Samuel Herrmann
email: herrmann@iecn.u-nancy.fr

Pierre Vallois
email: vallois@iecn.u-nancy.fr

From persistent random walks to the telegraph noise

We study a family of memory-based persistent random walks and we prove weak convergences after space-time rescaling. The limit processes are not only Brownian motions with drift. We have obtained a continuous but non-Markov process (Zt) which can be easely expressed in terms of a counting process (Nt). In a particular case the counting process is a Poisson process, and (Zt) permits to represent the solution of the telegraph equation. We study in detail the Markov process ((Zt, Nt); t ≥ 0).

1 The setting of persistent random walks.

1) The simplest way to present and define a persistent random walk with value in Z is to introduce the process of its increments (Yt, t ∈ IN). In the classical symmetric random walk case, this process is just a sequence of independent random variables satisfying IP(Yt = 1) = IP(Yt = -1) = 1 2 for any t ≥ 0. Here we shall introduce some short range memory in these increments in order to create the persistence phenomenon. Namely (Yt) is a {-1, 1}-valued Markov chain: the law of Yt+1 given Ft = σ(Y0, Y1, . . . , Yt) depends only on the value of Yt. This dependence is represented by the transition probability π(x, y) = IP(Yt+1 = y|Yt = x) with (x, y) ∈ {-1, 1} 2 :

π = 1 -α α β 1 -β 0 < α < 1, 0 < β < 1.
The persistent random walk is the corresponding process of partial sums:

Xt = t i=0
Yi with X0 = Y0 = 1 or -1.

(1.1)

Let us discuss two particular cases:

• If α + β = 1, then increments are independent and therefore the short range memory disappears. (Xt, t ∈ IN) is a classical Bernoulli random walk.

• The symmetric case α = β was historically suggested by Fürth [START_REF] Fürth | Schwankungerscheinungen in der Physik[END_REF] and precisely defined by Taylor [START_REF] Taylor | Diffusion by continuous movements[END_REF]. Goldstein [START_REF] Goldstein | On diffusion by discontinuous movements, and on the telegraph equation[END_REF] developed the calculation of the random walk law and clarified the link between this process and the so-called telegraph equation. Some nice presentation of these results can be found in Weiss' book [START_REF] Weiss | Aspects and applications of the random walk. Random Materials and Processes[END_REF] and [START_REF] Weiss | Some applications of persistent random walks and the telegrapher's equation[END_REF]. This particular short memory process is often called either persistent or correlated random walk or Kac walks (see, for instance, [START_REF] Eckstein | The mathematics of suspensions: Kac walks and asymptotic analyticity[END_REF]). An interesting presentation of different limiting distributions for this correlated random walk has been given by Renshaw and Henderson [START_REF] Renshaw | The correlated random walk[END_REF].

2) Recently, Vallois and Tapiero [START_REF] Vallois | Memory-based persistence in a counting random walk process[END_REF] studied the influence of the persistence phenomenon on the first and second moments of a counting process whose increments takes their values in {0, 1} instead of {-1, 1}. They obtained some nearly linear behaviour for the expectation.

Using the transformation y → 2y -1, it is easy to deduce that, in our setting, we have:

IE-1[Xt] := IE[Xt|X0 = Y0 = -1] = α -β 1 -ρ (t + 1) - 2α (1 -ρ) 2 (1 -ρ t+1 ).
(1.2)

IE+1[Xt] := IE[Xt|X0 = Y0 = +1] = α -β 1 -ρ (t + 1) - 2β (1 -ρ) 2 (1 -ρ t+1 ).
(1.3)

An application to insurance has been given in [START_REF] Vallois | A claims persistence process and insurance[END_REF].

It is actually possible to determine the moment generating function (see Proposition 6.4 in Section 6). Φ(λ, t) = IE[λ Xt ], (λ ∈ R * + ). However it seems difficult to invert this transformation; i.e. to give the law of Xt.

3) This leads us to investigate limit distributions. It is well-known that the correctly normalized symmetric random walk converges towards the Brownian motion. Let us define the time and space normalizations. Let α0 and β0 denote two real numbers satisfying:

0 ≤ α0 ≤ 1, 0 ≤ β0 ≤ 1.
(1.4)

Let ∆x be a positive small parameter so that:

0 ≤ α0 + c0∆x ≤ 1, 0 ≤ β0 + c1∆x ≤ 1, (1.5)
where c0 and c1 belong to R (see in subsection 6.2 the allowed range of parameters). Let (Yt, t ∈ IN) be a Markov chain whose transition probabilities are given by the matrix:

π ∆ = 1 -α0 -c0∆x α0 + c0∆x β0 + c1∆x 1 -β0 -c1∆x . (1.6)
Let (Xt, t ∈ IN) be the random walk associated with (Yt) (cf. (1.1)). Define the normalized random walk (Z ∆ s , s ∈ ∆t IN) by the relation:

Z ∆ s = ∆xX s/∆t , (∆t > 0, ∆x > 0).

(1.7) Set ( Z∆ s , s ≥ 0) the continuous time process obtained by linear interpolation of (Z ∆ s ). We introduce two essential parameters: ρ0 = 1 -α0 -β0 (the asymmetry coefficient), (1.8) η0 = β0 -α0.

(1.9)

In this paper, we will aim at showing the existence of a normalization (i.e. to express ∆t in terms of ∆x) which depends on α0, β0, so that ( Z∆ s ) converges in distribution, as ∆x → 0. Our main results and the organization of the paper will be given in Section 2.

The main results

2.1 Case : ρ 0 = 1 Obviously ρ0 = 1 implies that α0 = β0 = 0, and the transition probabilities matrix is given as

π ∆ = 1 -c0∆x c0∆x c1∆x 1 -c1∆x (c0, c1 > 0).
In order to describe the limiting process, we introduce a sequence of independent identically exponentially distributed random variables (en, n ≥ 1) with IE[en] = 1. We construct the following counting process:

N c 0 ,c 1 t = k≥1 1 {λ 1 e 1 +λ 2 e 2 +...+λ k e k ≤t} , (2.1)
where

λ k = 1/c0 if k is odd 1/c1 otherwise. (2.2)
Finally we define

Z c 0 ,c 1 t = t 0 (-1) N c 0 ,c 1 u du. (2.3)
For simplicity of notations, in the symmetric case (i.e. c0 = c1), N c 0 t (resp. Z c 0 t ) will stand for N c 0 ,c 0 t (resp. Z c 0 ,c 0 t ). The process (Z c 0 t ) has been introduced by Stroock (in [START_REF] Stroock | Lectures on topics in stochastic differential equations[END_REF] p. 37). It is possible to show that if we rescale (Z c 0 t ), this process converges in distribution to the standard Brownian motion. This property has been widely generalized. For instance Bardina and Jolis [START_REF] Bardina | Weak approximation of the Brownian sheet from a Poisson process in the plane[END_REF] have given weak approximation of the Brownian sheet from a Poisson process in the plane.

Theorem 2.1. Let ∆x = ∆t and Y0 = X0 = -1. Then the interpolated persistent random walk ( Z∆ s , s ≥ 0) converges in distribution, as ∆x → 0, to the process (-Z c 0 ,c 1 s , s ≥ 0). In particular if c0 = c1, then (N c 0 u ) is the Poisson process with parameter c0. If Y0 = X0 = 1 then the interpolated persistent random walk ( Z∆ s , s ≥ 0) converges in distribution, as ∆x → 0, to the process (Z c 1 ,c 0 s , s ≥ 0).

Proof. See Section 4.

Next, in Section 3, we investigate the process (Z c 0 ,c 1 t , N c 0 ,c 1 t ; t ≥ 0). In particular we prove that it is Markov, we determine its semigroup and the law of (Z c 0 ,c 1 t , N c 0 ,c 1 t ), t being fixed. This permits to prove, when c0 = c1, the well-known relation (cf. [START_REF] Weiss | Some applications of persistent random walks and the telegrapher's equation[END_REF], [START_REF] Eckstein | The mathematics of suspensions: Kac walks and asymptotic analyticity[END_REF], [START_REF] Goldstein | On diffusion by discontinuous movements, and on the telegraph equation[END_REF], [START_REF] Griego | Theory of random evolutions with applications to partial differential equations[END_REF]) between the solutions of the wave equation and the telegraph equation. For this reason the process (Z c 0 ,c 1 t ) will be called the integrated telegraph noise (ITN for short). We emphasize that our approach based on stochastic processes gives a better understanding of analytical properties. We will give in Section 5 below two extensions of Theorem 2.1 to the cases where (Yt) is 1) a Markov chain which takes its values in {y1, . . . , y k }, 2) a Markov chain with order 2 and valued in {-1, 1}.

Case : ρ 0 = 1

In this case, the limit process is Markov. We shall prove two kind of convergence results. The first one corresponds to the law of large numbers and the second one looks like functional central limit theorem. Recall that ( Z∆ t , t ≥ 0) is the linear interpolation of (Z ∆ t ) and ρ0 (resp. η0) has been defined by (1.8) (resp. (1.9)).

Theorem 2.2. 1) Suppose that r∆t = ∆x with r > 0. Then Z∆ t converges to the deterministic limit -rtη 0 1-ρ 0 when ∆x → 0. 2) Suppose that r∆t = ∆ 2

x with r > 0, then the process (ξ ∆ t , t ≥ 0) defined by

ξ ∆ t = Z∆ t + t √ rη0 (1 -ρ0) √ ∆t
converges in distribution to the process (ξ 0 t , t ≥ 0), as ∆x → 0, where

ξ 0 t = 2r -τ 1 -ρ0 + η0τ (1 -ρ0) 2 t + r(1 + ρ0) 1 -ρ0 1 - η 2 0 (1 -ρ0) 2 Wt, (2.4)
(Wt, t ≥ 0) is a one-dimensional Brownian motion, τ = (c0 + c1)/2 and τ = (c1 -c0)/2.
Proof. See Section 6.

Gruber and Schweizer have proved in [START_REF] Gruber | A diffusion limit for generalized correlated random walks[END_REF] a weak convergence result for a large class of generalized correlated random walks. However these results and ours can be only compared in the case α0 = β0.

Note that 1 -η 2 0

(1 -ρ0) 2 = 0 ⇐⇒ α0 = 0 or β0 = 0. Suppose for instance that α0 = 0. Then β0, c0 > 0 and

ξ ∆ t = Z∆ t + t √ r √ ∆t and ξ 0 t = 2rc0 β0 t.
Obviously, the diffusion coefficient of (ξ 0 t ) can also cancel when ρ0 = -1. Since ρ0 = -1 ⇐⇒ α0 = β0 = 1, then c0, c1 < 0 and

ξ ∆ t = Z∆ t and ξ 0 t = -rτ t.
This shows that, in the symmetric case (i.e. c0 = c1), we have ξ 0 t = 0. This means that the normalization is not the right one since the limit is null. Changing the rescaling we can obtain a non-trivial limit.

Proposition 2.3. Suppose α0 = β0 = 1, c0 = c1 < 0 and r∆t = ∆ 3

x with r > 0. The interpolated persistent walk ( Z∆ t , t ≥ 0) converges in law, as ∆x → 0, to ( √ -rc0Wt, t ≥ 0) where (Wt) is a standard Brownian motion.

Proof. See subsection 6.3

Organization of the paper

The third section presents few properties of the process (Z c 0 ,c 1 t , t ≥ 0) which has been defined by (2.3). Theorem 2.1 will be proven in Section 4. Section 5 will be devoted to two extensions of Theorem 2.1. In subsection 6.1 we determine the generating function of Xt (recall that Xt has been defined by (1.1)). This is the main tool which permits to prove Theorem 2.2 and Proposition 2.3 (see subsections 6.2 and 6.3).

Properties of the integrated telegraph noise

The aim of this section is to study the two dimensional process (Z c 0 ,c 1 t , N c 0 ,c 1 t ; t ≥ 0) introduced in (2.2) and (2.3). In the particular symmetric case c0 = c1, the study is simpler since the process (N c 0 t , t ≥ 0) is a Poisson process with rate c0 (IE(N c 0 t ) = c0t) and N c 0 0 = 0. However we shall study the general case.

