Thomas M Michelitsch

Federico Polito
email: federico.polito@unito.it

Alejandro P Riascos

Semi-Markovian discrete-time telegraph process with generalized Sibuya waiting times

Keywords: Non-Markovian random walk, telegraph (Cattaneo) process, generalized Sibuya distribution, discrete-time renewal process

In a recent work we introduced a semi-Markovian discrete-time generalization of the telegraph process. We referred this random walk to as 'squirrel random walk' (SRW). The SRW is a discrete-time random walk on the one-dimensional infinite lattice where the step direction is reversed at arrival times of a discrete-time renewal process and remains unchanged at uneventful time instants. We first recall general notions of the SRW. The main subject of the paper is the study of the SRW where the step direction switches at the arrival times of a generalization of the Sibuya discrete-time renewal process (GSP) which only recently appeared in the literature. The waiting time density of the GSP, the 'generalized Sibuya distribution' (GSD) is such that the moments are finite up to a certain order r ≤ m -1 (m ≥ 1) and diverging for orders r ≥ m capturing all behaviors from broad to narrow and containing the standard Sibuya distribution as a special case (m = 1). We also derive some new representations for the generating functions related to the GSD. We show that the generalized Sibuya SRW exhibits several regimes of anomalous diffusion depending on the lowest order m of diverging GSD moment. The generalized Sibuya SRW opens various new directions in anomalous physics.

Introduction

The telegraph process is an important model for transport where the velocity of the moving particle remains finite with a wide range of applications in physically existing and observable transport phenomena [START_REF] Giona | Extended Poisson-Kac Theory: A Unifying Framework for Stochastic Processes with Finite Propagation Velocity[END_REF]. The classical telegraph process (Poisson-Kac process) [START_REF] Goldstein | On diffusion by discontinuous movements and the telegraph equation[END_REF][START_REF] Kac | A stochastic model related to the telegrapher's equation[END_REF] is defined as a one-dimensional motion of a particle with constant velocity where the velocity direction is switched randomly at Poisson renewal times. The classical telegraph process is Markovian inheriting this feature from the Poisson process. Its time evolution is governed by the (hyperbolic) telegrapher's (also called Cattaneo) equation avoiding physically forbidden infinite propagation velocities of the moving particle as occurring in the parabolic standard diffusion equation. Meanwhile, a wide range of semi-Markovian variants of the telegraph process, including fractional generalizations were developed to model anomalous transport (see among others [START_REF] Ovidio | Time-changed processes governed by space-time fractional telegraph equations[END_REF][START_REF] Ovidio | Fractional Diffusion-Telegraph Equations and their Associated Stochastic Solutions[END_REF][START_REF] Garra | Fractional Klein-Gordon equations and related stochastic processes[END_REF][START_REF] Górska | Generalized Cattaneo (telegrapher's) equations in modeling anomalous diffusion phenomena[END_REF][START_REF] Masoliver | Fractional telegrapher's equation from fractional persistent random walks[END_REF][START_REF] Orsingher | Time-fractional telegraph equations and telegraph processes with Brownian time[END_REF]) as well as a tempered space-fractional generalization [START_REF] Beghin | The tempered space-fractional Cattaneo equation[END_REF]. Compte and Metzler considered phenomenological fractional generalizations [START_REF] Compte | The generalized Cattaneo equation for the description of anomalous transport processes[END_REF][START_REF] Compte | Stochastic foundation of normal and anomalous Cattaneo-type transport[END_REF] and related this model with the Montroll-Weiss continuous-time random walk (CTRW) framework. They found ballistic behavior for long times when the waiting time distribution has diverging mean, and enhanced non-ballistic transport in cases in which the waiting time distribution has a finite mean. Such behavior also occurs in the large-time asymptotics in our recent SRW model [START_REF] Michelitsch | Squirrels can little remember: A random walk with jump reversals induced by a discrete-time renewal process[END_REF] and in the model studied in the present paper. Further works considered the occurrence of random velocities [START_REF] Stadje | Telegraph processes with random velocities[END_REF], a relativistic model and analysis of occupation times, respectively [START_REF] Beghin | Probabilistic analysis of the telegrapher's process with drift by means of relativistic transformations[END_REF][START_REF] Bogachev | Occupation time distributions for the telegraph process[END_REF], Erlang distributed velocity reversals [START_REF] Crescenzo | On random motions with velocities alternating at Erlang-distributed random times[END_REF][START_REF] Di Crescenzo | On the Generalized Telegraph Process with Deterministic Jumps[END_REF], distribution of the maximum [START_REF] Cinque | On the distribution of the maximum of the telegraph process[END_REF] -consult also the references therein.

These works refer to continuous-time variants of the telegraph process. On the other hand many real-world datasets for instance in finance refer to discrete observation times [START_REF] Gregorio | Parametric estimation for the standard and geometric telegraph process observed at discrete times[END_REF]. Therefore, it appears natural to consider discrete-time variants of (generalized) telegraph type processes calling still for thorough analytical investigation. In a recent work [START_REF] Michelitsch | Squirrels can little remember: A random walk with jump reversals induced by a discrete-time renewal process[END_REF], we introduced a discrete-time semi-Markovian version of the telegraph process, the 'squirrel random walk' (SRW), which is also subject of the present paper. We chose that name since the SRW walker (the 'squirrel') in a sense has a 'weaker' memory as in walks with a full memory of their history such as the 'elephant' walker in the so called elephant random walk (ERW) [START_REF] Schütz | Elephants can always remember: Exact long-range memory effects in a non-Markovian random walk[END_REF].

Our paper is organized as follows. In Section 2 we give a brief account for discrete-time renewal processes and introduce pertinent generating functions for the present study. These generating functions will be used in Section 3 where we give an outline of basic notions of the SRW. Section 4 is devoted to a generalization of the Sibuya distribution which appeared in the literature only recently [START_REF] Kozubowski | A generalized Sibuya distribution[END_REF]. The speciality of the 'generalized Sibuya distribution' (GSD) is that it has existing integer order moments only up to a certain order. In this way the GSD covers a wide range of behaviors from narrow to broad. In Section 5 we analyze the SRW where the step direction is reversed at generalized Sibuya arrival times. We call this walk the 'generalized Sibuya SRW'. The (anomalous) diffusive features of this walk are analyzed in Section 6.

Discrete-time renewal process and related generating functions

First we consider a discrete-time counting (renewal) process as follows [START_REF] Michelitsch | On discrete time Prabhakar-generalized fractional Poisson processes and related stochastic dynamics[END_REF][START_REF] Michelitsch | Asymmetric random walks with bias generated by discrete-time counting processes[END_REF][START_REF] Pachon | On discrete-time semi-Markov processes[END_REF]:

N (t) = max(n ≥ 0 : J n ≤ t), N (0) = 0, t = 0, 1, 2 . . . (1)
The arrival times (renewal times) J n ∈ N = {1, 2, . . .} (time instants of events, arrivals) are characterized by the random variables

J n = n j=1 ∆t j , J 0 = 0, ∆t j ∈ N (2)
with IID (independent and identically distributed) strictly positive integer increments ∆t j ≥ 1

('interarrival times' or 'waiting times' in the renewal interpretation). The renewal chain ( 2) is a discrete version of a strictly increasing subordinator. We refer to the recent article [START_REF] Pachon | On discrete-time semi-Markov processes[END_REF] elaborating essential elements of the related theory of discrete-time semi-Markov processes.

The increments follow a discrete-time probability density function (PDF)

P(∆t = k) = ψ k , k = 1, 2, . . . ( 3
)
supported on positive integers k ∈ N with ψ 0 = 0 ensuring strictly positive waiting times. We employ the terms PDF and 'density' in both cases, discrete-time and continuous-time. The inverse of the renewal chain (2) is the discrete-time counting process (1) which counts the events (renewals) up to time t. In the present paper we extensively use generating functions (GFs). It is useful to introduce the GF of the waiting time density

u ∆t = ψ(u) = ∞ t=1 ψ t u t , |u| ≤ 1, (4)
which fulfills ψ(u) u=1 = 1 indicating normalization of (3) and be reminded that ψ(t) is supported on non-zero integers t ∈ N. Generally, the notation

f (u) = ∞ t=0 f (t)u t (5)
(with suitably chosen u) stands for the generating function (GF) of discrete functions f (t) supported on t ∈ N 0 . We employ E(. . .) = (. . .) as equivalent notations for expectation values where we will often use

f (∆t) = ∞ r=1 f (r)ψ r . (6)
Convenient is to introduce the indicator function

Θ(J n , t, J n+1 ) =      1 if J n ≤ t ≤ J n+1 -1, 0 otherwise (7)
which is one for N (t) = n and null else. Then, the 'state probabilities' (probabilities for n arrivals up to time t) are given by [START_REF] Michelitsch | Squirrels can little remember: A random walk with jump reversals induced by a discrete-time renewal process[END_REF] (and see [START_REF] Godrèche | Statistics of the Occupation Time of Renewal Processes[END_REF] for a related analysis for continuous time renewal processes)

P[N (t) = n] = Φ (n) (t) = Θ(J n , t, J n+1 ) . ( 8
)
A quantity of interest is the variable B n,t = t -J n containing information on the persistence of N (t) in state n and which gives a connection to the 'aged renewal process' [START_REF] Michelitsch | Squirrels can little remember: A random walk with jump reversals induced by a discrete-time renewal process[END_REF] (and consult [START_REF] Barkai | Aging Continuous-Time Random Walks[END_REF][START_REF] Godrèche | Statistics of the Occupation Time of Renewal Processes[END_REF][START_REF] Schulz | Aging Renewal Theory and Application to Random Walks[END_REF] for the continuous time cases). We have (read

