ε ∈ (0, ε 0 ). Theorem 2 is proved completely.

6 Proof of Theorem 3

On account of Corollary 2 it is sufficient to prove Theorem 3 for z = 0. Since τ y,0 = τ - y,0 + τ a.s., one has H x,y,0 (t) = H - x,y,0 * G(t), t ≥ 0. This immediately implies H x,y,0 (∞) = H - x,y,0 (∞), x ∈ Z d , y ∈ Z d \{0}. Furthermore, by formula [START_REF] Lawler | Intersections of Random walks[END_REF] we get that for λ > 0 (H - x,y,0 (∞) -H - x,y,0 )(λ) = (H x,y,0 (∞) -H x,y,0 )(λ)a -1 Ȟx,y,0 (λ).

In view of ( 9) -( 13) and Tauberian theorem (Theorem 4 in [START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF], Ch.13, Sec.5) it is easy to see that the first summand at the right-hand side of (84) makes the main contribution to the asymptotic behavior of the left-hand side of (84). More exactly, this is true if d ≤ 3, x ∈ Z d , y ∈ Z d \{0}, except for a simple random walk on Z in the cases 0 ≤ x < y, y < x ≤ 0,

x < 0 < y or y < 0 < x. Therefore, (H - x,y,0 (∞) -H - x,y,0 )(λ) ∼ (H x,y,0 (∞) -H x,y,0 )(λ), as λ → 0+, and due to (9)-( 13) and Tauberian theorem (Theorem 4 in [START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF], Ch.13, Sec.5) we prove Theorem 3 for the mentioned d, x and y.

When the random walk on Z is simple and 0 ≤ x < y or y < x ≤ 0, the estimate of H -

x,y,0 (∞) -H - x,y,0 (t), as t → ∞, is a direct consequence of Theorem 2 and the inequality H x,y,0 (∞) -H x,y,0 (t) ≥ H - x,y,0 (∞) -H - x,y,0 (t) used in the proof of Theorem 1 for d = 1. Moreover, for a simple random walk on Z when x < 0 < y or y < 0 < x we may assert that H - x,y,0 (t) ≡ 0 by virtue of the established identity H - x,y,0 (∞) = H x,y,0 (∞) and relation [START_REF] Karlin | A Second Course in Stochastic Processes[END_REF]. Thus, for these cases Theorem 3 is also proved.

To find the asymptotic behavior of H - x,y,0 (∞)-H - x,y,0 (t), as t → ∞, for d ≥ 4, we differentiate

d

  (λ; r) := K d (λ; r). Let us analyze for each r the asymptotic behavior of the n-th derivative (0 ≤ n ≤ [d/2] -1, n ∈ Z) of the function K d (λ; r) at point λ, as λ → 0+. To this end we need the relation implied by (3) and Theorem 31 in [26], namely,

  Moreover, employing Tauberian theorem (Theorem 2 in[START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF], Ch.13, Sec.5) we find that(-1) d/2-1 G (d/2-1) λ (0, r) ∼ γ d ln 1 λ , λ → 0+,only for even d. (65) As mentioned above, the numbers [(d -1)/2] -1 and [d/2] -1 coincide for uneven d whereas for even d one has [(d -1)/2] -1 = d/2 -2. Now for estimating the asymptotic behavior of the n-th order derivatives, 0 ≤ n ≤ [d/2] -1, of the function G λ (0, r)/(λ + a), as λ → 0+, it suffices to apply the Leibniz formula as well as relations (64) and (65). Indeed,

(d/2- 1 )

 1 = O ln 1 λ , λ → 0+, only for even d.

  only for even d. (75) Since all the basic relations are established, we turn to finding the asymptotic behavior of the function J ([d/2]-1) 1