First, we determine in Proposition 3.1 the conditional density of Z c 0 ,c 1 t given N c 0 ,c 1 t = n. As a by product we obtain the distribution of Z c 0 ,c 1 t (see Proposition 3.3). Second, we prove in Proposition 3.5 that (Z c 0 ,c 1 t , N c 0 ,c 1 t , t ≥ 0) is Markov and we determine its semi-group. We conclude this section by showing that the solution of the telegraph equation can be expressed in terms of the associated wave equation and (Z c 0 ,c 0 t ) t≥0 . For this reason, (Z c 0 ,c 1 t ) t≥0 will be called the integrated telegraph noise (ITN for short). Recall that:

τ = c0 + c1 2 , τ = c1 -c0 2 . (3.1) Proposition 3.1. 1) IP(N c 0 ,c 1 t = 0) = e -tc 0 and given N c 0 ,c 1 t = 0, we have Z c 0 ,c 1 t = t.
2) The counting process takes even values with probability:

IP(N c 0 ,c 1 t = 2k) = (c0c1) k α k (t) 2 2k k!(k -1)! e -τ t with α k (t) = t -t (t -z) k-1 (t + z) k e τ z dz, (3.2)
and the conditional distribution of Z c 0 ,c 1 t is given by

IP(Z c 0 ,c 1 t ∈ dz|N c 0 ,c 1 t = 2k) = 1 α k (t) (t -z) k-1 (t + z) k e τ z 1 [-t,t] (z) (k ≥ 1). (3.3)
3) The counting process takes odd values with probability:

IP(N c 0 ,c 1 t = 2k + 1) = c k+1 0 c k 1 αk (t) 2 2k+1 (k!) 2 e -τ t with αk (t) = t -t (t -z) k (t + z) k e τ z dz, (3.4)
and the conditional distribution of Z c 0 ,c 1 t is given by

IP(Z c 0 ,c 1 t ∈ dz|N c 0 ,c 1 t = 2k + 1) = 1 αk (t) (t -z) k (t + z) k e τ z 1 [-t,t] (z) (k ≥ 0). (3.5)
Corollary 3.2. In the particular symmetric case c0 = c1, the conditional density function of Z c 0 t given N c 0 t = n is the centered beta density, i.e.

for n = 2k, k ∈ N * : fn(t, z) = χ 2k (t + z) k (t -z) k-1 t 2k 1 [-t,t] (z), (3.6
)

for n = 2k + 1, k ∈ N : fn(t, z) = χ 2k+1 (t + z) k (t -z) k t 2k+1 1 [-t,t] (z), (3.7)
with

χ 2k+1 = χ 2k+2 = 1 2 2k+1 B(k + 1, k + 1) = (2k + 1)! 2 2k+1 (k!) 2 (k ≥ 0),
(B is the beta function (first Euler function): B(r, s) = Γ(r)Γ(s) Γ(r+s) ). Proof of Proposition 3.1. Associated with n ≥ 0 and a bounded continuous function f , we define ∆n(f Hence

) = IE f (Z c 0 ,c 1 t )1 {N c 0 ,c 1 t =n} . a) When n = 0, we obtain ∆0(f ) = IE f (Z c 0 ,c 1 t )1 {t<λ 1 e 1 } . If t < λ1e1, then Z c 0 ,c 1 t = t and ∆0(f ) = f (t) IP(t < λ1e1) = f (t)e -tc
Z c 0 ,c 1 t = λ1e1 -λ2e2 + λ3e3 + . . . + (-1) n-1 λnen + (-1) n (t -λ1e1 -. . . -λnen). (3.8) c) Evaluation of ∆ 2k (f ), k ≥ 1.
We introduce two sequences of random variables associated with (en):

ξ e k = e2 + . . . + e 2k , ξ o k = e1 + . . . + e 2k-1 , (k ≥ 1). ( 3
∆ 2k (f ) = 1 ((k -1)!) 2 Dt exp{-c0(t -y/c0 -x/c1)}f (t -2x/c1)e -x-y x k-1 y k-1 dx dy = c k 0 e -tc 0 k!(k -1)! tc 1 0 f (t -2x/c1)x k-1 (t -x/c1) k exp c0 c1 -1 x dx,
where Dt = R 2 + ∩ {y/c0 + x/c1 ≤ t}. Using the change of variable z = t -2x/c1, we obtain

x = c1 t-z 2 , t -x/c1 = t+z 2 and ∆ 2k (f ) = (c0c1) k 2 e -(c 0 +c 1 )t/2 k!(k -1)! t -t f (z) t -z 2 k-1 t + z 2 k exp{(c1 -c0)z/
Z c 0 ,c 1 t = 2ξ o k+1 /c0 -t; this implies ∆ 2k+1 (f ) = IE 1 {ξ o k+1 /c 0 +ξ e k /c 1 ≤t} exp -c1(t -ξ o k+1 /c0 -ξ e k /c1) f (2ξ o k+1 /c0 -t) .
Since ξ o k+1 and ξ e k are independent and gamma distributed with parameter k + 1 (resp. k), we get

∆ 2k+1 (f ) = c k+1 0 c k 1 2(k!) 2 e -(c 0 +c 1 )t/2 t -t f (z) t -z 2 k t + z 2 k exp (c1 -c0)z/2 dz. (3.11)
This leads directly to (3.4) and (3.5).

Let us recall the definition of the modified Bessel functions:

Iν(ξ) = m≥0 (ξ/2) ν+2m m!Γ(ν + m + 1) . Proposition 3.3. The distribution of Z c 0 ,c 1 t is given by IP(Z c 0 ,c 1 t ∈ dx) = e -c 0 t δt(dx) + e -τ t f (t, x)1 [-t,t] (x), (3.12)
where

f (t, x) = 1 2 c0c1(t + x) t -x I1 c0c1(t 2 -x 2 ) + c0I0 c0c1(t 2 -x 2 ) e τ x . (3.13)
Remark 3.4. Let us focus our attention to the symmetric case c0 = c1. We can introduce some randomization of the initial condition as follows: let ǫ be a {-1, 1}-valued random variable, independent from the Poisson process N c 0 t , with p := IP(ǫ = 1) = 1 -IP(ǫ = -1). It is easy to deduce from (3.12) that we have

IP(ǫZ c 0 t /t ∈ dx) = pδ1(dx) + (1 -p)δ-1(dx) + g(t, x)dx e -c 0 t , (3.14)
with g(t, x) = c0t 2 I0 c0t 1 -x 2 + 1 + (2p -1)x √ 1 -x 2 I1 c0t 1 -x 2 1 [-1,1] (x)
and δ1(dx) (resp. δ-1(dx)) is the Dirac measure at 1 (resp. -1).

In the particular case p = 1/2, x → g(t, x) is an even function. G.H. Weiss ([18] p.393) proved (3.14) using an analytic method based on Fourier-Laplace transform.

Proof of Proposition 3.3. The proof is a direct consequence of the expression of Proposition 3.1. Indeed, for each bounded continuous function ϕ we denote

∆ = IE[ϕ(Z c 0 ,c 1 t )] = ϕ(t)e -c 0 t + k≥1 ∆ 2k (ϕ) + k≥0 ∆ 2k+1 (ϕ) = ϕ(t)e -c 0 t + ∆e + ∆o, where ∆n(ϕ) = IE[ϕ(Z c 0 ,c 1 t )1 {N c 0 ,c 1 t =n} ]. Using (3.2) and (3.3) we get ∆e = e -τ t t -t ϕ(z)Se(z)e τ z dz, with Se(z) = 1 2 k≥1 (c0c1) k k!(k -1)! t -z 2 k-1 t + z 2 k = 1 2 √ c0c1 t + z t -z k≥0 1 k!(k + 1)! c0c1(t 2 -z 2 ) 2 2k+1 = 1 2 √ c0c1 t + z t -z I1 c0c1(t 2 -z 2 ) .
For the odd indexes, by (3.4) and (3.5) we get

∆o = e -τ t t -t ϕ(z)So(z)e τ z dz, with
So(z) = 1 2 k≥0 c k+1 0 c k 1 (k!) 2 t 2 -z 2 4 k = c0 2 I0 c0c1(t 2 -z 2 ) . Proposition 3.5. 1) (Z c 0 ,c 1 t , N c 0 ,c 1 t ; t ≥ 0) is a R × IN-valued Markov process. 2) Let s ≥ 0 and n ≥ 0. Conditionally on Z c 0 ,c 1 s = x and N c 0 ,c 1 s = n, (Z c 0 ,c 1 t+s , N c 0 ,c 1 t+s ), t ≥ 0 is distributed as      x + t 0 (-1) N c 0 ,c 1 u du, n + N c 0 ,c 1 t , t ≥ 0 when n is even, x - t 0 (-1) N c 1 ,c 0 u du, n + N c 1 ,c 0 t , t ≥ 0 otherwise.
Remark 3.6. Note that Propositions 3.5 and 3.1 permit to determine the semigroup of

(Z c 0 ,c 1 t , N c 0 ,c 1 t ), t ≥ 0 i.e. IP(Z c 0 ,c 1 t ∈ dx, N c 0 ,c 1 t = n|Z c 0 ,c 1 s = y, N c 0 ,c 1 s = m) where t > s, n ≥ m and y ∈ [-s, s].
Proof of Proposition 3.5. Let t > s ≥ 0. Using (2.3) we get

Z c 0 ,c 1 t = Z c 0 ,c 1 s + (-1) N c 0 ,c 1 s t-s 0 (-1) Ñs u du, where Ñ s u = N c 0 ,c 1 s+u -N c 0 ,c 1 s , u ≥ 0. Note that ( Ñ s u ; u ≥ 0) (d) = (N c 0 ,c 1 u ; u ≥ 0) if N c 0 ,c 1 s ∈ 2 IN and ( Ñ s u ; u ≥ 0) (d) = (N c 1 ,c 0 u ; u ≥ 0) if N c 0 ,c 1 s ∈ 2 IN +1. This shows Proposition 3.5.
Next, we determine (in Proposition 3.8 below) the Laplace transform of the r.v. Z c 0 ,c 1 t . It is possible to use the distribution of Z c 0 ,c 1 t (cf Proposition 3.3), but this method has the disadvantage of leading to heavy calculations. We develop here a method which uses the fact that (Z c 0 ,c 1 s ; s ≥ 0) is a stochastic process given by (2.3). The key tool is Lemma 3.7 below. Roughly speaking Lemma 3.7 gives the generator of the Markov process (Z c 0 ,c 1 t , N c 0 ,c 1 t ). Lemma 3.7 is an important ingredient in the proof of Proposition 3.11 besides.