P(A 1 |A 2 ) as the probability of A 1 conditional to A 2 ) f B (τ, t, n) = P[B n,t = τ |N (t) = n] = δ τ,t-Jn Θ(J n , t, J n+1 ) . ( 9
)
We consider its double GF fB (w

, u) = ∞ τ =0 ∞ t=0 f B (τ, t)w τ u t (|u| < 1, |w| ≤ 1) which yields fB (w, u, n) = w -Jn J n+1 -1 t=Jn (wu) t = u ∆t 1 +...+∆tn 1 -(uw) ∆t n+1 1 -uw = [ ψ(u)] n 1 -ψ(uw) 1 -uw ( 10
)
where fB (1,

u, n) = [ ψ(u)] n 1-ψ(u)
1-u recovers the GF of the state probabilities. In these derivations we always use the IID feature of the ∆t j and (4). The following two relations are related with the SRW propagator, namely

g(t; ζ 1 , . . . , ζ n ; ζ n+1 ) = ζ t-Jn+1 n+1 Θ(J n , t, J n+1 )ζ ∆t 1 -1 1 n j=2 ζ ∆t j j , n, t ∈ N 0 , |ζ j | ≤ 1 (11)
recovering for ζ j = 1 the state probabilities and

g(t, {ζ j }) = ∞ n=0 g(t; ζ 1 , . . . , ζ n ; ζ n+1 ). (12)
The

; ζ n+1 ) = ∞ t=0 u t f (t; ζ 1 , . . . , ζ n ; ζ n+1 ) = ζ -1 1 ζ n+1 n j=1 ζ ∆t j j J n+1 -1 t=Jn ζ t-Jn n+1 u t = ζ -1 1 ζ n+1 1 -ψ(uζ n+1 ) 1 -uζ n+1 n j=1 ψ(uζ j ) (|u| < 1) (13)
with ḡ(u, {ζ j })

ζ j =1,u=1 = 1
1-u corresponding to the normalization of the state probabilities g(t, {ζ j })

ζ j =1 = ∞ n=0 Φ n (t) = 1.
For the SRW the particular case when ζ j alternate as ζ 2j+1 = ζ 1 and ζ 2j = ζ 2 is pertinent. Then we have for [START_REF] Garra | Fractional Klein-Gordon equations and related stochastic processes[END_REF] the GF ḡn (u;

ζ 1 , ζ 2 ) =                1 -ψ(uζ 1 ) 1 -uζ 1 ψ(uζ 1 ) ψ(uζ 2 ) ℓ , n = 2ℓ ζ -1 1 ζ 2 1 -ψ(uζ 2 ) 1 -uζ 2 ψ(uζ 1 ) ψ(uζ 1 ) ψ(uζ 2 ) ℓ , n = 2ℓ + 1.
(ℓ = 0, 1, 2, . . .). [START_REF] Gel'fand | Generalized Functions[END_REF] Summation over n yields the GF of (12) as

ḡ(u; ζ 1 , ζ 2 ) = 1 -ψ(uζ 1 ) 1 -uζ 1 + ζ -1 1 ζ 2 ψ(uζ 1 ) 1 -ψ(uζ 2 ) 1 -uζ 2 1 1 -ψ(uζ 1 ) ψ(uζ 2 ) . ( 15
)
We will come back to these GFs in the context of the SRW propagator in the subsequent section.

The squirrel random walk -SRW

Here we give a brief outline of the 'squirrel random walk' (SRW), for an extensive study we refer to our recent work [START_REF] Michelitsch | Squirrels can little remember: A random walk with jump reversals induced by a discrete-time renewal process[END_REF]. The SRW is a discrete-time random walk X t ∈ Z where directed unit steps σ t ∈ {-1, 1} are performed at integer time instants (we denote with t ∈ N 0 the time variable)

X t = t r=1 σ r , t = 1, 2, . . . , X 0 = 0. (16)
The directions of steps are switched at arrival times of a discrete-time renewal process N (t).

A precise definition of the SRW is as follows:

(i) At uneventful time instants t, the squirrel performs a unit step σ t = σ t-1 in the same direction as at t -1 where this holds for t ≥ 2.

(ii) At arrival times t, the squirrel changes the step direction with respect to the previous step σ t = -σ t-1 .

(iii) We define that no step is performed at t = 0 in order to ensure the initial condition X 0 = 0. The first step is performed at t = 1 in the direction σ 1 = σ0 if t = 1 is uneventful and σ 1 = -σ 0 if there is an event at t = 1. The direction σ0 can be thought as either prescribed or randomly chosen.

In the following, we consider σ0 ∈ {-1, 1} as given. From the above it follows that the steps can be represented as

σ t = σ0 [(-1) N (t) -δ t0 ], t ∈ N 0 (17)
where the Kronecker-δ t0 ensures that no step is performed at t = 0. Therefore, given N (t) = n,

X t = σ0 -1 + ∆t 1 -∆t 2 + . . . + (-1) n-1 ∆t n + (-1) n (t -J n + 1) = X (+) t -X (-) t (18)
with initial condition X 0 = 0 and where X (+)

t , X (-) t cover the steps in σ0 -and in the opposite direction, respectively. Now we introduce the propagator (probability that the squirrel at time

t is sitting on X ∈ Z) as follows P[X t = X] = P (X, t) = δ X,Xt , X ∈ Z, t ∈ N 0 (19)
with the Kronecker symbol δ A,B . Now using

δ A,B = 1 2π π -π e iϕ(A-B) dϕ, A, B ∈ Z we have P (X, t) = 1 2π π -π
e iϕX e -iϕXt dϕ [START_REF] Kozubowski | A generalized Sibuya distribution[END_REF] and with ( 18) and ( 15) the characteristic function writes

P ϕ (t) = e -iϕXt = g(t; e -iϕσ 0 , e iϕσ 0 ), ϕ ∈ [-π, π]. (21)
In addition, the GF Pϕ (u) = ḡ(u; e -iϕσ 0 , e iϕσ 0 ) is useful where Pϕ (u)

ϕ=0 = 1
1-u tells us that the propagator P (X, t) is a (spatially) normalized PDF. Then we introduce

P(v, t) = v N (t) = t n=0 P[N (t) = n)]v n (22)
which is a polynomial of degree t ('state polynomial') since P[N (t) = n)] = 0 for n > t as N (t) ≤ t with initial condition P[N (t) = n)] = δ n0 . The feature P(1, t) = 1 reflects normalization of the state probabilities, and for v = -1 the average step is contained, namely

σ t = σ0 [ (-1) N (t) -δ t0 ] = σ0 [P(-1, t) -δ t0 ] (23)
where δ t0 takes into account that no step is performed at t = 0 maintaining the initial condition X 0 = 0. The GF of the average steps then takes

σ(u) = ∞ t=0 σ t u t = σ0 P(-1, u) -1 (24)
with the GF of the state polynomial

P(v, u) = 1 -ψ(u) (1 -u)[1 -v ψ(u)] , |u| < 1, |v| ≤ 1. ( 25
)
The GF of the expected position X t then reads

X(1) (u) = i ∂ ∂ϕ Pϕ (u) ϕ=0 = σ(u) 1 -u = [1 -ψ(u)]σ 0 (1 -u) 2 [1 + ψ(u)] - σ0 1 -u (26)
with the initial condition X(1) (u) u=0 = X 0 = 0. We will focus on the 'generalized Sibuya SRW' where the instants of the step reversals are drawn from a generalization of the Sibuya distribution which is subject of the subsequent section.

Generalized Sibuya counting process

Here we consider a discrete time counting process N λ (t) with IID generalized Sibuya waitingtimes. The resulting generalized Sibuya distribution (GSD) was to our knowledge first introduced and thoroughly studied by Kozubowski and Podgórski [START_REF] Kozubowski | A generalized Sibuya distribution[END_REF]. In the present section we recall the GSD in the light of discrete-time renewal processes and derive also some results which appear to be new.

We construct the 'generalized Sibuya counting process' (GSP) such that it has a discrete waiting-time PDF with finite moments up to order r ≤ m -1 (m ≥ 1) and diverging moments of orders r ≥ m. This implies that d r du r ψ(u)

u=1 is finite for r ≤ m -1 and d r du r ψ(u) u=1 → ∞ for r ≥ m (m ≥ 1)
. We derive the GSD by its waiting time GF as follows

ψλ (u) = Γ(1 -λ)Γ(m) Γ(m -λ) u 1-m [H λ (u) -(1 -u) λ ], 0 ≤ m -1 < λ < m ∈ N, |u| ≤ 1 (27)
with 1 m = ⌈λ⌉ and

H λ (u) = m-1 r=0 (-1) r λ r u r (28)
which removes the terms with alternating signs in the expansion -(1u) λ thus ψλ (u) contains all non-alternating orders u r for r ≥ m of this expansion. One can easily verify that the sign of these terms is (-1) m-1 = sign(H λ (1)) thus ( 27) contains only non-negative coefficients and is of the form

ψλ (u) = u 1-m (H λ (1)) -1 [H λ (u) -(1 -u) λ ] = u 1-m ḡλ (u)
where u 1-m shifts the distribution g λ (t) by m -1 to the left, ensuring that ψ λ (t) = g λ (t + m -1) is nonzero from t ≥ 1. The normalization factor is obtained as