∞ n=0 P

 n=0 (t < τ y,0 < ∞, N(t) = n| S(0) = x) ≤ ∞ n=0 P (0 < S k < y, 0 < k ≤ n, N(t) = n| S(0) = x) = ∞ n=0 P (0 < S k < y, 0 < k ≤ n| S(0) = x) P ( N(t) = n| S(0) = x) ≤ ∞ n=0

[d/ 2 ] 2 - 1 ∞ 0 e

 2210 -1 times the both sides of equality (84) w.r.t. λ > 0. The asymptotic relation for the ([d/2] -1)-th derivative of the first summand in (84) is given by formulae (81) and (82). Let us analyze the asymptotic properties of the ([d/2] -1)-th derivative of the second summand in (84). Integration by parts gives∞0 e -λt t [d/2]-1 dH x,y,0 (t) =λ ∞ 0 e -λt t [d/2]-1 (H x,y,0 (∞) -H x,y,0 (t)) dt + d -λt t [d/2]-2 (H x,y,0 (∞) -H x,y,0 (t)) dt. (85)

  for uneven d and is o(1/λ) for even d, respectively, as λ → 0+. Meanwhile, the rest summands have the order of growth O(1) for uneven d and o(ln 2 λ) for even d, respectively. Hence, taking into account (76) we conclude that for each r ∈ Z d

	o(1/	√ λ) J	), (72) and (73) we note that the first summand at the last sum is ([d/2]-1) 4 ([d/2]-1) (λ; y) = o J 1

The work is partially supported by RFBR grant 10-01-00266.

Lomonosov Moscow State University.

Obviously, the first summand in (85) is o ∞ 0 e -λt t [d/2]-1 (H x,y,0 (∞) -H x,y,0 (t)) dt , as λ → 0+. By formulae [START_REF] Caravenna | Random Walk Models and Probabilistic Techniques for Inhomogeneous Polymer Chains[END_REF], (81), (82) as well as Tauberian theorem (Theorem 2 in [START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF], Ch.13, Sec.5) we infer that the second summand in (85) is also o ∞ 0 e -λt t [d/2]-1 (H x,y,0 (∞) -H x,y,0 (t)) dt , as λ → 0+. Thus, the ([d/2]-1)-th derivative of the second summand in (84) does not make a contribution to the asymptotic behavior of the ([d/2]-1)-th derivative of the left-hand side of (84). Therefore,

]-1 (H x,y,0 (∞) -H x,y,0 (t)) dt where the asymptotic relation for the last expression is provided by formulae (81) and (82). Consequently, employing Corollary 43 in [START_REF] Vatutin | Harris Branching Processes[END_REF] we complete the proof of Theorem 3 when d ≥ 4. Theorem 3 is proved.


Cem Özatalay
email: cozatalay@gsu.edu.tr

How to Make the Economy "Embedded" in Turkey? One question, Two Contradictory

This paper aims to discuss the reasons and the consequences of the conflicts between actors with contradictory interests in the neo-liberal institution-building process in Turkey and privatization cases as one of the main components of the neo-liberalization process represents its primary focus. First, it will be outlined how the privatization program was delayed in Turkey until 2000 while it had been adopted in 1984. Secondly, on the basis of the implementation of some privatization deals after the 2000's and its effects, it will be shown (1) how the import-substitution periods' contending actors, that is to say 'traditional' employers' and workers' organizations, have became allies since the second half of the 1990s against the supporters of the neo-liberal transformation process; (2) how 'former' peasants and artisans who had been frustrated for a while by the import-substitution politics have -certainly through the proletarianization and/or the embourgeoisement processes due to neo-liberalization transformationhenceforth become supporters of the new mode of regulation. Finally, it will be shown that it is possible to distinguish the logic of action of these two camps with respect to the interpretation of the notion of embeddedness: while the first group pursues the Polanyian interpretation of the concept, for whom embeddedness refers to the inscription of economy in social, cultural and political rules that regulate goods and services production and circulation forms, the second group seems to go along with the definition made by the new economic sociology that reflects rather the integration of economic action in social and cultural networks.