Lemma 3.7. Let F : R × IN → R denote a bounded and continuous function such that z → F (z, n) is of class C 1 for all n. Then d dt IE[F (Z c 0 ,c 1 t , N c 0 ,c 1 t )] = IE ∂F ∂z (Z c 0 ,c 1 t , N c 0 ,c 1 t )(-1) N c 0 ,c 1 t + IE F (Z c 0 ,c 1 t , N c 0 ,c 1 t + 1) -F (Z c 0 ,c 1 t , N c 0 ,c 1 t ) × c11 {N c 0 ,c 1 t ∈2 IN +1} + c01 {N c 0 ,c 1 t ∈2 IN} . (3.15) Proof. Let us denote by ∆(t) = IE[F (Z c 0 ,c 1 t , N c 0 ,c 1 t )].
In order to compute the t-derivative we shall decompose the increment of t → ∆(t) in a sum of two terms:

∆(t + h) -∆(t) h = B h + C h , with B h = 1 h IE[F (Z c 0 ,c 1 t+h , N c 0 ,c 1 t+h )] -IE[F (Z c 0 ,c 1 t , N c 0 ,c 1 t+h )] , C h = 1 h IE[F (Z c 0 ,c 1 t , N c 0 ,c 1 t+h )] -IE[F (Z c 0 ,c 1 t , N c 0 ,c 1 t )] .
Since F (•, n) is continuously differentiable with respect to the variable z and t → Z c 0 ,c 1 t is differentiable (cf (2.3)), using the change of variable formula we obtain

1 h F (Z c 0 ,c 1 t+h , N c 0 ,c 1 t+h ) -F (Z c 0 ,c 1 t , N c 0 ,c 1 t+h ) = 1 h t+h t ∂F ∂z (Z c 0 ,c 1 u , N c 0 ,c 1 t+h )(-1) N c 0 ,c 1 u du.
Therefore lim

h→0 B h = IE ∂F ∂z (Z c 0 ,c 1 t , N c 0 ,c 1 t )(-1) N c 0 ,c 1 t . (3.16)
In order to study the limit of C h , we consider two cases:

N c 0 ,c 1 t ∈ 2 IN and N c 0 ,c 1 t ∈ 2 IN +1: C h = 1 h IE F (Z c 0 ,c 1 t , N c 0 ,c 1 t + Ñ c 1 ,c 0 h ) -F (Z c 0 ,c 1 t , N c 0 ,c 1 t ) 1 {N c 0 ,c 1 t ∈2 IN +1} + 1 h IE F (Z c 0 ,c 1 t , N c 0 ,c 1 t + Ñ c 0 ,c 1 h ) -F (Z c 0 ,c 1 t , N c 0 ,c 1 t ) 1 {N c 0 ,c 1 t ∈2 IN} , where Ñh = N c 0 ,c 1 t+h -N c 0 ,c 1 t . According to Proposition 3.5, conditionally on Z c 0 ,c 1 t and N c 0 ,c 1 t ∈ 2 IN (resp. N c 0 ,c 1 t ∈ 2 IN +1), Ñh is distributed as N c 0 ,c 1 h (resp. N c 1 ,c 0 h ). Note that Proposition 3.1 implies that IP(N c 0 ,c 1 h ≥ 2) = o(h) and IP(N c 0 ,c 1 h = 1) = c0 2 e τ h -e -τ h τ e -τ h = c0h + o(h). Consequently lim h→0 C h = c1 IE F (Z c 0 ,c 1 t , N c 0 ,c 1 t + 1) -F (Z c 0 ,c 1 t , N c 0 ,c 1 t ) 1 {N c 0 ,c 1 t ∈2 IN +1} + c0 IE F (Z c 0 ,c 1 t , N c 0 ,c 1 t + 1) -F (Z c 0 ,c 1 t , N c 0 ,c 1 t ) 1 {N c 0 ,c 1 t ∈2 IN} . (3.17)
Then, (3.16) and (3.17) clearly imply Lemma 3.7.

Let us introduce the two quantities:

Le(t) = IE e -µZ c 0 ,c 1 t 1 {N c 0 ,c 1 t ∈2 IN} and Lo(t) = IE e -µZ c 0 ,c 1 t 1 {N c 0 ,c 1 t ∈2 IN +1} , (t ≥ 0, µ ∈ R). (3.18) Since |Z c 0 ,c 1 t | ≤ t,
then Le(t) and Lo(t) are well defined for any µ ∈ R. Note that µ → Le(t) (resp. µ → Lo(t)) is a Laplace transform. We have mentioned the t-dependency only because it will play an important role in our proof of Proposition 3.8 below. Proposition 3.8. Let Le(t) and Lo(t) be defined by (3.18). Then

Le(t) = 1 √ E (-µ + τ ) sinh(t √ E) + √ E cosh(t √ E) e -τ t , (3.19
)

Lo(t) = c0 √ E sinh(t √ E)e -τ t , (3.20) IE[e -µZ c 0 ,c 1 t ] = 1 √ E (-µ + τ ) sinh(t √ E) + √ E cosh(t √ E) e -τ t , (3.21
)

where E = µ 2 -2τ µ + τ 2 .
Proof. Applying Lemma 3.7 with the particular function F (z, n) = e -µz 1 {n∈2 IN} , we have:

d dt Le(t) = -µ IE e -µZ c 0 ,c 1 t (-1) N c 0 ,c 1 t 1 {N c 0 ,c 1 t ∈2 IN} + E e -µZ c 0 ,c 1 t 1 {N c 0 ,c 1 t ∈2 IN +1} -1 {N c 0 ,c 1 t ∈2 IN} × c11 {N c 0 ,c 1 t ∈2 IN +1} + c01 {N c 0 ,c 1 t ∈2 IN}
We deduce

d dt Le(t) = -(µ + c0)Le(t) + c1Lo(t).
Similarly

d dt Lo(t) = -µ IE e -µZ c 0 ,c 1 t (-1) N c 0 ,c 1 t 1 {N c 0 ,c 1 t ∈2 IN +1} + E e -µZ c 0 ,c 1 t 1 {N c 0 ,c 1 t ∈2 IN} -1 {N c 0 ,c 1 t ∈2 IN +1} × c11 {N c 0 ,c 1 t ∈2 IN +1} + c01 {N c 0 ,c 1 t ∈2 IN} .
We get therefore

d dt Lo(t) = (µ -c1)Lo(t) + c0Le(t).
To sum up

d dt Le(t) Lo(t) = -µ -c0 c1 c0 µ -c1 Le(t) Lo(t) .
We deduce the expressions of Le(t) and Lo(t):

Le(t) = a+e λ + t + a-e λ -t Lo(t) = b+e λ + t + b-e λ -t , (3.22)
where λ± = -τ ± µ 2 -2τ µ + τ 2 = -τ ± √ E. The constants a± and b± are evaluated with the initial conditions:

Le(0) = IP(N c 0 ,c 1 0 ∈ 2 IN) = 1, Lo(0) = IP(N c 0 ,c 1 0 ∈ 2 IN +1) = 0, dLe dt (0) = -(µ + c0)Le(0) + c1Lo(0) = -µ -c0, dLo dt (0) = (µ -c1)Lo(0) + c0Le(0) = c0.
We obtain It is easy to deduce two direct consequences of Proposition 3.8. First, taking µ = 0 we obtain IP(N c 0 ,c 1 t ∈ 2 IN) and IP(N c 0 ,c 1 t ∈ 2 IN +1). Second, taking the expectation in (2.3) we get the mean of Z c 0 ,c 1 t . Corollary 3.9. We have:

a+ = 1 2 √ E (-µ + τ + √ E) and a-= 1 2 √ E (µ -τ + √ E), (3.23
IP(N c 0 ,c 1 t ∈ 2 IN) = 1 τ τ sinh(τ t) + τ cosh(τ t) e -τ t , IP(N c 0 ,c 1 t ∈ 2 IN +1) = c0 τ sinh(τ t)e -τ t ,
and

IE[Z c 0 ,c 1 t ] = τ τ t + c0 2τ 2 (1 -e -2τ t
). Remark 3.10. The Laplace transform with respect to the time variable can also be explicitly computed. We define

F (µ, s) = ∞ 0 e -st IE[e -µZ c 0 ,c 1 t ]dt. Integrating (3.21) with respect to dt we get F (µ, s) = 1 2 √ E √ E + (-µ + τ ) 1 s - √ E + τ + 1 2 √ E √ E -(-µ + τ ) 1 s + √ E + τ = ( √ E -µ + τ )(s + √ E + τ ) + ( √ E + µ -τ )(s - √ E + τ ) 2 √ E((s + τ ) 2 -E ) = 2s √ E + 4τ √ E -2µ √ E 2 √ E((s + τ ) 2 -E ) = s + 2τ -µ (s + τ ) 2 -E
In the symmetric case, E equals µ 2 + c 2 0 , then

F (µ, s) = s + 2c0 -µ s 2 + 2sc0 -µ 2 .
(3.25)

Let (Zt) be the symmetrization of (Z c 0 t ) which is defined by an initial randomization:

Zt = ǫZ c 0 t , t ≥ 0,
where ǫ is independent of Z c 0 t and IP(ǫ = ±1) = 1/2.

Relation (3.25) implies ∞ 0 e -st IE[e -µZt ]dt = s + 2c0 s 2 + 2sc0 -µ 2 .
This identity has been obtained by Weiss in [START_REF] Weiss | Some applications of persistent random walks and the telegrapher's equation[END_REF].

Let us now present a link between the ITN process and the telegraph equation in the particular symmetric case c0 = c1 = c > 0. Recall that (N c t ) is a Poisson process with parameter c. Let f : R → R be a function of class C 2 whose first and second derivatives are bounded. We define

u(x, t) = 1 2 f (x + at) + f (x -at) , x ∈ R, t ≥ 0.
Then (cf [START_REF] Eckstein | The mathematics of suspensions: Kac walks and asymptotic analyticity[END_REF]) u is the unique solution of the wave equation

     ∂ 2 u ∂t 2 = a 2 ∂ 2 u ∂x 2 , u(x, 0) = f (x), ∂u ∂t (x, 0) = 0.
Proposition 3.11. The function

w(x, t) = IE u x, t 0 (-1) N c s ds , (x ∈ R, t ≥ 0)
is the solution of the telegraph equation (TE)

       ∂ 2 w ∂t 2 + 2c ∂w ∂t = a 2 ∂ 2 w ∂x 2 , w(x, 0) = f (x), ∂w ∂t (x, 0) = 0.
This result can be proved using asymptotic analysis applied to difference equation associated with the persistent random walk [START_REF] Goldstein | On diffusion by discontinuous movements, and on the telegraph equation[END_REF] or using Fourier transforms [START_REF] Weiss | Some applications of persistent random walks and the telegrapher's equation[END_REF]. Here we shall present a new proof. Proof of Proposition 3.11. Applying twice Lemma 3.7 to (z, n) → u(x, z) and (z, n) → ∂u ∂t (x, z)(-1) n we obtain:

∂w ∂t (x, t) = IE ∂u ∂t x, t 0 (-1) N c s ds (-1) N c t .
and

∂ 2 w ∂t 2 (x, t) = IE ∂ 2 u ∂t 2 x, t 0 (-1) N c s ds -2c IE ∂u ∂t x, t 0 (-1) N c s ds (-1) N c t .
Since u solves the wave equation we have

∂ 2 w ∂t 2 (x, t) = a 2 ∂ 2 w ∂x 2 (x, t) -2c ∂w ∂t (x, t).
The function w is actually the solution of the telegraph equation. It is easy to prove that w satisfies the boundary conditions.

Let us note that Proposition 3.11 can be extended to the asymmetric case c0 = c1. In this general case the telegraph equation is replaced by a linear system of partial differential equations.

Remark 3.12. 1) In [START_REF] Eckstein | The mathematics of suspensions: Kac walks and asymptotic analyticity[END_REF], [START_REF] Griego | Theory of random evolutions with applications to partial differential equations[END_REF], an extension of Proposition 3.11 has been proved. Let A be the generator of a strongly continuous group of bounded linear operators on a Banach space. If w is the unique solution of this abstract "wave equation":

∂ 2 w ∂t 2 = A 2 w; w(•, 0) = f, ∂w ∂t (•, 0) = Ag (f, g ∈ D(A))
then u(x, t) = IE w x, t 0 (-1) N c s ds solves the abstract "telegraph equation":

∂ 2 u ∂t 2 = A 2 u -2c ∂u ∂t , u(•, 0) = f, ∂u ∂t (•, 0) = Ag.
2) In the same vein as [START_REF] Griego | Theory of random evolutions with applications to partial differential equations[END_REF], Enriquez [START_REF] Enriquez | Correlated processes and the composition of generators[END_REF] has introduced processes with jumps to represent solutions of some linear differential equations and biharmonic equations in the presence of a potential term. Moreover useful references are given in [START_REF] Enriquez | Correlated processes and the composition of generators[END_REF].