H λ (1) = 1 (m -1)! d m-1 du m-1 (1 -u) λ-1 u=0 = (-1) m-1 λ -1 m -1 = Γ(m -λ) Γ(1 -λ)Γ(m) = (1 -λ) m-1 (1) m-1 .
Let us remark that, although λ / ∈ N, integer values λ = m are admissible retrieving ψm (u) = u corresponding to the trivial (deterministic) counting process N m (t) = t and coinciding with the limit p → 1of a Bernoulli counting process where p indicates the probability of a Bernoulli success. By construction ψλ (u) = O(u) and we employ the Pochhammer symbol

(a) k = a(a + 1) . . . (a + k -1) =          Γ(a + k) Γ(a) , k ∈ N 1, k = 0
and we mention the useful features (a) k+r = (a) k (a + k) r . We notice for later use the property

d ℓ du ℓ H λ (u) = (-λ) ℓ H λ-ℓ (u) (29)
thus 1

H λ (1) d du H λ (u) u=1 = λ(m-1)
λ-1

(λ > 1) and H λ-ℓ (u) = 0 for ℓ ≥ m. Then we can expand (28) with respect to u -1 as follows

H λ (u) = m-1 ℓ=0 (u -1) ℓ ℓ! d ℓ du ℓ H λ (u) u=1 = Γ(m -λ) m-1 ℓ=0 λ ℓ (1 -u) ℓ Γ(1 + ℓ -λ)Γ(m -ℓ) . ( 30
)
Therefore, the GF [START_REF] Orsingher | Compositions, Random Sums and Continued Random Fractions of Poisson and Fractional Poisson Processes[END_REF] writes compactly as

ψλ (u) = u 1-m ḡλ (u) = u 1-m 1 - (1) m-1 (1 -λ) m-1 (1 -u) λ + m-1 ℓ=1 λ ℓ (m -ℓ) ℓ (1 -λ) ℓ (1 -u) ℓ . ( 31
)
The GSP waiting time PDF then writes

ψ λ (t) = Γ(1 -λ)Γ(m) Γ(m -λ) (-1) t+m λ t + m -1 = λΓ(m) Γ(m -λ) Γ(m -λ + t -1) Γ(m + t) (t ∈ N) (32)
where the positiveness of this expression is easily confirmed by accounting for λ = m -1 + µ, µ ∈ (0, 1) (see especially (37)). We refer the PDF (32) to as 'generalized Sibuya distribution' (GSD). In the large time limit we have (we employ symbol ∼ for asymptotic equality and use

Γ(t+a) Γ(t+b) ∼ t a-b as t → ∞) ψ λ (t) ∼ λΓ(m) Γ(m -λ) t -λ-1 (33)
which holds for any λ > 0. Thus the GSD (32) covers any power-law from narrow (large λ) to broad (small λ). Especially, for λ ∈ (0, 1) (m = 1) (32) recovers the fat-tailed (broad) standard Sibuya distribution with diverging first moment [START_REF] Pachon | On discrete-time semi-Markov processes[END_REF][START_REF] Sibuya | Generalized hypergeometric, digamma, and trigamma distributions[END_REF]. The long time asymptotics [START_REF] Whittaker | A Course of Modern Analysis[END_REF], by invoking Tauberian theorems, is obtained from the asymptotic expansion of (31) for u → 1- (1) (in which we can safely neglect the integer powers

with the relevant part ψλ (u) ∼ 1 -(1-u) λ H λ
(1u) n (n > 0) as they do not have distributions with a long tail).

It appears instructive to consider the GSP in the light of a sequential trial scheme which can be adopted for any discrete-time renewal process (see [START_REF] Michelitsch | Asymmetric random walks with bias generated by discrete-time counting processes[END_REF][START_REF] Pachon | On discrete-time semi-Markov processes[END_REF] for details). Perform a sequence of k = 1, 2, . . . ∈ N (GSP-) trials where each trial has two possible outcomes, "success" or "fail" where we introduce the random variables Z k ∈ {0, 1} (k ≥ 1) with Z k = 1 if the outcome is a success and Z k = 0 for a fail and Z 0 = 0 (no trial at t = 0). Then introduce the conditional probability α k = P[Z k = 1|{Z r = 0} r<k ] of a success in the kth trial given there was no success in earlier trials. Then performing at each integer time instant t a trial we have for the GSP counting variable

N λ (t) = t k=1 Z k . ( 34
)
Then the waiting time density ψ λ (t) has the interpretation as the probability of the first GSPsuccess at time t [START_REF] Michelitsch | Asymmetric random walks with bias generated by discrete-time counting processes[END_REF], i.e.

ψ λ (t) = α t (1 -α 1 ) . . . (1 -α t-1 ) = α t S t-1 . (35) Here S k = k r=1 (1 -α r ) = ∞ r=k+1 ψ λ (r)
is the probability of a sequence of k GSP fails (probability of no GSP success in k trials, 'survival probability'). We point out that any discrete PDF ψ(t) indeed can be represented by such a sequential trial scheme with α t = ψ(t)/[ ∞ r=t ψ(r)

(see [START_REF] Pachon | On discrete-time semi-Markov processes[END_REF] for details). Let us elaborate this structure for the GSP. Unlike in a Bernoulli trial process (characterized by the memoryless property α t = p independent of t) the GSD has a memory which is reflected by (35) containing the complete history up to this first GSP success. Then we can rewrite [START_REF] Stadje | Telegraph processes with random velocities[END_REF] in terms of Pochhammer symbols as follows

ψ λ (t) = λ (m -λ) t-1 (m) t . ( 36
)
Now, since (a) k = (a) k-1 (a + k -1), we have (m = ⌈λ⌉)

ψ λ (t) = λ m + t -1 (m -λ) t-1 (m) t-1 = λ m + t -1 1 - λ m 1 - λ m + 1 . . . 1 - λ m + t -2 (37)
coinciding with the representation which ad hoc was introduced by Kozubowski and Podgórski [START_REF] Kozubowski | A generalized Sibuya distribution[END_REF] and has clearly the structure (35) where we identify

α k = λ m + k -1 , m, k ≥ 1 (38) with α 1 = ψ λ (t) t=1 = λ m . Indeed m = 1 (λ = µ ∈ (0, 1
)) retrieves the standard Sibuya process. We notice that if λ = m ∈ N the trivial counting process N m (t) = t is recovered where each trial is a success with ψ m (t) = δ t0 . From (37) we obtain the 'survival probability', i.e. the probability of no event up to time t in a GSP (which we now denote with P[N λ (t

) = 0] = S t = Φ (0) λ (t)) Φ (0) λ (t) = t-1 r=0 1 - λ m + r = (m -λ) t (m) t = Γ(m) Γ(m -λ) Γ(m -λ + t) Γ(m + t) (39)
with initial condition Φ (0)

λ (t) t=0 = 1.
The large time asymptotics is obtained as

Φ (0) λ (t) ∼ Γ(m) Γ(m -λ) t -λ , (t → ∞) (40)
where for m = 1 and λ = µ ∈ (0, 1) these relations recover the case of the standard Sibuya

process.

For what follows we recall the Gauss hypergeometric function defined as [START_REF] Whittaker | A Course of Modern Analysis[END_REF] 2

F 1 (a, b; c; u) = ∞ r=0 (a) r (b) r r! (c) r u r , a, b, c ∈ R, c / ∈ Z ≤0 . ( 41
)
It is sufficient to consider here a, b, c > 0 with a + bc < 0 where for large r the coefficients decay with a power-law as

(a)r (b)r (c)rr! ∼ const r a+b-c-1 (r → ∞).
In this case (41) converges for |u| ≤ 1 and the Gauss summation theorem holds [START_REF] Whittaker | A Course of Modern Analysis[END_REF]:

2 F 1 (a, b; c; 1) = Γ(c)Γ(c -a -b) Γ(c -a)Γ(c -b) , (c > a + b). ( 42
)
The GF of the GSP survival probability (39) has then the form

Φ(0) λ (u) = ∞ r=0 (1) r r! (m -λ) r (m) r u r = 2 F 1 (1,
α k = ψ λ (k) ∞ r=k ψ λ (r) = 2 F 1 (1, k + m -1 -λ; k + m; 1) -1 = λ k + m -1 . ( 44
)
Figure 1: Generalized Sibuya distribution ψ λ (t). We depict the GSD from Eq. ( 32) for µ = 0.3 and different m ≥ 1. The inset shows the results in logarithmic scale for 1 ≤ t ≤ 100, we present with dashed lines the power-law relation ∝ t -m-µ associated to the asymptotic result in Eq. ( 33) for m = 1 and m = 10.

We evaluate the first moment of the random variable T having GSD [START_REF] Stadje | Telegraph processes with random velocities[END_REF] existing for m ≥ 2 (λ > 1), which yields

T λ = d du ψλ (u) u=1 = λ ∞ t=1 (m -λ) t-1 m t-1 t t + m -1 = λ ∞ t=1 (m -λ) t-1 m t-1 (1 - m -1 m -1 + t ) = λ Φ(0) (u) u=1 -(m -1) = λ 2 F 1 (1, m -λ; m; 1) -(m -1) = m -1 λ -1 , m ≥ 2, (λ > 1) (45)
also conveniently obtained by accounting for representation [START_REF] Sibuya | Generalized hypergeometric, digamma, and trigamma distributions[END_REF]. Note that, since ψ λ > 0 on positive integers it is necessarily T λ ≥ 1 which is fulfilled by this relation as m = ⌈λ⌉ ≥ λ.

We also observe that when we put λ = m the first moment is consistent with the behavior of the corresponding trivial counting process (with ψ(t) = δ t1 ). Further, for λ → ∞ we have

T λ = 1 1 + µ-1 m-1 → 1+
reflecting that, for large λ, the GSD becomes extremely narrow. Fig. 1 shows the GSD for different values of m. Notice that with increasing m the GSD becomes more narrow, and this is also reflected by the large time power-law scaling (33).