Introduction

Neo-liberalism has long been understood as a "monolithic bloc" whose logic of action is based on the principle "more market, less state". According to the critical version of this perception, since its emergence neo-liberalism has not given up his goal to dominate the world in the name of the transnational monopolistic capital as much in economic as in the political and the cultural areas. On the other hand, the version "not hostile" to neoliberalism, as equally totalizing as the critical version, has continued to defend that the more neo-liberalism succeeds in reducing the state, the more markets extend in parallel to individual liberties and political democracies.

Recently we have witnessed the publication of some works that now take into account the diversity of experiences during the implementation of neo-liberal programs in different countries and their various economic, social and political consequences. As this paper aims essentially to discuss the institutional change and transformation of power relations due to the implementation of neo-liberalism in Turkey, it will follow this recent approach focusing on the "variation", more precisely, on the "variegation" of the neo-liberalization processes.

The paper begins, above all, by describing some distinctive patterns of the neoliberalization process in Turkey, addressing in particular the case of the implementation of privatization. Secondly, the main actors of this controversial process and their attitudes will be considered. Finally, the article will try to demonstrate the parallelism between the variety of logics of action of these actors and the various ways of conceiving the notion of embeddedness in economic sociology.

The neo-liberalization "à la turca"

Taking into account local and national specificities and heterogeneity of neoliberalization processes has recently become a growing trend in the social sciences.

In this context, the comparative study conducted by Marion Fourcade-Gourinchas and Sarah Babb (Fourcade-Gourinchas and Babb, 2002: 533-579) on the various processes of the implementation of neo-liberalism in four countries (Chile, Great Britain, France and Mexico) provides a rich data. According to the results of this study there are three interrelated political-economic variables that distinguish each country's neo-liberalization process: (1) size of the annual GDP of each country during the Keynesian period, (2) the volume of social unrest and ability of each nation state to control these conflicts, (3) the rate of inflation. (Fourcade-Gourinchas and Babb, 2002: 538-539).

After analyzing the data according to these variables, Fourcade-Gourinchas and Babb concluded that in Chile and Great Britain, the low growth rates, ongoing social conflicts, and hyperinflation push large fractions of capital and labor to oppose the Keynesian state.

As these fractions gained control over the executive, whether through military (Pinochet) or democratic (Thatcher) way, they opened the state administration to new experts who were known by their militant stance against inflation-the monetarists. On the other hand, the Mexican and French moves to freer markets occurred later and in a much less revolutionary and more pragmatic manner.

When the case of Turkey is examined by referring to the variables used by Fourcade-Gourinchas and Babb, at first sight one can notice that Turkey's neo-liberalization experience is more similar with Britain and Chilean cases than French or Mexican's ones.

Because in Turkey the neo-liberalization process begins, just in the same way as the Chilean case, by a military coup-d'état carried out in order to put an end to political instability and social disorder related to struggles that was going on in the assembly (between the Kemalist-left and conservative-right parties) as well as in the streets (between revolutionary left and ultranationalist right). And following the 1980 coupd'état this process carried on, as in Great Britain, with a vigorous ideological attack on mass media against labor unions and social subsidies.

But on the other hand, in Turkey, import-substituting industrialization (ISI) was not defeated as crushingly as in Chile1 and, at least until 1980 coup-d'état, developmentalist ideology and Keynesian-ISI experience2 was not questioned by social and political actors with the same rigor as in Great Britain.