3) It is easy to deduce from Lemma 3.7 that the functions

we(x, t) = IE u x, t 0 (-1) N c 0 ,c 1 s ds 1 {N c 0 ,c 1 t ∈2 IN} , (x ∈ R, t ≥ 0) wo(x, t) = IE u x, t 0 (-1) N c 0 ,c 1 s ds 1 {N c 0 ,c 1 t ∈2 IN +1} , (x ∈ R, t ≥ 0)
are solutions of the general telegraph system (TS)

               ∂ 2 we ∂t 2 = (c0c1 -c 2 0 )we + (c0c1 -c 2 1 )wo -2c0 ∂we ∂t + a 2 ∂ 2 we ∂x 2 , ∂ 2 wo ∂t 2 = (c0c1 -c 2 0 )we + (c0c1 -c 2 1 )wo -2c1 ∂wo ∂t + a 2 ∂ 2 wo ∂x 2 , we(x, 0) = f (x), wo(x, 0) = 0 ∂we ∂t (x, 0) = -c0f (x) ∂wo ∂t (x, 0) = c0f (x).
4 Convergence of the persistent walk to the ITN Suppose ρ0 = 1. The aim of this section is to prove the convergence of the interpolated persistent random walk towards the generalized integrated telegraph noise (ITN) i.e. Theorem 2.1. Let us start with preliminary results. First, let us recall that (Xn, n ∈ IN) is the persistent random walk starting in 0 defined by the increments process (Yn, n ∈ IN) (see Section 1) with transition probabilities

π ∆ = 1 -c0∆t c0∆t c1∆t 1 -c1∆t .
Let (T k ; k ≥ 1) be the sign changes sequence of times :

   T1 = inf{t ≥ 1 : Yt = Y0} T k+1 = inf{t > T k : Yt = YT k }; k ≥ 1. (4.1)
We put T0 = 0 and

A k = T k -T k-1 k ≥ 1 (4.2)
Let Nt be the number of times over [0, t] so that the sign of (Yn) changes:

Nt = j≥1 1 {T j ≤t} (4.
3)

The definition of Nt implies that:

Nt = k ⇐⇒ T k ≤ t < T k+1
We suppose in this subsection that Y0 = -1.

We deduce from the identities above:

Xt = k j=1 (-1) j Aj + (-1) k+1 (t -T k + 1)
where k = Nt. We introduce the normalization of (Xn; n ∈ IN) given by (1.7) with ∆x = ∆t:

Z ∆ s = ∆tX s/∆t (s/∆t ∈ IN). (4.6)
Let us define:

N ∆ s = j≥1 1 { j i=1 ∆tA j ≤s} s ≥ 0. (4.7)
Let us note that N ∆ s = N s/∆t if s/∆t ∈ IN . That permits to extend the definition of Z ∆ s to any s ≥ 0 by setting

Z∆ s = k j=1 (-1) j (∆tAj) + (-1) k+1 (s -∆tT k + ∆t) k = N ∆ s . (4.8)
Obviously Z∆ s = Z ∆ s if s/∆t ∈ IN. In order to study the asymptotic behaviour of ( Z∆ s ) as ∆t → 0, we shall first prove the convergence in distribution of (∆tAj) j≥1 and (N ∆ s ) s≥0 . We recall that some random variable ξ is exponentially distributed with parameter λ > 0 if its density is given by 1 λ e -x/λ 1 {x≥0} . Lemma 4.1. The random variables (A k ) are independent and ∆tA 2k (resp. ∆tA 2k+1 ) converges in distribution, as ∆t → 0, to the exponential law with parameter 1 c 1 (resp. 1 c 0 ).

Proof. Since (Yn) is a Markov chain, then the (A k ) are independent. First let us study the convergence in distribution of the sequence ∆tA 2k . We use the Laplace transform of ∆tA 2k : ϕ(µ) = IE[e -µ∆tA 2k ], µ ≥ 0. Since A 2k is geometrically distributed with parameter c1∆t, we obtain

ϕ(µ) = ∞ j=1 e -µ∆tj (1 -c1∆t) j-1 c1∆t = c1∆t e µ∆t -(1 -c1∆t) = c1∆t (µ + c1)∆t + o(∆t) = c1 µ + c1 + o(∆t) (4.9)
The function ϕ(µ) converges for any µ ≥ 0 to the Laplace transform of some exponential law with parameter c -1 1 . This proves the convergence in distribution of ∆tA 2k . Concerning A 2k-1 the arguments are similar.

Let us recall that the counting process (N c 0 ,c 1 t , t ≥ 0) has been defined through the sequence of jumps (en; n ≥ 1) via (2.1), and (en; n ≥ 1) are i.i.d. and exponentially distributed. where ⌊a⌋ denotes the integer part of a.

2) Set k ≥ 1. The event {N ∆ s = k} can be decomposed as follows:

{N ∆ s = k} = ∆t k j=1 Aj ≤ s ∩ ∆t k+1 j=1 Aj > s .
This identity imply existence of a bounded Borel function ψ k : R k+1 → R so that

Φ k (∆tA1, . . . , ∆tA k )1 {N ∆ s =k} = Φ k (∆tA1, . . . , ∆tA k )1 {∆t k j=1 A j ≤s} 1 {∆t k+1 j=1 A j >s} = ψ k (∆tA1, ∆tA2, . . . , ∆tA k+1 ).
Since Φ k is continuous, the discontinuity points of ψ k are included in:

U = x ∈ R k+1 : k j=1 xj = s ∪ x ∈ R k+1 : k+1 j=1 xj = s .
By Lemma 4.1, (∆tA1, . . . , ∆tA k+1 ) converges in distribution towards (λ1e1, . . . , λ k+1 e k+1 ) as ∆t → 0. Since the Lebesgue measure of U is null, the limit law does not charge U. We can conclude evoking for instance Theorem 14 p.247 in [START_REF] Brancovan | Probabilités -Cours et exercices corrigés[END_REF]).

Let us formulate a straightforward generalization of Lemma 4.2.

E∆(k) = E f k j=1 (-1) j λjej + (-1) k+1 s - k j=1 λj ej 1 {N c 0 ,c 1 s =k} = IE[f (-Z c 0 ,c 1 s )1 {N c 0 ,c 1 s =k} ].
Moreover since f is bounded by M , we get

|E∆(k)| ≤ M IP(N ∆ s = k).
Suppose that k ≥ 1. Then, using the Markov inequality and the independence property of the random sequence (An, n ≥ 0), we obtain where ϕj(µ) = IE[e -µ∆tA j ]. Since (Yn) is a Markov chain starting at Y0 = -1, for any j ≥ 1, A2j-1 (resp. A2j ) is geometrically distributed with parameter c0∆t (resp. c1∆t). According to (4.9) we get

IP(N ∆ s = k) = IP ∆t k j=1 Aj ≤ s < ∆t
ϕ2j (1) = c1∆t e ∆t -1 + c1∆t ≤ c1∆t ∆t + c1∆t = c1 1 + c1 < 1.
By the same way, we have:

ϕ2j-1(1) ≤ c0 1 + c0 < 1.
As a result, there exists 0 < r < 1 so that

IP(N ∆ s = k) ≤ e s r k . (4.11)
We are now allowed to apply the dominated convergence theorem:

lim ∆t→0 IE[f ( Z∆ s )] = k≥0 lim ∆t→0 E∆(k) = k≥0 IE[f (-Z c 0 ,c 1 s )1 {N c 0 ,c 1 s =k} ] = IE[f (-Z c 0 ,c 1 s )].
Proposition 4.5. For any (s1, . . . , sn) ∈ R n + such that s1 ≤ s2 ≤ . . . ≤ sn, the random vector

( Z∆ s 1 , . . . , Z∆ sn ) converges in distribution to (-Z c 0 ,c 1 s 1 , . . . , -Z c 0 ,c 1 sn
), as ∆t tends to 0.

Proof. We follow the approach developed in the proof of Proposition 4.4. Let f : R n → R be a bounded and continuous function. We have:

IE f ( Z∆ s 1 , . . . , Z∆ sn ) = k 1 ,...,
{N c 0 ,c 1 s 1 =k 1 ,...,N c 0 ,c 1 sn =kn} ].
According to the definition of the process Z c 0 ,c 1 s , we may deduce :

lim ∆t→0 E∆(k1, . . . , kn) = IE[f (-Z c 0 ,c 1 s 1 , . . . , -Z c 0 ,c 1 sn )1 {N c 0 ,c 1 s 1 =k 1 ,...,N c 0 ,c 1 sn =kn} ].
In order to obtain that

lim ∆t→0 IE[f ( Z∆ s 1 , . . . , Z∆ sn )] = IE[f (-Z c 0 ,c 1 s 1 , . . . , -Z c 0 ,c 1 sn )],
it suffices (cf the proof of Proposition 4.4) to prove that We are now able to complete the proof of Theorem 2.1. Since ( Z∆ s ) and (Z c 0 ,c 1 s ) are both continuous processes, the convergence of the process ( Z∆ s ) to the process (-Z c 0 ,c 1 s ) will be proved as soon as the following measure tension criterium (cf Theorem 8.3 p.56 in [START_REF] Billingsley | Convergence of probability measures[END_REF]) holds : for all ε > 0 and η0, there exists some constants δ ∈]0, 1[ and µ > 0 such that

1 δ IP sup s≤u≤s+δ | Z∆ u -Z∆ s | ≥ ε ≤ η0, for any ∆t ≤ µ. (4.12)
Since ( Z∆ s , s ≥ 0) is the interpolated persistent random walk, its slope is always equal to 1 or -1. Hence we obtain for any (u,

s) ∈ R 2 + , | Z∆ u -Z∆ s | ≤ |u -s|. Consequently sup s≤u≤s+δ | Z∆ u -Z∆ s | ≤ δ.
By choosing δ = ε/2 we get the tension criterium and so the convergence of the process ( Z∆ s ) to the process (-Z c 0 ,c 1 s ).

Two extensions of Theorem 2.1

First of all, the extensions presented in this section concerns the regime ∆x = ∆t.

The case when (Y t ) takes k values.

Let us introduce our parameters. Let k ≥ 2, y1, . . . , y k denote k real numbers, and (c(i, j); 1 ≤ i, j ≤ k) a matrix so that c(i, j) ≥ 0 for any i = j, c(i, i) = 0, k l=1 c(i, l) > 0 ∀i.

(5.1)

We directly consider the asymptotic regime. Let (Yt) be a {y1, . . . , y k }-valued Markov chain, with transition probability matrix:

π ∆ (yi, yj) =    c(i, j)∆t i = j 1 - k l=1 c(i, l) ∆t i = j, (5.2)
where ∆t > 0 is supposed to be small so that

c(i, j)∆t ≤ 1, k l=1 c(i, l) ∆t < 1.
Similarly to the case k = 2 and y1 = -1, y2 = 1, we are interested in the linear interpolation ( Z∆ s ; s ≥ 0) of the process (Z ∆ s ; s ≥ 0) defined by (1.7). Theorem 5.1. Suppose Y0 = yi. Then ( Z∆ s ; s ≥ 0) converges in distribution, as ∆t → 0, to the process t 0 Rsds; t ≥ 0 where (Rs) is a {y1, . . . , y k }-valued continuous-time Markov chain starting at level yi, whose dynamic is the following: (Rt) stays on level yi an exponential time with parameter 1/ k l=1 c(j, l) and jumps to y j ′ (j ′ = j) with probability c(j, j ′ )/ k l=1 c(j, l) . Remark 5.2. In the case k = 2, y1 = -1 and y2 = 1, then ((-1) N c 0 ,c 1 t ; t ≥ 0) (cf (2.1)) may be chosen as a realization of (Rt) when it starts at R0 = -1.

Proof of Theorem 5.1. We proceed as in the proof of Theorem 2.1 developed in Section 4. Let (Tn) n≥1 be the sequence of stopping times defined by (4.1). Then:

Xt = yi(t + 1) 0 ≤ t < T1 yiT1 + YT 1 (t -T1 + 1) T1 ≤ t < T2.
Recall that (Z ∆ s ; s/∆t ∈ IN) has been defined by (4.5). From the relations above, it is easy to deduce:

Z ∆ s = yi(s + ∆t) 0 ≤ s ≤ ∆tT1 yi(∆tT1) + YT 1 (s -∆tT1 + ∆t) ∆tT1 ≤ s < ∆tT2.
Let us determine the limit distribution of (∆tT1, YT 1 ) as ∆t → 0. Set

V ∆ (λ, j) = IE e -λ∆tT 1 1 {Y T 1 =y j } , λ > 0, j = i.
Proceeding as in the proof of Lemma 4.1, we obtain:

V ∆ (λ, j) = e -λ∆t c(i, j)∆t 1 -1 - k l=1 c(i, l) ∆t e -λ∆t
.