Bernoulli time-changed with GSP and scaling limits

Before we return to the SRW it appears instructive to highlight some connections of the GSP with pertinent counting processes which have recently appeared in the literature and to consider scaling limits to continuous-time. To this end we introduce the composed counting process N B (N λ (t)) where N B is a Bernoulli process and N λ a GSP independent of N B . This composition is a Bernoulli counting process time-changed with a GSP where Bernoulli trials are performed at arrival times of the GSP (which describes a random clock). For outlines on such compositions we refer to [START_REF] Michelitsch | On discrete time Prabhakar-generalized fractional Poisson processes and related stochastic dynamics[END_REF][START_REF] Michelitsch | Asymmetric random walks with bias generated by discrete-time counting processes[END_REF][START_REF] Orsingher | Compositions, Random Sums and Continued Random Fractions of Poisson and Fractional Poisson Processes[END_REF]. The GF of the waiting time PDF of this composition is given by

χλ,ξ (u) = ψB [ ψλ (u)] = ξ ψλ (u) ξ + 1 -ψλ (u) (46)
with the Bernoulli waiting time GF ψB (u) = ξu ξ+1-u with ξ = p 1-p where p denotes the probability of success in each single Bernoulli trial. The limit p → 1-(i.e. ξ → ∞) N B (N λ (t)) → N λ (t) retrieves the GSP. In the Sibuya case m = 1 the composed process N B (N λ (t)) contains the so called 'fractional Bernoulli counting process' (of type A). The fractional Bernoulli counting process was introduced in [START_REF] Pachon | On discrete-time semi-Markov processes[END_REF] and has the waiting time GF

χλ,ξ (u) = ψB [ ψλ (u)] = ξ ξ + (1 -u) λ [1 -(1 -u) λ ],
λ ∈ (0, 1).

(

) 47
Evoking Tauberian arguments, the long-time asymptotics of the waiting time density of the composed process can be obtained by expanding (46) for u → 1and considering only the lowest non-integer order in 1u (see ( 31)), namely χλ,ξ (u) = ψλ (u)

1 + 1 ξ (1 -ψλ (u)) ∼ 1 - 1 p (1 -ψλ (u)) ∼ 1 - 1 pH λ (1) (1 -u) λ (48)
where 1/p is the mean waiting time in a Bernoulli process. We skip in this asymptotic relation all integer orders in 1u. Hence, we get the density of the composed process

χ λ,ξ (t) = - (-λ) t t! pH λ (1) ∼ λΓ(m) pΓ(m -λ) t -λ-1 , (t → ∞) (49)
having the same tail of the GSD (33) up to the multiplier 1/p (Bernoulli mean). Now we can define a well-scaled limit to continuous time t ∈ hN 0 → R + where (see [START_REF] Michelitsch | On discrete time Prabhakar-generalized fractional Poisson processes and related stochastic dynamics[END_REF] for a thorough outline of such continuum limit procedures)

χλ,ξ 0 (s) = lim h→0 χλ,ξ(h) (e -hs ) = lim h→0 ξ 0 h η ψλ (e -hs )

ξ 0 h η + 1 -ψλ (e -hs ) (50)
with the scaling assumption ξ(h) = ξ 0 h η (ξ 0 > 0 is an arbitrary constant independent of the time increment h and of the physical dimension sec -η ). The scaling exponent η has to be chosen such that this limit exists. Then, accounting for [START_REF] Sibuya | Generalized hypergeometric, digamma, and trigamma distributions[END_REF] with u = e -hs → 1we have the asymptotic relation

ψλ (e -hs ) ∼      1 -h λ s λ + o(h λ ), (λ ∈ (0, 1)) 1 -h T λ s + o(h), (λ > 1) (51)
with T λ given by (45). Therefore, there exist only two possible limits for (50), namely for m = 1 (standard Sibuya case) we have η = λ ∈ (0, 1) thus we obtain for that limit χλ,ξ 0 (s) = ξ 0 ξ 0 + s λ , λ ∈ (0, 1)

(52)
which is the Laplace transform of the Mittag-Leffler density, obtained in [START_REF] Pachon | On discrete-time semi-Markov processes[END_REF] as the continuous time limit of fractional Bernoulli to the fractional Poisson renewal process (see e.g. [START_REF] Laskin | Fractional Poisson process[END_REF] among the many papers on the subject). For m > 1 this limit exists only if we choose η = 1 thus we get (by introducing the new constant time scale constant

ζ 0 = ξ 0 / T λ ) χλ,ξ 0 (s) = ζ 0 ζ 0 + s , λ > 1 (53)
which is the Laplace transform of the exponential density χ λ,ξ 0 (t) = ζ 0 e -ζ 0 t of the standard Poisson process. For m > 1 the composition N B (N λ (t)) converges in the above defined scaling limit to the standard Poisson process. These features also come into play when we consider the scaling limit of the expectation of the rescaled GSP renewal chain n -ρ J (λ) n → J λ for n → ∞ (see Eq. ( 2)) and choose exponent ρ such that this limit exists

e -sJ λ = lim n→∞ exp (- sJ (λ) n n ρ ) = lim n→∞ [ ψλ (e -s n ρ )] n =        e -s λ λ ∈ (0, 1) e -s T λ λ > 1 (54)
where we use the IID feature of the interarrival times together with (4) and we have to choose ρ = 1 λ for m = 1 (standard Sibuya) and ρ = 1 for m > 1. Hence n -ρ J (λ) n → J λ is a stable subordinator.

5 Generalized Sibuya SRW

Large time asymptotics of the expected squirrel position

Here we explore the diffusive features of the generalized Sibuya SRW where the step directions are switched at GSP arrival times. To this end consider first the large time asymptotics of the expected position. From [START_REF] Sibuya | Generalized hypergeometric, digamma, and trigamma distributions[END_REF] we have in the asymptotic expansion three cases

ψλ (u) =                        1 -(1 -u) λ , 0 < λ < 1 1 -T λ (1 -u) - 1 H λ (1) (1 -u) λ + o I [(1 -u)] + o[(1 -u) λ ], 1 < λ < 2 1 -T λ (1 -u) + B (λ) 2 (1 -u) 2 - 1 H λ (1) (1 -u) λ + o[(1 -u) 2 ], λ > 2 (u → 1-). (55)
We denote with the symbol o I [(1-u)] = a 2 (1-u) 2 +a 3 (1-u) 3 .. a power series in (1-u) containing solely integer powers of orders larger than one. Further we use o

[(1-u) λ ] ≪ (1-u) λ as u → 1-. The constant B (λ)
2 is positive (existing for λ > 2) and yields with ( 31)

B (λ) 2 = 1 2! d 2 du 2 ψλ (u) u=1 , (λ > 2, m ≥ 3) = 1 2! (1 -m)(-m)u -m-1 ḡλ (u) + 2(1 -m)u -m d du ḡλ (u) + u 1-m d 2 du 2 ḡλ (u) u=1 = (m -1)(m -λ) (λ -1)(λ -2) = T λ m -λ λ -2 , ( 56
)
where the non-negativeness of

B (λ) 2 can be seen from m = ⌈λ⌉. Since B (λ) 2 = ∞ t=1 t(t-1) 2 ψ λ (t)
this coefficient contains also the second moment of the GSD

T 2 λ = 2B (λ) 2 + T λ = T λ (2m -λ -2) λ -2 = (m -1)(2m -λ -2) (λ -1)(λ -2) , (λ > 2) (57)
where T λ was determined in (45). Further of interest is the variance existing for λ > 2 (m ≥ 3) which yields

V λ = T 2 λ -[ T λ ] 2 = λ(m -1)(m -λ) (λ -1) 2 (λ -2) = λ λ -1 B (λ)
2 .

(

The GSP variance (58) coincides with the expression given in [START_REF] Kozubowski | A generalized Sibuya distribution[END_REF] (see there Definition 1 with Remark 4 and Eq. ( 22) in that paper and identify m = ν + 1, λ = α in their notation). Recall that we mainly consider λ / ∈ N and bear in mind that integer values λ = m are admissible defining the deterministic counting process N m (t) = t where for λ → m-(m ≥ 3) the variance (58) exists and is vanishing. Now with ( 26) and (55) we obtain for the GF of the expected squirrel position the asymptotic relation

Xλ (u) =              σ0 2 (1 -u) λ-2 + o[(1 -u) λ-2 ], 0 < λ < 1 σ0 T λ 2 -1 (1 -u) -1 + σ0 (1 -u) λ-2 2H λ (1) + o I [(1 -u) -1 ] + o[(1 -u) λ-2 ], λ > 1.
(59) The first line in this relation corresponds to standard Sibuya. In order to capture the leading contributions for large times, by Tauberian arguments, we have respectively picked up the lowest integer and non-integer orders in 1u 2 which is consistent with our previous result [START_REF] Michelitsch | Squirrels can little remember: A random walk with jump reversals induced by a discrete-time renewal process[END_REF] -see Eq. ( 26) and identify with For λ ∈ (0, 1) (standard Sibuya) the squirrel escapes to infinity along the direction of σ 0 by a t 1-λ -power law. Physically this can be interpreted by the occurrence of very long waiting times between the step reversals. For narrower GSDs with λ > 1 (shorter waiting times with existing mean T λ ) the squirrel remains trapped close to the departure site where the value X λ (∞) = σ0 2 ( T λ -2) is approached by a t -(λ-1) -power law term which has opposite sign to σ0 (see (61)). For λ = 3 2 we have T 1.5 = 2 and X 1.5 (∞) = 0 where the walk is in the large time limit unbiased (in the average any second step is reversed). For λ < 3 2 (i.e. m = 2 and µ < 0.5 with T λ > 2) the waiting times between the step reversals are still relatively long (the GSD being relatively broad) where X λ (∞) has the same sign as σ0 . In this case the squirrel does not escape in σ0 -direction, but in the average remains trapped on the same side of the departure site (sign( X λ (∞) ) = σ0 ). This behavior changes for λ > 3 2 which means shorter waiting times between the step switches and narrower GSD ( T λ < 2): the sign of X λ (∞) changes and becomes opposite to σ 0 . We can see this more closely if we rewrite (60) for λ > 1 (λ = m -1 + µ and µ ∈ (0, 1)) as