While the growth rates of Chilean and Britain economies between 1961 and 1974 were respectively 2.3 percent and 2.7 percent the Turkish economy has grown up between 1963 and 1977 with an average annual rate 6.7 percent (Sönmez, 1982: 61). This increasing trend of Turkish ISI model has turned bad only on the three last years of the 70s, when Turkish people were more preoccupied with politic instability than the defeat of the ISI model. Also, the military and civilian bureaucracy in Turkey has never been willing to leave the economy in the hands of the neo-liberal politicians and technocrats at least until the 2000's. This fact clearly shows that there is a discrepancy between the views of Turkish and Chilean armies concerning the neo-liberal transformation process. So, it is known that General Pinochet had transferred the direction of the economy entirely to the Chicago Boys after 1975. Therefore, Chile completed his neo-liberal transformation process in six years, while Turkey was still struggling with privatizations 25 years after the 1980 coup-d'état.

All this does not mean that the neo-liberalization process has not begun in Turkey after the 1980 coup-d'état, when the import-substituting industrialization strategy was replaced by export-oriented industrialization, but only to say that the Turkish army, although it adopted the neo-liberal agenda in broad terms, still had some reservations about the implementation of the transition. The army's reserve was especially about the privatization which means the transformation of power relations to the detriment of the given status quo. And these reserves will become more pronounced, especially after the mid-1990 when neo-liberalism is linked with the process of "globalization".

Two moments of neoliberalization and privatizations in Turkey

Jamie Peck and Adam Tickell (2002: 38) in their well known article identify two phases of neo-liberalization: qualified as "roll-back" neo-liberalism the first phase is characterized by the "destruction and discreditation of Keynesianwelfarist and socialcollectivist institutions"; and the second one called as "roll-out" neo-liberalism, is characterized by "neoliberalized economic management and market-friendly state building process".

Although there are some criticisms about the ambiguity of defining the roll-back and rollout distinction (Graefe,97), I found it quite heuristic to demonstrate the characteristics of the Turkish neo-liberalization process and especially its privatization pillar.

Privatizations in Turkey during the roll-back moment of neoliberalization: Resistance of the state bureaucracy

In Turkey, contrary to the opinion of the junta, the post-puch 1983 parliamentary election is carried by Motherland Party then led by Turgut Özal. The latter was without doubt a Chicago Boy, a strong advocate of monetarism, a rigorous believer of cuts in public expenditure and also a supporter of de-unionization. In other words, he was the man par excellence of "roll-back" neo-liberalism. Turgut Özal remained in power until 1991 ensuring 5.2 percent of average economic growth. As a believer of neoclassical liberalism, he gave great importance to sweep away the cultural patterns shaped by the "decommodification areas" -or market-free areas-of the Turkish capitalism's corporatist structure. In order to accomplish this task, Özal governments always strove to replace the "clientelism" to "state paternalism" as the "individual interest" to "collective passions" in the sphere of social relations.

In the same mindset, the privatization program is seen by the Özal government as a crucial tool to reduce the strong state presence in the economy. Because the presence of the Turkish state in the industrial sectors, especially so-called "strategic" ones, amounted to a rate of about 40 percent in 1960 and 50 percent in 1970 (Boratav, 1986 : 125).

Therefore, privatization should be placed in the heart of the neo-liberalization process in Turkey and in the wake of the cliché "less state, more market" the implementation of the privatization program is triggered with the adoption of a first legal framework relating the privatization (Law No. 2983) in 1984.

While the first privatization sale is carried out in 1986, just after the building of the Privatization Administration (PA), the implementation of the privatization program has been proceeding very slowly, at least until the 2000s. Between 1986 and 2000, only a few small and medium state companies had been privatized by producing an income of $ 4.6 billion.

As a result, we can note that a major reason of the delay of the privatization process in Turkey was the state bureaucracy's -with its military and civilian pillars-tendency to resist against the privatization in order to avoid losing control over the direction of the economy and as a result of the country.

Undoubtedly the most effective opponent actor of this period was the Supreme Court. As a result of the weakness of the existing legal and institutional framework necessary for privatization, governments' steps had been mostly canceled and blocked by supreme courts.

But it is clear that behind the opposition of the civilian and military bureaucracy to the privatization, there was a historical aversion to the market, or to be precise, to the domination of logic of market.