Using standard analysis, we deduce that (∆tT1, YT 1 ) converges in distribution as ∆t → 0 to (e ′ 1 , U1) where:

IE e -λe ′ 1 1 {U 1 =j} = c(i, j) λ + k l=1 c(i, l)
.

As a result, e ′ 1 and U1 are independent, e ′ 1 is exponentially distributed with parameter 1/ k l=1 c(i, l) and

IP(U1 = j) = c(i, j) k l=1 c(i, l)
.

Using the approach developed in Section 4, we can prove Theorem 5.1. The details are left to the reader.

5.2

The case when (Y t ) is a Markov chain of order 2.

Let (Yt) be a Markov chain with order 2. For simplicity we suppose that it takes its values in {-1, 1}. Obviously (Yt, Yt+1) t≥0 is a Markov chain with state space E = {(-1, -1), (-1, 1), (1, -1), (1, 1)}.

Let π ∆ be the transition probability matrix:

π ∆ =     1 -c0∆t c0∆t 0 0 0 0 1 -p0 p0 p1 1 -p1 0 0 0 0 c1∆t 1 -c1∆t     (5.3)
where ∆t, c0, c1, p0, p1 > 0 and c0∆t, c1∆t, p0, p1 < 1.

Let us introduce:

vi = pi 1 -(1 -p0)(1 -p1) , c ′ i = civi, i = 0, 1. (5.4)
Recall that (Z ∆ t ) and ( Z∆ t ) have been defined by (4.6), resp. (4.8), (N c 0 ,c 1 t

) is the counting process defined by (2.1), and

Z c 0 ,c 1 t = t 0 (-1) N c 0 ,c 1 u du, t ≥ 0. Theorem 5.3. 1) Suppose that Y0 = Y1 = -1 (resp. Y0 = Y1 = 1) then ( Z∆ s ; s ≥ 0) converges in distribution, as ∆t → 0, to (-Z c ′ 0 ,c ′ 1 s ; s ≥ 0) (resp. (Z c ′ 1 ,c ′ 0 s ; s ≥ 0)). 2) Suppose Y0 = 1 and Y1 = -1 (resp. Y0 = -1, Y1 = 1) then ( Z∆ s ; s ≥ 0) converges in distribution, as ∆t → 0, to (ǫ -1) s 0 (-1) N c ′ 0 ,c ′ 1 u du + ǫ s 0 (-1) N c ′ 1 ,c ′ 0 u du; s ≥ 0 where ǫ is independent from (N c ′ 0 ,c ′ 1 u ), (N c ′ 1 ,c ′ 0 u
) and

IP(ǫ = 0) = 1 -IP(ǫ = 1) = v1 (resp. IP(ǫ = 1) = 1 -IP(ǫ = 0) = v0).

The moment generating function of X t

Let us recall that the increments process (Yt, t ∈ N) is a Markov chain valued in the state space E = {-1, 1}. Its transition probability is given by

π = 1 -α α β 1 -β 0 < α < 1, 0 < β < 1.
The persistent random walk (Xt, t ∈ N) is defined by the partial sum:

Xt = t i=0
Yi with X0 = Y0 = 1 or -1.

Lemma 6.1. Let us define the functions at and bt: at(j) = IP(Xt = j, Yt = -1) and bt(j) = IP(Xt = j, Yt = 1). (

Then, at+1(j) = (1 -α)at(j + 1) + βbt(j + 1) (6.2) bt+1(j) = αat(j -1) + (1 -β)bt(j -1). (

Proof. Using the Markov property of (Yt) we have:

at+1(j) = IP(Xt+1 = j, Yt+1 = -1, Yt = -1) + IP(Xt+1 = j, Yt+1 = -1, Yt = 1) = IP(Xt = j + 1, Yt+1 = -1, Yt = -1) + IP(Xt = j + 1, Yt+1 = -1, Yt = 1)
= (1 -α)at(j + 1) + βbt(j + 1).

The second recursive formula involving (bt(j)) can be obtained similarly.

Let us define the moment generating function Φ(λ, t) = IE[λ Xt ], (λ > 0). We decompose Φ(λ, t) as Φ(λ, t) = Φ-(λ, t) + Φ+(λ, t), (

Φ-(λ, t) = IE[λ Xt 1 {Yt=-1} ], Φ+(λ, t) = IE[λ Xt 1 {Yt=1} ]. 6.4) with
Lemma 6.2. 1) Φ-(λ, 0) = 1 λ IP(Y0 = -1) and Φ+(λ, 0) = λ IP(Y0 = 1).

2) The moment generating function verifies the following induction equations:

Φ-(λ, t + 1) = 1 -α λ Φ-(λ, t) + β λ Φ+(λ, t) (6.6)
Φ+(λ, t + 1) = αλΦ-(λ, t) + (1 -β)λΦ+(λ, t) (6.7)
Proof. Definition (6.1) implies that

Φ-(λ, t) = j∈Z λ j at(j) = t+1 j=-t-1 λ j at(j). Hence, Φ-(λ, t + 1) = j λ j at+1(j) = (1 -α) j λ j at(j + 1) + β j λ j bt(j + 1) = (1 -α) 1 λ j λ j+1 at(j + 1) + β λ j λ j+1 bt(j + 1) = 1 -α λ Φ-(λ, t) + β λ Φ+(λ, t).
The proof of (6.7) is similar.

Lemma 6.3. Let f (λ, t) be equal to either Φ-(λ, t) or Φ+(λ, t), then

f (λ, t + 2) - 1 -α λ + (1 -β)λ f (λ, t + 1) + (1 -α -β)f (λ, t) = 0. (6.8)
Proof. By (6.6), we get

Φ+(λ, t) = Φ-(λ, t + 1) - 1 -α λ Φ-(λ, t) λ β . (6.9)
Replacing t by t + 1 in (6.9), we obtain

Φ+(λ, t + 1) = Φ-(λ, t + 2) - 1 -α λ Φ-(λ, t + 1) λ β . (6.10)
Using successively (6.7), (6.10) and (6.9), we have:

αλΦ-(λ, t) = Φ+(λ, t + 1) -(1 -β)λΦ+(λ, t) = λ β Φ-(λ, t + 2) - 1 -α λ Φ-(λ, t + 1) - 1 -β β λ 2 Φ-(λ, t + 1) - 1 -α λ Φ-(λ, t) . Finally Φ-(λ, t + 2) - 1 -α λ + (1 -β)λ Φ-(λ, t + 1) + (1 -α)(1 -β) -αβ Φ-(λ, t) = 0.
The proof concerning f (λ, t) = Φ-(λ, t) is similar and is left to the reader.

In order to obtain the explicit form of Φ-(λ, t) and Φ+(λ, t) in terms of λ and t, it suffices to compute the roots ϑ-and ϑ+ of the following polynomial

ϑ 2 - 1 -α λ + (1 -β)λ ϑ + 1 -α -β = 0 (6.11)
Its discriminant equals

D = 1 -α λ + (1 -β)λ 2 -4(1 -α -β). (6.12)
It is clear that

D = 1 -α λ + (1 -β)λ + 2 √ ρ 1 -α λ + (1 -β)λ -2 √ ρ = 1 λ 1 -α λ + (1 -β)λ + 2 √ ρ (1 -β)λ 2 -2 √ ρλ + 1 -α . (6.13) Since the discriminant of λ → (1 -β)λ 2 -2 √ ρλ + 1 -α is equal to -4αβ then D > 0 for any λ > 0.
Consequently the roots of (6.11) are:

ϑ± = 1 2 1 -α λ + (1 -β)λ ± √ D . (6.14)
We deduce the following result. Moreover, using (6.6) and (6.7) with t = 0, we get

Φ(λ, 1) = Φ+(λ, 1) + Φ-(λ, 1) = 1 -α λ + αλ Φ-(λ, 0) = 1 -α λ 2 + α = a+ϑ+ + a-ϑ-.
It is clear that Lemma 6.3 and Φ(λ, t) = Φ+(λ, t) + Φ-(λ, t) implies that Φ(λ, t) satisfies (6.8). Then (6.15) follows by standard arguments. The second case X0 = Y0 = 1 can be proved in a similar way.

Proof of Theorem 2.2

We keep the notations given in Section 1. Let α0 and β0 be two real numbers in [0, 1]. Let ∆x be a small space parameter so that:

0 ≤ α0 + c0∆x ≤ 1, 0 ≤ β0 + c1∆x ≤ 1,
where c0 and c1 belong to R. Note that α0 > 0 (resp. β0 > 0) implies that α0 + c0∆x > 0 (resp. β0 + c1∆x > 0) when ∆x is small enough. If α0 < 1 (resp. β0 < 1), similarly α0 + c0∆x < 1 (resp. β0 + c1∆x < 1) as soon as ∆x is small. In the case α0 = 1 (resp. β0 = 1) c0 (resp. c1) has to be chosen in ] -∞, 0]. We assume that the coefficients of the transition probability matrix π ∆ of the Markov chain (Yt) satisfy:

α = α0 + c0∆x, β = β0 + c1∆x (6.16)
i.e. π ∆ is given by (1.6). (Xt) is defined by (1.1) and (Z ∆ s ) is the normalized persistent random walk:

Z ∆ s = ∆xX s/∆t , (∆t > 0, ∆x > 0, s ∈ ∆t IN). ( Z∆ t ; t ≥ 0) denotes the linear interpolation of (Z ∆ t ). Recall that ρ0 = 1 -α0 -β0 and η0 = β0 -α0. Note that ρ0 = 1 ⇐⇒ α0 + β0 = 0 Proposition 6.5. Let ρ0 = 1, 1) if r∆t = ∆x with r > 0 then Z∆ t converges towards the deterministic limit -rtη 0 1-ρ 0 as ∆x tends to 0.

2) if r∆t = ∆ 2

x with r > 0, Z∆ t + t √ rη 0

(1-ρ 0 ) √ ∆t converges in distribution to the Gaussian law with mean m = rt -c 1 -ρ0 + η0c (1 -ρ0) 2 (6.17)

and variance

σ 2 = r(1 + ρ0) 1 -ρ0 1 - η 2 0 (1 -ρ0) 2 t, (6.18)
where c = c0 + c1 and c = c1 -c0. (6.19)

Proof. We shall prove the statement under the condition X0 = Y0 = -1. If X0 = Y0 = +1, the limit is obtained by changing the sign and replacing c0 (resp. c1) by c1 (resp. c0). 1) Let Φ(λ, t) be the generating function associated with Xt. In order to determine the limit distribution of Z ∆ t , let us introduce: Applying (6.12) with α = α0 + δ and β = β0 + δ we have:

φ(µ, t) = E-1[e -µ Z∆ t ], ( 6
D = (1 -α0 -δ)e µ∆x + (1 -β0 -δ)e -µ∆x 2 -4(1 -α0 -β0 -δ -δ).
By (6.22) we get

D = (2 -α0 -β0) + ∆x µ(β0 -α0) -c0 -c1 (6.23) +∆ 2 x µ 2 2 (2 -α0 -β0) + µ(c1 -c0) + o(∆ 2 x ) 2 - 4 1 -α0 -β0 -∆x(c0 + c1) (6.24)
It is clear that D admits the following asymptotic expansion, as ∆x → 0:

D = A0 + A1∆x + A2∆ 2 x + o(∆ 2
x )

It is usefull to note that α0 and β0 can be expressed in terms of η0 and ρ0:

α0 = 1 -η0 -ρ0 2 and β0 = 1 + η0 -ρ0 2 .
Let us compute A0, A1 and A2 using standard analysis:

A0 = (2 -α0 -β0) 2 -4(1 -α0 -β0) = α 2 0 + β 2 0 + 2α0β0 = (α0 + β0) 2 = (1 -ρ0) 2 . A1 = 2(2 -α0 -β0) µ(β0 -α0) -(c0 + c1) + 4(c0 + c1) = 2µ(2 -α0 -β0)(β0 -α0) -4(c0 + c1) + 2(α0 + β0)(c0 + c1) + 4(c0 + c1) = 2 µ(2 -α0 -β0)(β0 -α0) + (α0 + β0)(c0 + c1) = 2 µη0(1 + ρ0) + c(1 -ρ0) . (6.25) A2 = 2(2 -α0 -β0) µ 2 2 (2 -α0 -β0) + µ(c1 -c0) + µ(β0 -α0) -(c0 + c1) 2 = µ 2 (2 -α0 -β0) 2 + (β0 -α0) 2 +2µ (2 -α0 -β0)(c1 -c0) -(β0 -α0)(c0 + c1) + (c0 + c1) 2 = 2µ 2 (α0 -1) 2 + (β0 -1) 2 + 4µ (1 -β0)c1 -(1 -α0)c0 + (c0 + c1) 2 = µ 2 (η 2 0 + (1 + ρ0) 2 ) + 2µ (1 + ρ0)c -η0c + c 2 . (6.26)
Under the condition ρ0 = 1, we have

√ D = (1 -ρ0) 1 + A1 (1 -ρ0) 2 ∆x + A2 (1 -ρ0) 2 ∆ 2 x + o(∆ 2 x ) Hence √ D = B0 + B1∆x + B2∆ 2 x + o(∆ 2 x ) with B0 = 1 -ρ0, B1 = 1 2 A1 1 -ρ0 = 1 1 -ρ0 µ(2 -α0 -β0)(β0 -α0) + (α0 + β0)(c0 + c1) = µ η0(1 + ρ0) 1 -ρ0 + c B2 = 1 2 A2 1 -ρ0 - 1 8 A 2 1 (1 -ρ0) 3 . (6.27)
As a result, B2 is a second order polynomial function with respect to the µ-variable:

B2 = µ 2 B22 + µB21 + B20.
Identities (6.25), (6.26) and (6.27) imply:

B20 = c 2 2(1 -ρ0) - 2c(1 -ρ0) 2 8(1 -ρ0) 3 = 0 B21 = 1 2 2 (1 + ρ0)c -η0c 1 -ρ0 - 1 8 8η0c(1 -ρ0)(1 + ρ0) (1 -ρ0) 3 = c 1 + ρ0 1 -ρ0 - 2η0c (1 -ρ0) 2 B22 = 1 2 η 2 0 + (1 + ρ0) 2 1 -ρ0 - 1 8
4η 2 0 (1 + ρ0) 2 (1 -ρ0) 3 = 1 2 η 2 0 + (1 + ρ0) 2 (1 -ρ0) 2 -η 2 0 (1 + ρ0) 2 (1 -ρ0) 3 = (1 + ρ0) 2 2(1 -ρ0) - 2η 2 0 ρ0 (1 -ρ0) 3 . Consequently √ D = 1 -ρ0 + µ η0(1 + ρ0) 1 -ρ0 + c ∆x + µ 2 (1 + ρ0) 2 2(1 -ρ0) - 2η 2 0 ρ0 (1 -ρ0) 3 +µ c 1 + ρ0 1 -ρ0 - 2η0c (1 -ρ0) 2 ∆ 2 x + o(∆ 2
x ). (6.28)

2) The first order development suffices to determine the limit of φ(µ, t) as ∆x → 0. Indeed Consequently, the second term in (6.32) tends to 0. It is important to note that the initial condition X0 = Y0 = -1 disappears. Let us study the first term in the right hand side of (6.32). Note that lim∆ x→0 ϑ+ = 1, then if ∆x is small enough, we can take the logarithm of ϑ+. From (6.30) a straightforward calculation gives

√ D = 1 -ρ0 + ∆x 1 -ρ0 µη0(1 + ρ0) + c(1 -ρ0) + o(∆x). ( 6
log ϑ+ = ∆x µη0 1 -ρ0 + o(∆x)
Choosing r∆t = ∆x and using (6.32), (6.34) and (6.31), we obtain the following limit:

lim ∆x→0 φ(µ, t) = exp{ rµη0t 1 -ρ0 }.
Since the convergence holds for any µ ∈ R, we can conclude (cf Theorem 3 in [START_REF] Curtiss | A note on the theory of moment generating functions[END_REF]) that

lim ∆x→0 IE-1[exp(iu Z∆ t )] = exp - iurη0t 1 -ρ0 , for any u ∈ R.
Thus Z∆ t converges in distribution, as ∆x → 0, to the Dirac measure at -rη 0 t 1-ρ 0 . 3) Next, we consider the convergence of the process

ξ ∆ t = Z∆ t + tη0 √ r (1 -ρ0) √ ∆t .
Hence we define

ψ(µ, t) = IE-1[e -µξ ∆ t ] = e - µtη 0 √ r (1-ρ 0 ) √ ∆ t φ(µ, t).
To determine the limit of ψ(µ, t) as ∆t, ∆x → 0, from (6.32) and (6.34) we may deduce that it suffices to compute the second order development of the root ϑ+. Using (6.14) and (6.28) we get:

ϑ+ = 1 2 (1 + ρ0) + ∆x 2 (µη0 -c) + ∆ 2 x 2 µ 2 (1 + ρ0) 2 + µc + 1 -ρ0 2 + ∆x 2 µη0(1 + ρ0) 1 -ρ0 + c + ∆ 2 x 2 µ 2 (1 + ρ0) 2 2(1 -ρ0) - 2η 2 0 ρ0 (1 -ρ0) 3 + µ c 1 + ρ0 1 -ρ0 - 2η0c (1 -ρ0) 2 + o(∆ 2 x ).
As a result

ϑ+ = 1+∆x µη0 1 -ρ0 +∆ 2 x µ 2 2 1 + ρ0 1 -ρ0 - 2η 2 0 ρ0 (1 -ρ0) 3 +µ c 1 -ρ0 - η0c (1 -ρ0) 2 +o(∆ 2
x ). (6.35)

We take r∆t = ∆

σ 2 = r(1 + ρ0) 1 -ρ0 1 - η 2 0
(1 -ρ0) 2 t. (6.38) 4) Since (6.36) holds for any µ ∈ R, this implies that ξ ∆ t converges in distribution, as ∆x → 0, to the Gaussian distribution with mean m and variance σ 2 . (see Theorem 3 in [START_REF] Curtiss | A note on the theory of moment generating functions[END_REF]) Proposition 6.6. Assume that ρ0 = 1 and r∆t = (∆x) 2 . Let us denote ξ ∆ the process defined by

ξ ∆ t = Z∆ t + t √ rη0
(1 -ρ0) √ ∆t .

Then (ξ ∆ t 1 , ξ ∆ t 2 , . . . , ξ ∆ tn ) converges in distribution, as ∆x → 0, towards (ξ 0 t 1 , ξ 0 t 2 , . . . , ξ 0 tn ) where ξ 0 is given by

ξ 0 t = r -c 1 -ρ0 + η0c (1 -ρ0) 2 t + r(1 + ρ0) 1 -ρ0 1 - η 2 0
(1 -ρ0) 2 Wt.

(Wt, t ≥ 0) is the one-dimensional Brownian motion starting at 0.

Proof. The proof is only presented in the case n = 2. For simplicity let s = t1 < t2 = t. We are interested in the limit of the random vector (ξ ∆ s , ξ ∆ t ). Let us then compute the two dimensional Fourier transform = IE exp{iµ(ξ 0 t -ξ 0 s ) + iλξ 0 s }

We are now able to end the proof of Theorem 2.2 (item 2). We may apply, without any change, the measure tension criterium used in the proof of convergence of (Z ∆ t ) in the case α0 = β0 = 1 (see the end of Section 4). This, and Proposition 6.6 show that (ξ ∆ t ) t≥0 converges in distribution as ∆x → 0 to the Brownian motion with drift (ξ 0 t ) t≥0 .

Proof of Proposition 2.3

We suppose α0 = β0 = 1, c1 = c0 < 0 and r∆t = ∆ 3 x where r > 0. We briefly sketch the proof of Proposition 2.3. The approach is similar to the one developed in the case 2) of Theorem 2.2. We only prove that Z∆ t converges to the Gaussian distribution with 0-mean and variance equals -rc0t. Using Theorem 3 in [START_REF] Curtiss | A note on the theory of moment generating functions[END_REF], it is equivalent to show , ∀µ ∈ R.

We have already observed that we may reduce to the case t/∆t ∈ IN; in this case we have Z∆ t = Z ∆ t and IE-1 e -µZ ∆ t = Φ e -µ∆x , t ∆t where Φ(λ, t) is the moment generating function associated with (Xt) (see the beginning of subsection 6.1). Recall that Φ(λ, t) is given by identity (6.15). Note that: α = α0 + c0∆x = 1 + c0∆x, β = β0 + c0∆x = 1 + c0∆x.

Since α and β have to belong to [0, 1], this implies that c0 < 0. Recall that D, ϑ+ and ϑ-are the real numbers which have been defined by (6.12) resp. (6.14) (with λ = e -µ∆x ). We have:

D = 4c 2 0 ∆ 2
x cosh 2 (µ∆x) + 4(1 + 2c0∆x), ϑ± = -c0∆x cosh(µ∆x) ± c 2 0 ∆ 2 x cosh 2 (µ∆x) + 1 + 2c0∆x. Using classical analysis we get:

√ D/2 = 1 + 2c0∆x + c 2 0 ∆ 2 x + o(∆ 3 x ) = 1 + c0∆x + o(∆ 3 x ), ϑ+ = 1 - c0µ 2 2 ∆ 3 x + o(∆ 3
x ), ϑ-= -1 -2c0∆x + o(∆x). Relation (6.15) implies that the variable Z ∆ t is asymptotically normal distributed with variance -rc0t.

1 t) 1 = λ 1 e 1 0(- 1 ) 2 λ 1 e 1 (

 111121 0 . b) When n ≥ 1, using (2.1) we obtain ∆n(f ) = IE f (Z c 0 ,c {λ 1 e 1 +...+λnen≤t<λ 1 e 1 +...+λ n+1 e n+1 } . If λ1e1 + . . . + λnen ≤ t < λ1e1 + . . . + λn+1en+1 then Z c 0 ,c 1 t 0 du + λ 1 e 1 +λ 2 e -1)du + . . . + λ 1 e 1 +...+λnen λ 1 e 1 +...+λ n-1 e n-1 (-1) n-1 du + t λ 1 e 1 +...+λnen (-1) n du.

  22), (3.23) and (3.24), Proposition 3.8 follows.

2 )

 2 we obtain:T k = A1 + . . . + A k k ≥ 1.(4.5) Hence the equations (4.3), (4.4) and (4.5) permit to emphasize the bijective correspondence between (Xn; n ∈ IN) and (A k ; k ∈ IN).