[Remark: For 1 < λ < 2 since -1 < λ -2 < 0 we have o I (1 -u) -1 = c 0 + c 1 (1 -u) + . . . ∈ o(1 -u) λ-
B λ = -1/H λ (1) > 0 for λ ∈ (1, 2)]. This yields X λ (t) ∼                σ 0 2 (2 -λ) t t! → σ0 2Γ(2 -λ) t 1-λ , 0 < λ < 1 σ0 2 [ T λ -2] + σ0 (2 -λ) t 2H λ (1)t! → σ0 2 [ T λ -2] + σ0 t 1-λ 2H λ (1)Γ(2 -λ) λ > 1 (60)
X λ (t) ∼ - σ0 2(m -2 + µ) m -3 + 2µ + Γ(t + 3 -m -µ) Γ(t + 1) Γ(m) Γ(1 -µ) → - σ0 2(m -2 + µ) m -3 + 2µ + Γ(m) Γ(1 -µ) t -(m-2+µ) m = ⌈λ⌉ ≥ 2 (61)
This relation is plotted in Figure 2 for three values of λ ∈ (1, 2) including the asymptotically unbiased case λ = 3 2 where the squirrel approaches the departure site with a -σ 0 (πt) -1

Anomalous diffusive features

In this section we analyze the mean square displacement (MSD) which we denote with X2 λ (t) (with respect to the initial position X λ (0) = 0) and especially focus on the large time asymptotics. The MSD is given by

X 2 λ (t) = t r 1 =1 t r 2 =1 σ r 1 σ r 2 = 2 t r=1 r s=1 σ r σ s - t r=1 σ 2 r = 2K λ (t) -t = -t + 2 t r=1 t s=r (-1) N λ (r;s-r) = -t + 2 t r=0 t-r k=0 (-1) N λ (r;k) -2 t k=0 (-1) N λ (0;k) t ∈ N (62)
where comes into play the new quantity

N λ (r; s -r) = N λ (s) -N λ (r), s ≥ r ≥ 0 (63)
of the so called aged (generalized Sibuya) counting process N λ (r; k) and N λ (0; k) recovers the original GSP. Aged renewal processes have been introduced and analyzed for continuous-time renewal processes [START_REF] Barkai | Aging Continuous-Time Random Walks[END_REF][START_REF] Godrèche | Statistics of the Occupation Time of Renewal Processes[END_REF][START_REF] Schulz | Aging Renewal Theory and Application to Random Walks[END_REF] and only recently for discrete time counting processes [START_REF] Michelitsch | Squirrels can little remember: A random walk with jump reversals induced by a discrete-time renewal process[END_REF]. We emphasize that the aged renewal process N (r; k) (apart of the Markovian cases, Bernoulli and Poisson) depends on the 'aging parameter' r and is different from the original counting process N (k) reflecting non-markovianity of the latter. To evaluate (62) (see [START_REF] Michelitsch | Squirrels can little remember: A random walk with jump reversals induced by a discrete-time renewal process[END_REF] for more details) it is useful to consider first the GF 2 :

ḡv (w, u) = ∞ τ =0 ∞ t=0 w τ u t v N (τ ;t) , |u|, |w| < 1, |v| ≤ 1. (64)
We further introduce the auxiliary function h v (r, t) = Θ(tr) t-r k=0 v N (r;k) and its double GF

hv (w, u) = ∞ r=0 ∞ t=0 w r u t Θ(t -r) t-r k=0 v N (r;k) = ∞ s=0 u s s k=0 ∞ r=0 (wu) r v N (r;k) = ḡv (uw, u) 1 -u (65)
where in the second line we substitute t = r + s and introduced the discrete Heaviside step function Θ(s) = 1 for s ≥ 0 and Θ(s) = 0 else (especially Θ(0) = 1). We then can write for the MSD GF

X(2) (u) = 2 K(u) - u (1 -u) 2 (66) we have d ψλ (u) du =                          λ(1 -u) λ-1 , 0 < λ < 1 T λ + λ H λ (1) (1 -u) λ-1 + o I (1) + o[(1 -u) λ-1 ], 1 < λ < 2 T λ -2B (λ) 2 (1 -u) + λ H λ (1) (1 -u) λ-1 + o I (1) + o[(1 -u) λ-1 ], λ > 2 (u → 1-) (73)
where o I (1) is an expansion containing only integer powers (1 -u) n with n ≥ 1. Thus we obtain (u → 1-)

Kλ (u) = 1 (1 -u) 3 - 2u (1 -u) 2 (1 -ψλ (u))(1 + ψλ (u)) d ψλ (u) du - 1 (1 -u) 2 1 -ψλ (u) 1 + ψλ (u) =                                            1 (1 -u) 3 1 - λ [1 -1 2 (1 -u) λ ] + λ(1 -u) [1 -1 2 (1 -u) λ ] - 1 2 (1 -u) λ-2 [1 -1 2 (1 -u) λ ] = (1 -λ)(1 -u) -3 + o[(1 -u) -3 ], λ ∈ (0, 1) 1 - T λ 2 (1 -u) -2 - λ -1 H λ (1) T λ (1 -u) λ-4 + o[(1 -u) -2 ], λ ∈ (1, 2) (1 -u) -2   1 + B (λ) 2 T λ - T λ 2   + o[(1 -u) -2 ], λ > 2
(74) where only the first two terms in (72) contain the relevant orders. Inversion of (74) yields

(t → ∞) K λ (t) ∼                                (1 -λ) 2 t 2 , 0 < λ < 1 1 - T λ 2 t - λ -1 H λ (1) T λ Γ(4 -λ) t 3-λ → - λ -1 H λ (1) T λ Γ(4 -λ) t 3-λ , 1 < λ < 2   1 + B (λ) 2 T λ - T λ 2   t, λ > 2 (75) The MSD (62) then scales as X 2 λ (t) = 2K λ (t) -t ∼                            (1 -λ)t 2 , 0 < λ < 1 -2 λ -1 H λ (1) T λ Γ(4 -λ) t 3-λ , 1 < λ < 2   1 + 2B (λ) 2 T λ -T λ   t = t T λ T 2 λ -( T λ ) 2 , λ > 2 (76)
where all quantities are non-negative and with the GSD variance V λ = T 2 λ -( T λ ) 2 determined in (58) and the mean waiting time T λ in (45). Hence this relation writes

X 2 λ (t) ∼                          (1 -λ)t 2 , 0 < λ < 1 2(λ -1) Γ(4 -λ) t 3-λ , 1 < λ < 2 λ(m -λ) (λ -1)(λ -2) t , λ > 2. (t → ∞) (77)
In view of the power-laws governing the expected position (60) one can see that X 2 λ (t) ≫ X 2 λ (t) 2 . Therefore, the MSD (77) dominates the large-time asymptotics of the spatial variance of the squirrel motion. The normal diffusive behavior occurring for λ > 2 brakes down at the limits λ = m-(m ≥ 3) where we have V m-= 0 (see ( 56) -( 58)) with deterministic oscillatory squirrel motions. Contrarily to these cases the limit λ = 2is non-deterministic which is expressed by X 2 2-(t) = 2t (see (77)) corresponding to persistent normal diffusion (Brownian motion) of the squirrel with spatial Gaussian limiting distribution of propagator [START_REF] Kozubowski | A generalized Sibuya distribution[END_REF]. The limiting cases λ = nand λ = n+ (n ∈ N) exhibiting respectively distinct behaviors are considered in the Appendix more closely.

We identify three different diffusive large-time regimes for the generalized Sibuya SRW:

(i) A ballistic superdiffusive regime when the GSD is broad with 0 < λ < 1 (standard Sibuya) with a t 2 -law.

(ii) A superdiffusive regime for 1 < λ < 2 with a t 3-λ -law with scaling exponent 1 < 3λ < 2.

(iii) A normal diffusive regime when the GSD is narrow for λ > 2 with emergence of Brownian motion.

These results are consistent with those obtained in our recent paper [START_REF] Michelitsch | Squirrels can little remember: A random walk with jump reversals induced by a discrete-time renewal process[END_REF] by considering general asymptotic features of discrete-time renewal processes. Superdiffusive large time regimes of these types were also reported for continuous time Cattaneo transport models [START_REF] Compte | The generalized Cattaneo equation for the description of anomalous transport processes[END_REF][START_REF] Compte | Stochastic foundation of normal and anomalous Cattaneo-type transport[END_REF]. The generalized Sibuya SRW of the present study covers for different ranges of λ the whole spectrum from anomalous-ballistic (i), over anomalous (ii) to normal (Brownian) diffusion (iii).

Conclusions

In the present paper we have studied a semi-Markovian discrete-time generalization of the telegraph (Cattaneo) process where the waiting times between the step reversals follow the generalized Sibuya distribution -GSD. We called this walk the generalized Sibuya SRW. It turns out that the presented model has a large flexibility to cover a wide range of behaviors including superdiffusive-ballistic, superdiffusive and normal diffusive transport. We have shown that these features are solely governed by the "broadness" of the GSD waiting time density. For follow-up research an interesting subject is the analysis of scaling limits to continuous time and space which define new semi-Markovian generalizations of telegraph (Cattaneo) processes.