In fact, the "marketophobia" of the Turkish state bureaucracy goes back to 1930s. At this time, the Kemalist single -party regime substituted 1923-29 liberal mode of economic regulation by a centrally planned "dirigiste" one. This change has occurred without a doubt out of obligation due to the 1929 Great Depression. In other words, the reaction of Turkey against "The Great Transformation" was the institutionalization of anti-market approach within the state bureaucracy.

To depict the intellectual climate of the 1930s in Turkey, the characterization of Wilhelm Röpke who is one of the founders of the "social market economy approach" is exemplary. In his article published in 1934, Röpke described Turkish intellectuals who gathered around the Kadro magazine in the 30s in this way: "The current trend of industrialization in Turkey (…) is strongly influenced by the view of an intellectual circle in Ankara whose members are gathered around the "Kadro magazine" that seem to "Die Tat" circle in Germany. But unlike "Die Tat" circle, "Kadro" is at the same time inspired by the Russian anti-liberal tradition. In other words, the Turkish Kadro movement is extremely nationalistic and anti-capitalist just like "Die Tat" circle but at the same time, it has been also inspired of the Russian model which is both technicalrationalist and anti-romantic."

The aversion to markets proper to the 30s sprit is reflected in the distinction made by Werner Sombart in 1915 between the Hero and the Merchant. According to Sombart's distinction, unlike the selfish logic of action of "the Merchant" who never stops talking about the "rights", "the Hero", is always putting forward the virtues such as courage, obedience and altruism insists on the priority of the "tasks".

It is obvious that senior members of the Turkish state bureaucracy, especially those of the Supreme Court have resisted against the intrusion of the market during the neoliberalization process with a kind of heroic motivation. And this heroic resistance shapes its ideological discourse by an introversive nationalistic rhetoric which assumes that privatization is part of an attack directed by global actors and its domestic collaborators against Turkey and its independent national state.

Since the neo-liberalization process is implemented in this period by setting up a standard "structural adjustment" program imposed top-down by the IMF, the rhetoric used by the anti-privatization coalition had found support mainly among some of the population who had been already accustomed to the requirements of Keynesian-ISI mode of regulation. But, while this resistance strategy has been proved to be successful, at least across the "roll-back" moment of neo-liberalization process, it is failed with the transition to "rollout" neo-liberalism. Because, following the last years of 90's decade, the contradiction between the "double movement" of the capitalism has taken a new shape in Turkey.

Transition to roll-out neo-liberalism and the emergence of Islamic entrepreneurship

The metamorphosis of neo-liberalization process was not only a consequence of the failure of the top-down "structural adjustment" program of the "roll-back" neoliberalization process, but occurred also due to the globalization process and its intercommunity-based global economic integration tendency.

Taking into account this new reality the managers of "global institutions" had to redesign the neo-liberalization program and for its implementation [they had] to find out the new local partners whose emergence was in fact much to do with the results of the roll-back moment of neo-liberalism. In the case of Turkey, new partners of "roll-out" neoliberalism were represented mainly by the "Anatolian Tigers" composed essentially by small and medium enterprises (SMEs) whose owners by a majority has adopted more or less Islamic religious values.

Due to the fact that the economic activities of SME employers are embedded in business and social community networks in the Anatolian cities, a process of production for external markets in parallel with a capital accumulation process were triggered with a high speed. Anatolian SME employers simultaneously invented an "Islamic ethos" in accordance with the notion of "social capital" proposed for a while by global institutions.

This new ethos is based on the notion of "homo islamicus" which contains both accents "anti-statist" and "anti-oligarchic" echoed even among new proletarians who came in big cities with their traditional and religious values as a result of the massive rural exodus in the 1980s.

The main actor that shaped and spread out this new "Islamic ethos" in Turkey was without a doubt the MUSIAD, acronym for the Independent Industrialists and Businessmen Association. This entrepreneurial organization was established in 1990 as the collective organization of the mostly family-owned SMEs and with its some 3000 members has become within a short time Turkey's largest and widespread business association.