Lemma 4 . 2 . 1 s 1 s

 4211 Let s > 0, k ≥ 1 and Φ k : R k → R be a bounded continuous function. Then1) lim ∆t→0 IP(N ∆ s = 0) = IP(N c 0 ,c k (∆tA1, ∆tA2, . . . , ∆tA k )1 {N ∆ s =k} ] = IE[Φ k (λ1e1, λ2e2, . . . , λ k e k )1 {N c 0 ,c=k} ], where λ k has been defined by (2.2). Proof. 1) Statement 1) follows from: IP(N ∆ s = 0) = IP(N ⌊s/∆t⌋ = 0) = IP(T1 ≥ ⌊s/∆t⌋) = IP(A1 ≥ ⌊s/∆t⌋) = IP(∆tA1 ≥ ∆t⌊s/∆t⌋)

Lemma 4 . 3 . 1 =k 1 1 =k 1 ,Proposition 4 . 4 .(- 1 )

 431111441 Let n ∈ IN, (k1, . . . , kn) ∈ IN n such that k1 ≤ k2 ≤ . . . ≤ kn and (s1, . . . , sn) ∈ R n + with s1 ≤ s2 ≤ . . . ≤ sn. Let Φ : R kn → R be a bounded and continuous function. Then lim ∆t→0 IE[Φ(∆tA1, ..., ∆tA kn )1 {N ∆ s ,...,N ∆ sn =kn} ] = IE[Φ(λ1e1, ..., λ kn e kn )1 {N c 0 ,c 1 s The random variable Z∆ s converges in distribution towards -Z c 0 ,c 1 s , for any s > 0, as ∆t → 0. Proof. Let f : R → R be a continuous function which is bounded by M . Identities (4.8) and (4.5) imply that IE[f ( Z∆ s )] = ∞ k=0 E∆(k), with E∆(k) = IE f k j=1 j ∆tAj + (-1) k+1 s -∆t k j=1 Aj + ∆t 1 {N ∆ s =k} Applying Lemma 4.2 and (3.8), we obtain for any k ≥ 0, lim ∆t→0

Aj

  ≤ s = IP exp -∆t k j=1 Aj ≥ e -s ≤ e s IE exp -∆t k j=1 Aj = e s k j=1 ϕj (1)

k 1 ,

 1 ...,k n-1 sup ∆t |E∆(k1, . . . , kn)| < ∞. Since f is bounded, |E∆(k1, . . . , kn)| ≤ M IP(N ∆ sn = kn) Using moreover (4.11) we get k 1 ,...,kn |E∆(k1, . . . , kn)| ≤ M e sn kn (kn) n-1 r kn < ∞ since r < 1.

Proposition 6 . 4 . 1 )- 1 -

 6411 The moment generating function Φ(λ, t) satisfiesΦ(λ, t) = a+ϑ t + + a-ϑ t α + λ(λα -ϑ-) λ 2 √ D and a-= 1 λ -a+ if X0 = Y0 = -1 and a+ = (1 -β)λ 2 + β -λϑ-√ D and a-= λ -a+ if X0 = Y0 = 1.Proof. Suppose that X0 = Y0 = -1. Let us first determine the values of the generating function at time t = 0 and t = 1: Φ(λ, 0) = Φ+(λ, 0) + Φ-(λ, 0) = 1 λ IP(Y0 = -1) + λ IP(Y0 = 1) = 1 λ = a+ + a-

  .20) where IE-1 denotes the expectation when Y0 = -1. Observe thatφ(µ, t) = Φ(e -µ∆x , t ∆t ) = E-1[e -µ∆xX(t/∆t) ], (6.21) when t/∆t ∈ IN. According to Proposition 6.4, when t/∆t ∈ IN, φ(µ, t) can be expressed in terms of a+, a-and √ D. First let us study the asymptotic expansion of the discriminant D as ∆x → 0. It is convenient to set: δ = c0∆x and δ = c1∆x. (6.22)

Ψ= e (iµm-σ 2 2 µ 2 )

 22 ∆ (µ, λ) = IE-1 e iµ(ξ ∆ t -ξ ∆ s ) e iλξ ∆ s , (λ, µ ∈ R).Since the process (Xt, Yt) is Markovian, we obtainΨ ∆ (µ, λ) = IE-1 e iµξ ∆ t-s IE-1 1 {Y (s/∆t)=-1} e iλξ ∆ s + IE+1 e iµξ ∆ t-s IE-1 1 {Y (s/∆t)=+1} e iλξ ∆ s ,when s/∆t and t/∆t belongs to IN.Note that |ξ∆ u -ξ ∆ u ′ | ≤ ∆x∆t when u ′ = u ∆t ∆t. Consequently Ψ ∆ (µ, λ) ∼ ∆x→0 IE-1 e iµξ ∆ t ′ -s ′ IE-1 1 {Y (s ′ /∆t)=-1} e iλξ ∆ s ′ + IE+1 e iµξ ∆ t ′ -s ′ IE-1 1 {Y (s ′ /∆t)=+1} e iλξ ∆ s ′ , (s ′ = ⌊s/∆t⌋∆t, t ′ = ⌊t/∆t⌋∆t). (t-s)where m and σ 2 are defined by (6.37), resp. (6.38). Then we can deduce:lim ∆x→0 Ψ ∆ (µ, λ) = e (iµm-σ 2 2 µ 2 )(t-s) lim ∆x→0 IE-1 e iλξ ∆ s ′ = e (iµm-σ 2 2 µ 2 )(t-s) lim ∆x→0 IE-1 e iλξ ∆ s= e (iµm-σ 2 2 µ 2 )(t-s) e (iλm-σ 2 2 λ 2 )s

  lim ∆x→0 IE-1 e -µ Z∆ t = e -rc 0 tµ 2 2

Remark 5.4. 1) Note that (Yt)t∈IN is a Markov chain if and only if 1 -c0∆t = p1 and 1 -c1∆t = p0. If we replace formally p0 (resp. p1) by 1 -c1∆t (resp. 1 -c0∆t) in (5.4) and take the limit ∆t → 0, we obtain vi = pi and c ′ i = ci. We recover Theorem 2.1.

2) The fact that (Yt) is a Markov chain with order 2 does not modify drastically the limit. The limit process can be expressed in terms of processes of the type (Z α,β s ; s ≥ 0).

Proof of Theorem 5.3. 1) We only consider the case Y0 = Y1 = 1. Let us define T1, T2 and T3 as follows:

Using the definition (cf (1.1)) of (Xt) we easely obtain:

where Xt equals either -1 or 0. Moreover, when T2 ≤ t < T3, we have:

According to (1.7), we can deduce:

2) a) Proceeding as in the proof of Theorem 2.1, we can prove that ∆tT1 converges in distribution, as ∆t → 0, to e ′ 1 , where e ′ 1 is exponentially distributed with parameter 1/c1. Then ( Z∆ s ; 0 ≤ s ≤ ∆tT1)

, as ∆t → 0. b) The distribution of T2 -T1 does not depend on ∆t. Moreover | X•| ≤ 1, then the limit of the length of the interval [∆tT1, ∆tT2] is null. We have

c) Using the strong Markov property, we easely show that ( Z∆ s+∆tT 2 ; 0 ≤ s ≤ ∆t(T3-T1))

, as ∆t → 0, where (e ′ 1 , YT 1 ) (resp. (e ′ 1 , e ′ 2 )) are independent r.v.'s and conditionally on YT 2 = 1 (resp. YT 2 = -1) e ′ 2 is exponentially distributed with parameter 1/c1 (resp. 1/c0). d) Let us summarize the former analysis. We have proved that ( Z∆ s ; s ≥ 0)

, where ( Ru) is a continuous-time Markov chain which takes its values in {-1, 1} and R0 = 1. Moreover the dynamic of ( Ru) is the following: ( Ru) stays in 1 (resp. -1) an exponential time with parameter 1/c1 (resp. 1/c0) and moves to -1 (resp. 1) with probability v1 (resp. v0). Note that ( Ru) is allowed to stay in the same site. It is classical (cf [START_REF] Ross | Introduction to probability models[END_REF]) to prove that ( Ru)u≥0

) u≥0 where c ′ 0 and c ′ 1 are defined by (5.4).

6 Convergence of the persistent random walk towards the Brownian motion with drift

In subsection 6.1 below we determine the generating function of Xt, where Xt is the persistent random walk defined by (1.1). This allows to prove Theorem 2.2 and Proposition 2.3 in subsections 6.2, 6.3.


Vasil Penchev
email: vasildinev@gmail.com

The "noncausal causality" of quantum information

Keywords: Hilbert arithmetic, probabilistic interpretation of quantum mechanics; quantum information, quantum-information conservation, qubit Hilbert space, reverse causality

The paper is concentrated on the special changes of the conception of causality from quantum mechanics to quantum information meaning as a background the revolution implemented by the former to classical physics and science after Max Born's probabilistic reinterpretation of wave function. Those changes can be enumerated so: (1) quantum information describes the general case of the relation of two wave functions, and particularly, the causal amendment of a single one; (2) it keeps the physical description to be causal by the conservation of quantum information and in accordance with Born's interpretation; (3) it introduces inverse causality, "backwards in time", observable "forwards in time" as the fundamentally random probability density distribution of all possible measurements of any physical quantity in quantum mechanics; (4) it involves a kind of "bidirectional causality" unifying (4.1) the classical determinism of cause and effect, (4.2) the probabilistic causality of quantum mechanics, and (4.3) the reversibility of any coherent state; (5) it identifies determinism with the function successor in Peano arithmetic, and its proper generalized causality with the information function successor in Hilbert arithmetic.

I BACKGROUND: THE REVOLUTION OF QUANTUM CAUSALITY

The concept of causality is fundamental for science constituting it in opposition to religion and often, even to common sense. Any observable phenomenon has its cause though it can be hidden. Science legitimates its existence also due to finding those hidden causes.

At first glance, quantum mechanics deprived physics of causality. This reason may explain the stubborn resistance and non-acceptance of quantum mechanics during decades even by great scientists such as Einstein.

The elucidation followed Max Born's interpretation, after which wave function could be interpreted as the characteristic function of a certain probability density distribution of any quantity in quantum mechanics and substituted the unambiguously determined single value of physical quantity featured in classical mechanics and physics. That replacement is fatal to causality as determinism, for which Einstein coined the metaphor of "God playing dice" [START_REF] Einstein | Letter to Max Born, 16 December 1926[END_REF]) rejecting it.

However, Max Born himself pointed out that the change of wave function is rigorously deterministic though the result of a single measurement is fundamentally random (as that of any finite set of measurements, what all experiments are). This corresponds to the holism of quantum mechanics conserving determinism to the whole (of all possible experiments), but not to any finite subset of them.

Furthermore, the discussion about causality in quantum mechanics was rather abstract and metaphysical corresponding to the mentality of the "founding fathers" of quantum mechanics, but not to that of their inheritors, whose slogan became "Shut up and calculate!" [START_REF] Mermin | Could Feynman Have Said This?[END_REF][START_REF] Kaiser | History: Shut up and calculate![END_REF]. So, the problem about quantum causality was delivered to the philosophers of science, who in turn abandoned it gradually and left to sink into oblivion.

II HOW QUANTUM INFORMATION DESCRIBES QUANTUM CAUSALITY

The specific subject of the paper is the interpretation and elucidation of quantum causality, implemented by theory of quantum information implicitly in virtue of its interpretation of quantum mechanics, merging it with information by its generalization to quantum information.

Indeed and in fact, "qubit" is defined as an "atom" (literally, "inseparable" in Greek) or a "quantum" (also metaphorically) of quantum causality: where

α⊥ 0 + β⊥ 1 . That is: the superposition of two α, β ∈ 𝐶: α | | 2 + β | | 2 = 1; ⊥ 0 ,⊥ 1 ⊆ Ⓗ: (⊥ 0 ) ⊥ (⊥ 1 )
orthogonal subspaces (not necessarily to be convex to each other) normed unitarily and unambiguously by two complex numbers. Then, can be interpreted correspondingly as α⊥ 0 , β⊥ 1 "cause" and "effect", at that normed to their joint physical action: i.e. interaction.

The commutativity of the operation of addition "+" notates the reversibility of quantum causality. Furthermore, the mapping is a function but not a

[(α, β) → β | |] ⊂[(𝐶 ⊗ 𝐶) → 𝑅] bijection, and thus, the reverse mapping is not a function: it is [(α, β) ← β | |] ⊂[(𝐶 ⊗ 𝐶) ← 𝑅]
an ambiguous mapping so that more values than one of the mapping correspond to a certain value of its variable. The former mapping can be interpreted as the "normal" deterministic causality and "forwards in time", and the latter, as the specifically quantum, reverse causality and "backwards in time". Each of both as well as both together follow from the introduction of quantum information and its unit, a qubit.