Moreover, variants of SRW models in multidimensional spaces appear to be interesting directions. For instance in problems where a walker is moving with constant velocity in a Ddimensional infinite space and changing its velocity direction randomly at the renewal times of a discrete-time counting process such as the GSD or others. The class of generalized Sibuya SRW and similar models open various new directions in random walk theory, general fractional calculus and non-Markovian dynamics in complex systems.

has a 1/ǫ-singularity.

Considering now again n = 2 the two limits λ = 2and λ = 2+, their difference becomes clear when we look at the tails of the GSD [START_REF] Whittaker | A Course of Modern Analysis[END_REF] where m(2+) = 3 = m(2-) + 1. (91)

Then we further have

T 2+ǫ = 2 1+ǫ → 2-≪ T 2 2+ǫ thus V 2+ǫ T 2+ǫ ∼ T 2 2+ǫ T 2+ǫ ∼ 2 ǫ (92)
bringing us back to (78).

Figure 2 : 2 (

 22 Figure 2: Large time behavior of the expected position (61) for λ < 3 2 , λ = 3 2 , and λ > 3 2 . The

  to ψ 2-0 (t) the PDF (90) remains broad for ǫ → 0. Therefore,

⌈λ⌉ indicates the ceiling function, producing the smallest integer larger than or equal to λ.

-law.

We suppress here λ in N λ (t) to emphasize that this deduction holds for any discrete-time renewal process.

Acknowledgments

F. Polito has been partially supported by INdAM/GNAMPA.

where u (1-u) 2 = ∞ t=1 tu t and with (65) we have

To evaluate this relation we need to determine ḡv (w, u) which is the GF of the state polynomial of the aged counting process

where Φ(m) w (u) stand for the double GF of the state probabilities P[N (τ ; t) = m] of the aged process N (τ ; t) which we determine as

e. that a state n at time τ still persists at time t+τ . Using the IID feature of the ∆t and Θ(J n , t+τ,

, m = 0.

(70)

Now we can evaluate (68) to arrive at

This relation contains the GF of the state polynomial ( 25) ḡv (0, u)

Then we obtain for (67) which determines the MSD GF (66) the expression

Now consider the large-time asymptotics of the MSD for the case when N (t) = N λ (t) is the GSP where we denote then (72) with Kλ (u). Using (55) for u → 1 and d ψλ (u)

A Appendix

Let us discuss here the behaviors emerging in the limiting cases λ → nand λ → n+ (n ∈ N), respectively.

(a) λ = 2 + 0 (λ = 2 + ǫ with ǫ → 0+ and m = 3):

i.e. for λ = 2+ the MSD (77)

is singular where the average position X 2+ǫ (t) ∼ -σ0 ǫ 1+ǫ → 0, see (61). In the limit λ = 2+

emerges for large observation times Brownian diffusion (according to case (iii)) where the squirrel position is in the average on the departure site, but with extremely large fluctuations.

We observe in (77) that the normal diffusive behavior is not singular at the limits λ = n+ (n ∈ N) for n > 2 and it is also different at the limit λ = 2which we consider next.

(b) λ = 2 -0 (λ = 2ǫ with ǫ → 0+ and m = 2): Let us compare this limit with (a). We then have H

where with (61) we see that X 2- 2 = 1 4 ≪ X 2 2-ǫ . This limit corresponds to Brownian motion of the squirrel and is different from the (deterministic) trivial oscillatory motion of the case λ = 2 and is also different from the fast Brownian motion emerging in the limit λ = 2+ of (a).

It is worthy of mention that relation (79) can be re-derived in the following different way. GF [START_REF] Orsingher | Compositions, Random Sums and Continued Random Fractions of Poisson and Fractional Poisson Processes[END_REF]

with small ǫ > 0. For u → 1we have

Now consider the second derivative bǫ (u) = 1 2

Although the second derivative does not exist at u = 1 we can define it in a distributional (Gel'fand-Shilov) sense [START_REF] Gel'fand | Generalized Functions[END_REF] to define constant B (λ) 2 for the ('forbidden') limit

We hence have for the GSD variance

which takes us with (76) and T 2-= 1+ back to relation (79).

In fact what we are using in (81) is that the asymptotic expansion of ψ2-ǫ (u) for u → 1captures the dominating contribution of the GSD power-law tail ψ2-ǫ (t) ∼

where we use G'elfand-Shilov distributional relation τ ǫ-3 Γ(ǫ-2) → d 2 dτ 2 δ + (τ ) which only captures the information of the highest moment T 2 2-ǫ . The contribution (85) is in a sense due to the power law tail of τ ǫ-3 Γ(ǫ-2) of ψ 2-ǫ (t) which is dying out for ǫ → 0+ and which is null for the deterministic case with the exact value λ = 2.

In the same way we can consider the mth moment in the limit λ = mfor any m ∈ N. We then have for the tail [START_REF] Whittaker | A Course of Modern Analysis[END_REF] the distributional relation

which leads to the finite limiting value for the mth moment

For m = 1 this yields T 1-= 1 and is different from singular limiting case T 1+ǫ = 1 ǫ → ∞. For a further discussion of the (standard Sibuya) limit λ = 1-, we refer to our recent paper [START_REF] Michelitsch | Squirrels can little remember: A random walk with jump reversals induced by a discrete-time renewal process[END_REF]. Now it is instructive to compare λ = nwith λ = n+ where n ∈ N. In the latter case we have m = ⌈n+⌉ = n + 1 thus (33) takes the form ψ n+ǫ (t) ∼ (n + ǫ)Γ(n + 1) Γ(1ǫ) t -n-1-ǫ (88) which remains 'broad' behaving as t -1-n when ǫ → 0+ (contrarily to the limit λ = n-, see (86) for n = m).


Marianne Jover

Andréa Huau

A Dimensional Approach to Comorbidity Between Developmental Coordination Disorder and Dyslexia: A Study of Graphomotor Abilities

Keywords: developmental coordination disorder, dyslexia, comorbidity, neurodevelopmental disorders, graphomotor abilities, dimensional model

The present study examined the comorbidity between developmental coordination disorder and dyslexia, adopting a dimensional approach. Participants were 120 children aged 8-12 years, with or without developmental disorders. After assessment with a reading test (Alouette) and motor test (MABC), children performed handwriting and graphomotor tasks on a graphic tablet. GAM analyses were used to test the contribution of the children's reading and motor test scores to their handwriting and graphomotor performances. Results revealed a correlation between reading and motor scores, despite considerable heterogeneity within the groups. The variables measured during the handwriting and graphomotor tasks often depended on the children's scores on the motor and reading tests. Results support the hypothesis of shared risk factors for DCD and dyslexia, and confirm the relevance of adopting a dimensional approach to neurodevelopmental disorders.

A Dimensional Approach to Comorbidity Between Developmental Coordination Disorder and Dyslexia: A Study of Graphomotor Abilities

Comorbidity refers to the co-occurrence of two diseases within the same patient [START_REF] Feinstein | The pre-therapeutic classification of co-morbidity in chronic disease[END_REF]. In the case of developmental disorders, it is commonly described as the rule rather than the exception. A number of clinicians and researchers have tackled this challenging subject in the past decade [START_REF] Kaplan | The term comorbidity is of questionable value in reference to developmental disorders: Data and theory[END_REF][START_REF] Pennington | Diagnosing learning disorders. A neuropsychological framework[END_REF][START_REF] Smits-Engelsman | DCD and comorbidity in neurodevelopmental disorder: How to deal with complexity[END_REF].

Frequent comorbidity between two given disorders raises questions about the links between their etiology and pathogenesis, as well as about the quality of their diagnostic classifications [START_REF] Angold | Comorbidity[END_REF]. According to this view, "comorbidity may not imply the presence of multiple disorders or dysfunctions but could reflect our inability to apply a single diagnosis to account for all symptoms" (Dewey, 2018, p. 5). Disorders should therefore be regarded as patterns of underlying impairment on a continuum or set of dimensions, rather than distinct entities or categories. We can thus make a distinction between categorical and dimensional approaches to developmental disorders (Sonuga- [START_REF] Sonuga-Barke | Categorical models of childhood disorder: A conceptual and empirical analysis[END_REF]. The former, classic approach is commonly employed by clinicians using DSM (American Psychiatric Association, 2013) or ICD (World Health Organization, 2019-2021) criteria. The latter can best be described as revolutionary [START_REF] Sonuga-Barke | People get ready': Are mental disorder diagnostics ripe for a Kuhnian revolution[END_REF]. For instance, Thapar et al. (2017) recommend modeling neurodevelopmental disorders as traits, to account for the considerable heterogeneity in clinical profiles, and the overlap and comorbidity between diagnoses. For example, inattention and impulsivity, assessed according to DSM-5 criteria for attention deficit hyperactivity disorder (ADHD), could constitute a trait. More recently, the Research Domain Criteria project set up a cross-diagnostic and dimensional program to document and understand typical and atypical neurodevelopment. By transcending existing diagnostic boundaries, its researchers aim to identify the fundamental components of complex clinical profiles, predict development trajectories, and guide individualized intervention approaches [START_REF] Cuthbert | Le cadre de travail des RDoC: Faciliter la transition de la CIM et du DSM vers des approches dimensionnelles qui intègrent les neurosciences et la psychopathologie[END_REF][START_REF] Licari | The brain basis of comorbidity in neurodevelopmental disorders[END_REF]. Current classifications are still mainly based on the categorical approach, but also incorporate elements of the dimensional approach. For example, in the DSM-5 (APA, 2013), specific learning disorders are no longer divided into three diagnostic categories (reading disorder, mathematics disorder, disorder of written expression; APA, 1994) as they were in the DSM-IV, but are viewed as being on a single dimension with specifiers of learning impairment in three domains: reading, writing, and mathematics [START_REF] Peterson | How specific are learning disabilities?[END_REF].