MUSIAD also has proposed that the Islamic ethos of work should be based on the concept of "homo islamicus ". According to Yarar, the founding president of MÜSİAD, (…) homo islamicus is said to be both entrepreneurial and moral (Adas, 2006: 127). In his recent article, Ziya Öniş point out to the fact that 2001 crisis is one of the most important factor accelerating the Turkey's mass-privatization experience and he states: "The 2001 crisis is a deep crisis which has a far-reaching impact in terms of empowering external actors such as the IMF, World Bank and the EU and of breaking down resistance to privatization" (Öniş, 2011: 716). On the other hand, domestically, the process of political stabilization is being consolidated with the establishment of a singleparty government by Justice and Development Party after the 2002 elections. From the pragmatic 'globalist' perspective of the AKP government, privatization of public enterprises was not an ideological issue and Turkey would benefit from closer integration with the global markets. In a similar vein, the MÜSİAD, as an association of religious employers whose emergence was due to export-oriented development strategy, and Hak-İş, as a trade union confederation whose members are mostly conservative and recently proletarianized due to the massive rural exodus of the 1980s., "respond to the challenges of the contemporary world economy and criticize the statist model associated with past development experience of Turkey" (Buğra, 2002: 192).

In such a situation, in the eyes of the "anti-privatization coalition", it was apparently not possible henceforward to delay or block the course of privatization. The main issue of privatization during the post-2001 period was therefore, "to whom to sell" the public enterprises in the industries of strategic importance rather than to keep them as such. And in the absence of a "right" candidate privatization would be canceled by putting forward legal problems. In the presence of a "wrong" candidate, as seen especially in the early years of 2000, legitimate motives -in one way or another -would be explored or invented in order to cancel the auction and to endeavor to train a "right" candidate for the next auction.

The privatization case of Tüpraş, the seventh European refinery, shows clearly how the new strategy of state bureaucracy works during the post-2001 period. First privatization auction concerning "bulk sale" of 65.76 percent of the public shares of Tüpraş held "roll-out" neo-liberalism, that is to say Privatization Administration and Justice and Development Party government with the support of SMEs entrepreneurs and the new proletariat, have united in order to the accomplishment in any case of privatization, whose effect would probably weaken the oligarchic status quo inherited from the importsubstituting development regime.

Conclusion

All along this paper, I tried to show that the neo-liberalization process and privatization in particular has been a matter of disputes in Turkey between the actors who are for neoliberal "institutional change" and those who are for protecting the status quo inherited. But if we look more closely at these two groups in conflict, we see that their goals are formally the same: to make the economy embedded in society. Because both consider the current situation as an anomaly: for the opponents of neoliberal transformation, the economy was in fact embedded in the nation-state's institutions since the 30s, but the neoliberal turn has made it increasingly more disembedded. As for the advocates of the neoliberal transformation, interventionist policies were carried out after 30s, themselves in fact have made the economy disembedded in favor of a minority out of touch with society. For the latter, it is the neo-liberalization process that makes the economy more and more embedded in society.

But it is also clear that, although both camps have agreed for embedding the economy in society, they differ with respect to wherein really embedding it. According to the opposition camp, it would be embedded in the institutional framework proper to the planned economy period when markets depended on the state. In this manner, use of "embeddedness" corresponds largely to Polanyian use of the term. That is to say, it corresponds to the use of "embeddedness as a variable that point out the changing place of the economy in society" (Gemici, 2007: 11). In contrast, according to the actors of the neo-liberalism-friendly camp the economy must be increasingly embedded in cultural and social networks. We see in this case that the Polanyi's second and methodological use of the concept of "embeddedness", which is also inherited by new economic sociology, is interpreted such as "an analytical proposition". Therefore, today's rise of Anatolian