The mapping can be interpreted also meaning that wave

[(α, β) → β | |] ⊂[(𝐶 ⊗ 𝐶)
→ 𝑅] function after Born's interpretation is to be a characteristic function of a binary probability distribution of the two possible results of the measurement of a qubit and corresponding 𝑅 0 , 𝑅 1 bijectively to . Since the zero index is granted conventionally to "cause", and the ⊥ 0 , ⊥ 1 ⊆ Ⓗ unit index, to "effect", a qubit as unit of quantum causality represents the general case of the physical interaction of cause and effect unlike the particular case of determinism reducing quantum causality to the extreme particular case of causal (or temporal) irreversibility: cause acts physically upon effect (conventionally, while effect does not physically upon cause β = 1) (following the same convention, . α = 0) If a qubit represents an elementary quantum state, e.g. the coherent state of the two possible spins of an electron, would notate the probabilities for either the one or the other values

α | | 2 , β | | 2
of the electron spin at issue to be measured following Born's interpretation and shared (but complemented) by the qubit interpretation of quantum information.

III QUANTUM-INFORMATION CONSERVATION AND QUANTUM CAUSALITY Energy conservation in turn conserved in quantum mechanics in virtue of quantum conservation needs to be generalized to the latter (Penchev 2020 October 5) therefore forecasting nontemporal physical phenomena or interpreting all those of entanglement as the former furthermore sharing reverse causality. Thus, quantum information implying energy conservation implies also the "normal" deterministic causality "forwards in time", but along it, reverse causality featuring all phenomena of entanglement as well as even their inseparability in any coherent quantum state being reversible.

The total "sum" of all entanglement in the universe following from quantum-information conservation (Penchev 2020 October 5) can be projected anyway on the temporal screen of the "normal" deterministic causality where is admissible to be depicted only as "shadows" of matter and energy (Penchev 2021June 8); that is: as the experimentally corroborated "dark matter" and "dark energy", estimately about 20 times more than the visible ones.

In other words, the reverse causality of entanglement prevails crucially in the universe, and physical cognition supported only by energy conservation and respectively, by the "normal" deterministic causality forwards "in time" is hopelessly outdated.

The essential sense of quantum-information conservation as to quantum causality generalizing classical one consists in a new option (at that, very often met in the universe though not crucially necessary for human survival, however rather seemingly), namely: the transformation of the standard deterministic causality into reverse causality (the latter being representable as probability "veer of causes forwards in time") as well as vice versa by the necessary mediation of the coherent-state reversibility. Just the conservation of quantum information is able to manage that transformation unambiguously establishing how many (i.e. not "much") of classical causality (respectively, classical information) is to be equated to how much (i.e. not "many") of reverse causality (respectively, quantum information)1 .

Since reverse causality can be identified as randomness or fundamental randomness, deterministic causality, as necessity, and the equation of both as actuality or reality, following the "sleng of philosophy", quantum-information conservation rules quantitatively the mutual transformation between randomness and necessity as reality.

IV QUANTUM CAUSALITY AS INVERSE CAUSALITY Though, quantum causality (in a wider meaning as above) includes the usual deterministic causality of classical science as well its transformation into reverse causality, it (in a narrow meaning discussable now) means properly just the latter, reverse causality. Being directed "backwards in time", it is generalized by the fussy term or concept of randomness or fundamental randomness 2 following the empirical time arrow of our experience "forwards in time".

Modern society believes that the influence of randomness can and must be restricted more and more, not being subject to human action and thus uncontrollable and unmanageable: threat for humankind. That diminishing of randomness is partly real (for the cognition of new and new variables influencing "flipping a coin" and due to science), partly seeming (for no cognition of any variables can resolve the problem about the state of "Schrödinger's cat").

The former and latter can be considered also as underlying the opposition of science and religion. Indeed religion of any kind all over the world declares hidden, divine reality to which the religion at issue is relevant (or even, the only relevant one). On the contrary, science (at least until now) is devoted to the visible, profane reality of empirical and experimental experience. However, the divine (or sacral) reality of religion can be identified (interpretatively) as the sphere of fundamental randomness and thus obeying reverse causality (in detail in: Penchev 2021 April 12).

If that interpretation is correct or can be accepted to be valid under additional conditions, religion can be considered noy only "als strenge Wissenschaft" (what was Husserl's intention in relation to philosophy), but even as quantitative and experimental, investigating and transforming the mutual transformation between randomness and necessity as reality.

Of course, the belief, rites and rituals of religion would be useless and absolutely irrelevant to that research just as they are to any theory and experiments for gravitation (for example). The conflict of science and religion would continue (or even increase) probably, due to the redline of whether fundamental randomness is accessible to scientific study or on the contrary, this is the tabooed area of God's will.

V BIDIRECTIONAL CAUSALITY IN THREE HYPOSTASES

The conclusion of the three hypostases of quantum causality being bidirectional, though eventually valid as to quantum phenomena, can met the following decisive objection and resistance to be applied to the macroscopic experiments of classical physics and science or to the immediate experience shared by all human beings (as the "invasion" of quantum mechanics into the sacral divine area of God's will suggests), namely:

Reverse causality is relevant only to the microscopic quantum phenomena where both Heisenberg uncertainty and Planck constant are essential and able to influence the observed experimental results considerably. However, it is absolutely inapplicable to the macroscopic human experience since the corresponding quantities of physical actions are much, much greater 2 The term and concept of fundamental randomness distinguishes the randomness involved by quantum mechanics from the usual one of common sense. It can be visualized opposing "Schrödinger's cat" versus "flipping a coin" both sharing a space of events isomorphic to a bit of classical information ("dead/ alive" and "heads/ tails'', accordingly). Nonetheless, the determinist prediction of the latter is possible meaning a finite set of relevant variables. On the contrary, this is impossible to the former according to the [START_REF] Kochen | The problem of hidden variables in quantum mechanics[END_REF] theorem: thus, guessing the state of "Schrödinger's cat" is a "not-P, but NP" problem (following the formulation of the "P vs NP problem"), but "flipping a coin is a "P and NP problem" [START_REF] Penchev | A Class of Examples Demonstrating that 'P ≠ NP' in the "P vs NP' Problem[END_REF].

than the magnitude of the Planck constant. Speaking loosely, the size of matter and energy does matter: thus, only the microscopic quantum world is unstable under both uncertainty and reverse causality. Ergo, the world of our experience is inherently deterministic and predictable, on the one hand, or if one insists that randomness or God's will is essential for it, only religion, but not science, can relate to them, on the other hand.

A frequent response to that disagreement pays attention to the option for any quantum events to be amplified by dissipativity and thus to become noticeable macroscopically. In particular, this is the most common method of measurement in quantum mechanics. Then, any quantum phenomenon can call a fundamental random and unpredictable effect even crucial or fatal as after "Schrödinger's cat". Human brain -mind can be interpreted as such a dissipative amplifier or preamplifier of quantum randomness (or "God's Will" theologically), by which immaterial thoughts can cause physical actions after the neuron impulses be amplified additionally by the human body and its muscles.

Another and much more interesting kind of answer to the same objection explores the identification of the local space obeying quantum uncertainty with the global space of the apparatus and classical mechanics, which is due to the idempotency of the two dual Hilbert spaces of quantum mechanics or gauge symmetries meant by the Standard model. Then, the idempotency of both time arows (respectively, that of both deterministic and reverse causality) is interpreted to be isomorphic to the former as well as dependable on the choice of an either internal or external reference frame. Meaning their relativity investigated in another paper (Penchev 2021April 2012) one can speak of quantum causality as relative, or invariant to deterministic and reverse causality.

Then, the "three hypostases" of quantum causality can be unified in virtue of the same relativity or invariance meant also by quantum-information conservation.

VI HILBERT ARITHMETIC AND BIDIRECTIONAL CAUSALITY Deterministic causality can be represented mathematically as a well-ordering in time therefore implying the axiom of induction in arithmetic and both axiom of choice and well-ordering "theorem" in set theory. Though a well-ordering as in arithmetic as in set theory means the same structure isomorphic to that of natural numbers they are inconsistent to each other as the [START_REF] Gödel | Über formal unentscheidbare Sätze "Principia mathematica" und verwandter Systeme I[END_REF] incompleteness theorems demonstrate. As far as they can be interpreted as the deterministic and quantum causality correspondingly, the latter two ones are inconsistent to each other just as the former two ones: deterministic causality is either incomplete or inconsistent to quantum causality, thus not being able to offer any complete description of reality.

On the contrary, the theorem of the absence of hidden variables in quantum mechanics (Neumann 1832;[START_REF] Kochen | The problem of hidden variables in quantum mechanics[END_REF]) can be interpreted as the completeness of the mathematical formalism itself (namely that of the separable complex Hilbert space being able to be represented as the qubit Hilbert space, and thus, as "quanta" of quantum causality), and then, as the completeness of quantum causality (which is not true as to deterministic causality in virtue of the cited Gödel theorems).

The completeness of quantum mechanics, as confirmed experimentally very well, can serve to build an arithmetic generalizing the standard Peano arithmetic, but unlike the latter, to be consistent with set theory. That generalization is Hilbert arithmetic just doubling Peano arithmetic by an anti-isometric twin therefore implying all the pairs of "number-sake" triples, the one of which belongs to the one counterpart of Peano arithmetic, the second one, to the other, twin and anti-isometric counterpart of Peano arithmetic, and the third one, to the coherent superposition of the former two ones, therefore deprived of any well-ordering. Each of those triple is isomorphic to a bit of classical information.

The structure of Hilbert arithmetic can be defined from the qubit Hilbert space as follows. A unit of Hilbert arithmetic is the class of equivalence of all possible values of a qubit, or in other words, an "empty" qubit of quantum information. Thus, Hilbert arithmetic is able to share the property of unitarity of the separable complex (or qubit) Hilbert space, which in turn underlies the completeness of both structures. Even they can be identified as the same under the additional condition that the qubits be defined as transfinite ordinal numbers, but less than the countable ordinal numbers and thus being the countable equivalent of inaccessible ordinals (respectively, inaccessible countable cardinals). The same ones can be granted definitively to be "transfinite natural numbers" implied by the complement of Peano arithmetic to Hilbert arithmetic following the way described above.

Hilbert arithmetic (being isomorphic or complementary to the qubit Hilbert space) is able to demonstrate all properties of quantum causality as well as its relation to deterministic causality properly and only mathematically. Its completeness together with the incompleteness of deterministic causality implies the existence of physical phenomena being inherently, or fundamentally random: at least, they should be at least the "half of the world". Particularly, "dark matter" and "dark energy" are to be enumerated among them.

VII CONCLUSIONS

The transition from quantum mechanics to quantum information reflects on a wider understanding of quantum causality, the features of which are enumerated sequentially section by section above. The unit of quantum information can be interpreted as a quantum of quantum causality: as a ratio (defined by the coefficients ) of the interaction of deterministic and α, β reverse causality. Quantum causality generalizes deterministic causality and relies on quantum-information conservation in turn generalizing energy conservation, to which deterministic causality is linked closely.

Particularly, the existence of nontemporal physical phenomena can be predicted and the very well corroborated experimentally "dark matter" and "dark energy" are to be enumerated among them.

Quantum causality can be defined in a narrow sense, as reverse causality, and then, three "hypostases" of quantum causality in a wide meaning being linked to irreversibility or reversibility of time: the standard deterministic causality irreversible and "forwards in time"; the proper quantum (or reverse) causality also irreversible, but "backwards in time"; at last the reversible coherent state sharing simultaneously both former irreversible ones. That bidirectional causality in three hypostases can be formalized as a fundamental mathematical structure and generalizing Peano arithmetic in a way to be consistent to set theory and therefore overcoming the restriction of the Gödel incompleteness. It can be called Hilbert arithmetic and can be identified either as complementary or as equivalent to the qubit Hilbert space in turn equivalent to the separable complex Hilbert space of quantum mechanics. Transfinite natural numbers can be defined in the framework of Hilbert arithmetic so that a bijection exists between all transfinite natural numbers and all elements (wave functions) of the separable complex Hilbert space.

The same equation can be realized as the "relativity of classical and quantum information, of bit and qubit"(Penchev

April 12).