The work carried out by Pennington and his team around the multiple deficit model, recently renamed the multiple factor model, is part of this dimensional approach [START_REF] Mcgrath | The multiple deficit model: Progress, problems, and prospects[END_REF][START_REF] Pennington | From single to multiple deficit models of developmental disorders[END_REF][START_REF] Pennington | Diagnosing learning disorders. From science to practice[END_REF]. This model rejects the classic hypothesis of a single deficit explaining a given neurodevelopmental disorder. Instead, it postulates that the etiology of neurodevelopmental disorders lies at the intersection of multiple genetic and/or environmental risk and protective factors (susceptibility locus on a chromosome, premature birth, family stimulation, etc.).

These shape the development of the central nervous system, the mediator of cognitive functions, and can trigger a cascade of behavioral disorders. Given the interactions between risk and protective factors, the plasticity of the central nervous system, and compensation between cognitive modules, a single risk factor cannot give rise to a developmental disorder.

For the authors, there are necessarily several, possibly interrelated, predictors that contribute in a probabilistic (and not deterministic) way to the expression of a given neurodevelopmental disorder. Several risk factors must therefore coexist for a neurodevelopmental disorder to appear. This model explains both the comorbidity and the variety of clinical profiles and subtypes in neurodevelopmental disorders. Finally, this model is compatible with the hypothesis of a continuum of severity, whereby a disorder appearing in a severe form is frequently associated with another comorbid disorder [START_REF] Jongmans | Consequences of comorbidity of developmental coordination disorders and learning disabilities for severity and pattern of perceptual-motor dysfunction[END_REF][START_REF] Kaplan | Comorbidity, cooccurrence, continuum: What's in a name? Child: Care[END_REF][START_REF] Schoemaker | Is severity of motor coordination difficulties related to co-morbidity in children at risk for developmental coordination disorder[END_REF].

The comorbidity between developmental coordination disorder (DCD) and dyslexia (DYS) is an extensively documented example of comorbidity between neurodevelopmental disorders. It concerns 10-70% of children with DYS, and 29-70% of children with DCD (for a review, see [START_REF] Jover | Les troubles moteurs chez les enfants dyslexiques: Revue de travaux et perspectives[END_REF]. Beyond the epidemiological research, many studies have specifically examined motor impairments in children with DYS, and reading impairments in children with DCD. Their results show that children with DYS frequently (but not always) exhibit motor difficulties in postural, motor learning and coordination tasks (e.g., [START_REF] Fawcett | Impaired performance of children with dyslexia on a range of cerebellar tasks[END_REF][START_REF] Haslum | Motor performance and dyslexia in a national cohort of 10year-old children[END_REF][START_REF] Moore | Bimanual coordination in dyslexic adults[END_REF]. These motor difficulties are often specific, and rarely affect all motor areas. By the same token, children with DCD regularly have a lower reading level than control children, even if it does not reach the threshold required for a diagnosis of DYS (e.g., [START_REF] Alloway | A comparison of working memory skills and learning in children with developmental coordination disorder and moderate learning difficulties[END_REF][START_REF] Dewey | Developmental coordination disorder: Associated problems in attention, learning, and psychosocial adjustment[END_REF]Kadesjö & Gillberg, 1999).

Despite discrepancies between research results [START_REF] Chaix | Motor impairment in dyslexia: The influence of attention disorders[END_REF][START_REF] Rochelle | Impaired balance in developmental dyslexia? A metaanalysis of the contending evidence[END_REF] and the heterogeneity within neurodevelopmental disorders (e.g., for DYS: [START_REF] Norton | Functional neuroanatomical evidence for the double-deficit hypothesis of developmental dyslexia[END_REF][START_REF] Zoubrinetzky | New insights on developmental dyslexia subtypes: Heterogeneity of mixed reading profiles[END_REF]and for DCD: Gomez, & Sirigu, 2015;Tallet, & Wilson, 2020), some authors have suggested that DCD and DYS share a partly common etiology [START_REF] Kaplan | DCD may not be a discrete disorder[END_REF][START_REF] Nicolson | Procedural learning difficulties: Reuniting the developmental disorders?[END_REF]. The methodologies required to examine this hypothesis are complex. Several recent studies have compared the performances of children with DYS, or DCD, or both (DYSDCD) in the same task. The performances of the children in each group were analyzed to identify commonalities and specificities linked to each diagnosis [START_REF] Geuze | Tapping a rhythm: A problem of timing for children who are clumsy and dyslexic?[END_REF],and the characteristics of the children with DYSDCD were analyzed to check whether they resembled children with DCD or children with DYS, presented a more severe condition, or even exhibited a different profile. The results of these studies showed that children with DYSDCD sometimes have a profile similar to that of children with DYS [START_REF] Biotteau | Procedural learning and automatization process in children with developmental coordination disorder and/or developmental dyslexia[END_REF], or that of children with DCD [START_REF] Bellocchi | Exploring the link between visual perception, visual-motor integration, and reading in normal developing and impaired children using DTVP-2[END_REF], and sometimes exhibit the difficulties of both these groups [START_REF] Woods | Effect of comorbid reading disorder on oculomotor behavior in children with DCD: A study with the Developmental Eye Movement test[END_REF][START_REF] Biotteau | Neuropsychological status of French children with developmental dyslexia and/or developmental coordination disorder: Are both necessarily worse than one[END_REF][START_REF] Downing | Prevalence and cognitive profiles of children with comorbid literacy and motor disorders[END_REF][START_REF] Ho | Reading-related cognitive deficits in developmental dyslexia, attention-deficit/hyperactivity disorder, and developmental coordination disorder among Chinese children[END_REF][START_REF] Maziero | Influence of comorbidity on working memory profile in dyslexia and developmental coordination disorder[END_REF]. Children with comorbid DYS and DCD sometimes have greater difficulties than children with just one disorder [START_REF] Cignetti | Feedforward motor control in developmental dyslexia and developmental coordination disorder: Does comorbidity matter[END_REF][START_REF] Jongmans | Consequences of comorbidity of developmental coordination disorders and learning disabilities for severity and pattern of perceptual-motor dysfunction[END_REF], but this is not always the case [START_REF] Woods | Effect of comorbid reading disorder on oculomotor behavior in children with DCD: A study with the Developmental Eye Movement test[END_REF][START_REF] Maziero | Influence of comorbidity on working memory profile in dyslexia and developmental coordination disorder[END_REF]. However, the classic categorical approach on which this research is based has the disadvantage of relying on strict inclusion criteria and of focusing on children who are often relatively unrepresentative of the clinical population, possibly explaining the contradictory results.

The aim of the present research was to document the comorbidity between DYS and DCD in an innovative way, using a dimensional approach. [START_REF] Visser | Learning difficulties in children with symptoms of DCD and/or ADHD: Analyses from a categorical and a continuous approach[END_REF] recently showed that, depending on whether a categorical or a dimensional approach is adopted, the study of the comorbidity between ADHD and DCD can yield different results. We therefore focused on graphomotor and handwriting difficulties, which are frequently reported in both children with DYS (e.g., [START_REF] Afonso | Writing impairments in Spanish children with developmental dyslexia[END_REF][START_REF] Alamargot | Handwriting delay in dyslexia: Children at the end of primary school still make numerous short pauses when producing letters[END_REF][START_REF] Berninger | Writing problems in developmental dyslexia: Under-recognized and under-treated[END_REF][START_REF] Germano | Characterization of Brazilians students with dyslexia in Handwriting Proficiency Screening Questionnaire and Handwriting Scale[END_REF][START_REF] Gosse | Do children with dyslexia present a handwriting deficit? Impact of word orthographic and graphic complexity on handwriting and spelling performance[END_REF][START_REF] Iversen | Motor coordination difficulties in a municipality group and in a clinical sample of poor readers[END_REF][START_REF] Lam | Chinese handwriting performance of primary school children with dyslexia[END_REF][START_REF] Pagliarini | Dyslexic children fail to comply with the rhythmic constraints of handwriting[END_REF] and children with DCD (e.g., Huau, Velay, & Jover, 2015;[START_REF] Jolly | Analysis of cursive letters, syllables, and words handwriting in a French second-grade child with developmental coordination disorder and comparison with typically developing children[END_REF]O'Hare & Khalid, 2002;Prunty & Barnett, 2020;[START_REF] Prunty | Handwriting speed in children with developmental coordination disorder: Are they really slower?[END_REF], 2014;[START_REF] Rosenblum | Handwriting process and product characteristics of children diagnosed with developmental coordination disorder[END_REF][START_REF] Rosenblum | Handwriting features of children with developmental coordination disorder-Results of triangular evaluation[END_REF][START_REF] Smits-Engelsman | Fine motor deficiencies in children with developmental coordination disorder and learning disabilities: An underlying open-loop control deficit[END_REF]. A literature review revealed deficits in the quality of the writing: both the process, (i.e., way of producing the gesture) and the end-product in children with DCD or DYS [START_REF] Rosenblum | Product and process evaluation of handwriting difficulties[END_REF]. At the product level, children with DCD exhibited letter malformation and macrography, which was not the case for children with DYS. At the process level, studies based on graphic tablet recordings showed that children with DYS or DCD had a lower writing speed than controls, mainly owing to more writing pauses [START_REF] Prunty | Handwriting speed in children with developmental coordination disorder: Are they really slower?[END_REF]Sumner, Connelly, & Barnett, 2013). Children with DCD had significantly lower handwriting fluency than that of typically developing children, with more spikes in speed. This variable was not measured in studies among children with DYS. Finally, significant intra-individual variability was found in both disorders. In sum, graphomotor dysfunctions were detected in both children with DCD and children with DYS. The remaining question concerns the origin of these dysfunctions, and their similarity or dissimilarity in these two neurodevelopmental disorders.