  The rise of the new Islamic bourgeoisie led to the establishment of a new political party and The Justice and Development Party (AKP) was well founded as representative of ascending Islamic business circles by a group of politicians originating from fundamentalist Islamic movement. So, to sum up, at the turn of 2000 the anti-privatization coalition was facing a new situation. As well as the inevitable rise of Islamic business, just on the eve of the 2002 elections the outburst of an economic crisis in 2001 had apparently led the antiprivatization coalition to change tactic. Privatization boom after 2000s and Tüpraş case Between 2000 and 2010, the privatization gains have reached $ 37 billion in Turkey and $ 34.2 billion of this gain made during the Justice and Development Party governments.

Anyway, Allende was come into power in Chile thanks to support of the people affected by the defeat of Chilean ISI model.

Although ISI was not Keynesian per se,Hirschman (1981) has argued that, by emphasizing the role of state investment in economic development, it drew its inspiration from Keynesian thinking(Fourcade- Gourinchas & Babb, 2002: 537).

Tigers and Islamic Work Ethic, is, for this camp, no other thing than of the expression of embeddedness of the economy in society in Turkey.

Finally, to conclude my presentation, I can say that the "scientific" and "ordinary" uses of the "embeddedness" notion, which is apparently proper especially to "great transformation" periods in the history of capitalism, must be treated on their own as "embedded" in the logic of the regime of capital accumulation whom the actors are part.

No need to mention, we have keep in mind the embeddedness of the latter's position in the power relations. Consequently as our case shows privatization process is considered by some actors in Turkey as a way to make more embedded economy in the society, but for the rest as a reason of that the economy becomes more and more disembedded. And these two contradictory "embeddedness" approaches provides a vivid expression of Polanyi's "double movement" during the process of neo-liberalization in the Turkish context.

January 13, 2004. The tender was awarded to a Turkish-Russian consortium (Zorlu -Tafnet / Efremov Kautschuk GMBH ) with a bid of $ 1.3 billion. Turkish partner of the winning consortium, Zorlu Holding was considered one of the most powerful of the "Anatolian Tigers" and the most distinctive feature of the conglomerate was to have close ties with the Justice and Development Party, while in government.

But the achievement of the Tüpraş privatization process was disrupted by the lawsuit brought by the enterprise's representative trade union, Petrol-İş. Due to procedural irregularities, the Ankara Administrative Court first decided to suspend the execution and soon after, to cancel of the sale. As this case shows, the labor union -Petrol-İş -and the judicial authority played a decisive role during the cancellation of the first Tüpras privatization auction.

After the failure of the first attempt concerning "bulk sale" of 65.76 percent of the public shares of Tüpraş, a second attempt was made at 2005. The second tender was awarded to the Koç-Shell consortium, whose 90 percent shares belong to Turkish Koç Holding. What is interesting about this auction was the debate that preceded it about "selling strategic public enterprises to foreigners," which took place in the media until the day of the auction. And in this debate Koç Holding was always considered as one of the "correct candidates" by the advocates of selling public enterprises to the "national capitalists".

And since Tüpraş was bought by Koç Holding which is considered "correct candidate," we have seen the outbreak of the "anti-privatization coalition": while the vast majority of the coalition was satisfied with the result of the privatization auction because of the buyer's identity, only the trade-union of Petrol-İş continued to try to stop the implementation of the privatization. But this time, lawsuits sued by the trade-union in order to cancel the implementation of the privatization were rejected by the Courts. In the same vein with the Tüpraş privatization case, among the 34 public enterprises privatized throughout the year 2005, 33 were bought by consortiums dominated entirely or predominantly by domestic companies, which were credible in the eyes of the state elite.

This case shows that in the post-2000 era, how the actors "in conflict" of the importsubstituting development regime became allies in order to counter institutional change generated by the neo-liberalization process. But on the other hand, the new partners of the