According to [START_REF] Downing | Understanding writing difficulties amongst children with neurodevelopmental disorders. The cases of dyslexia and/or developmental coordination disorder (DCD)[END_REF], the graphomotor difficulties of children with DYS are mainly due to fluency problems, whereas those of children with DCD arise from problems of letter formation (legibility). The reduced fluency in DYS, reflected in longer pauses during the handwriting process, can be explained by slow decoding of the words increasing working memory load and by the visual back and forth between the model and the sheet when copying.

Handwriting difficulties are therefore the result of spelling and reading difficulties here [START_REF] Arfé | The effects of script and orthographic complexity on the handwriting and spelling performance of children with dyslexia[END_REF][START_REF] Martínez-García | Handwriting production in Spanish children with dyslexia: Spelling or motor difficulties?[END_REF]Suárez-Coalla, Afonso, Martínez-García, & Cuetos, 2020;Sumner, Connelly, & Barnett, 2014). By contrast, children with DCD tend to have difficulty setting up the handwriting motor program and performing the graphic movement (Barnett & Prunty, 2020;Huau et al., 2015;Jolly, Danna, & Jover, in preparation).

The aim of the present research was to use a dimensional approach to improve our understanding of the comorbidity between DCD and DYS, as well as the writing difficulties observed in children with these disorders. Accordingly, after recruiting a group of children with DCD, DYS, or both (DYSDCD), together with a group of age-matched controls (CTL), we administered standardized tests to measure their levels of reading and motor skills. Based on the multiple factor model, we expected children's motor and reading skills to lie on a continuum of impairment, with considerable heterogeneity in performances on both tests [START_REF] Pennington | From single to multiple deficit models of developmental disorders[END_REF][START_REF] Pennington | Diagnosing learning disorders. From science to practice[END_REF]. In addition, we administered a graphomotor precision task and a word writing task with or without speed constraint, as well as a writing test (BHK; Hamstra-Bletz, [START_REF] Hamstra-Bletz | Beknopte Beoordelingsmethode voor Kinderhandschriften [A concise method for the evaluation of children's handwriting[END_REF]. The aim was to measure the proportion of variance in performances on the graphomotor and writing tasks (process and product) explained by the children's reading and motor skills scores. To this end, we calculated generalized additive models (GAMs), which investigate nonlinear relationships and interactions between variables without a priori assumptions on their form. We expected the children's motor level to contribute more than their reading level to the variability in performances, especially on the graphomotor (vs. handwriting) tasks. We expected children with a low reading level to make longer pauses in writing tasks. Finally, performances on long or constrained tasks would depend more heavily on the children's motor skills and reading performances. The variables related to the writing product would be impacted more by motor performance, and those related to the writing process by reading performance, at least when these performances were impaired. The dimensional approach would therefore make it possible to identify the specific and possibly nonlinear effects of reading and motor difficulties associated with DYS and/or DCD on graphomotor and handwriting performances.

Methods

Participants

Participants were 130 children (Table 1), divided into two groups. The first group (n = 65 children, age range: 8-12 years) had exhibited difficulty acquiring reading and/or coordination, leading professionals to establish a diagnosis of DYS and/or DCD with their usual tools (DYS, DCD and DYSDCD groups). All the children were receiving treatment for one and/or the other of these disorders. They were contacted through a professional network (psychomotor therapists, speech therapists) or through a local learning disability reference center (Timone Marseille Hospital, Aix-en-Provence Hospital). We did not control for the presence of other comorbidities, except for ADHD. Children with ADHD, together with those who exhibited excessively impaired graphomotor skills, (spontaneous capital letter writing) were excluded from the sample.

The second group (n = 65) comprised children matched for age who had never attended a consultation for developmental difficulties (CTL group). These children were recruited from a primary school and a junior high school in the Marseille region. Children who had repeated a grade were excluded from the sample. A questionnaire attached to the information and consent form that was sent to parents allowed us to identify the children who needed to undergo a speech and/or psychomotor assessment. These children were excluded from the sample.

This study was carried out in accordance with the Declaration of Helsinki, and parents and children all signed a consent form after being informed of the study's objectives and content.

Procedure and Material

All the children in the final sample were assessed with the Alouette reading test and the Movement Assessment Battery for Children (MABC;Henderson & Sugden, 1992), and performed graphomotor and handwriting tasks. The study took place either in the laboratory or at the children's school.

The French adaptation [START_REF] Soppelsa | Manuel de la Batterie d'Evaluation du Mouvement chez l'Enfant (MABC)[END_REF] of the MABC was used to assess the motor skills of children aged 4-12 years in three domains: manual dexterity, ball skills, and static and dynamic balance. This battery contains items specific to four different age groups (4-6 years, 7-8 years, 9-10 years, 11-12 years). For each item, the instruction is accompanied by a demonstration and then a training trial, followed by formal trials. The test yields a score for each domain and a total impairment score. In the present study, the total score, expressed as a percentile, was used to characterize each child's level of motor performance.

The Alouette reading test (Lefavrais, 1965;revised version, 2005) is widely used in France to assess the accuracy and speed of reading aloud in children from elementary school to middle school. The text comprises 265 words, and the reading time is limited to 3 minutes.

The sentences are not integrated into an overall structure such as a narrative, and drawings around the text depict words from the text or words that are phonologically related. Children are told to read the text and are informed that no comprehension questions will be asked. We used the corrected speed-accuracy reading score, which takes into account both reading accuracy and speed, and the z score was converted into a percentile to characterize each child's reading level.

The BHK is a handwriting test for children (French adaptation: [START_REF] Charles | BHK -Echelle d'évaluation rapide de l'écriture chez l'enfant[END_REF][START_REF] Hamstra-Bletz | Beknopte Beoordelingsmethode voor Kinderhandschriften [A concise method for the evaluation of children's handwriting[END_REF] or adolescents [START_REF] Soppelsa | BHK Ado. Echelle d'évaluation rapide de l'écriture chez l'adolescent[END_REF] that consists in copying a text for 5 minutes. The first five sentences comprise monosyllabic words encountered in first grade, but the text then becomes more complex, and the size of the characters decreases. It is copied on a white unlined A4 sheet. Writing quality is scored on a set of criteria (max. 13 for the Child version; 9 for the Adolescent version). A speed score is also calculated from the number of letters copied in 5 minutes. The quality and speed scores were z-transformed according to age and sex, and used as the dependent variables.

The graphomotor control task consisted in tracing the displacement of a mouse toward a piece of cheese (Fig. 1). The first step was to connect five targets measuring 2.5 x 3 mm that were spaced 50 mm apart, and the second was to draw a line between two lines spaced 2 mm apart. The model was printed on a sheet of paper positioned on the graphic tablet. This task was performed either at the speed chosen by the children (normal speed condition) or as quickly as possible (fast speed condition). The numbers of missed targets and line crossings were summed to use as a dependent variable (Precision). The graphomotor control task and the Lapin word writing task were performed on a Wacom Intuos 4L graphic tablet. Its A4 format recording surface (29.7 x 21 cm) allows pen movements to be recorded with good spatial (0.25 mm) and temporal (200 points / s) precision. Children wrote with an Inking Pen, a ballpoint pen of classic size and heft. The tablet was placed on the school desk in the landscape orientation. Data processing was carried out with the Ecriture Suite 2 software [START_REF] Gilhodes | Ecriture Suite 2[END_REF]. The Segment Run tool was used offline to clean up the recorded data. The Stats Trace tool allowed us to extract various indices from the recordings. The variables recorded by the tablet and used in this study were 1) trace length (mm; i.e., length of the child's production), where the trace is unwound like a thread to provide an index of production, 2) mean velocity (mm / s), a kinematic indicator of the writing movement, 3) pause duration (s), namely the total time spent still, without moving or lifting the pen, accounting for lack of fluency in the movement.

The precision, corrected trials and trace length variables provided indices of the result/ product of the graphomotor movement. The mean velocity and stop duration variables accounted for the motor processes during task performance.

Statistical Analysis

To describe the sample, we compared the scores of the children of the CTL, DCD, DYS and DYSDCD groups) with a Kruskal-Wallis ANOVA, and carried out post hoc analyses using the Dwass-Steel-Critchlow-Fligner test.

The following analyses were performed on the data of the whole sample. The motor and reading assessment scores were subjected to a correlation analysis (Spearman's rho).

Generalized additive models (GAMs) were used to identify the contributions of motor skills and reading performance (as assessed by the MABC and the Alouette), and their possible interaction, to children's graphic and writing tasks. GAMs make it possible to model the link between a variable to be explained and several continuous predictors. The objective is to maximize the prediction of a dependent variable from several predictors and achieve the best fit to real-world data by considering the possible nonlinear relationships between explanatory variables. An F test is used to gauge the significance of the contribution of the explanatory variables (p < .05), and R 2 indicates the variance explained by the model. Graphical representations are provided to interpret the effects of the predictors. They make it possible to identify whether the interaction effects are absent, simple, linear, or nonlinear, based on the color gradient and the orientation of the isolines representing the linear term. The standard error is shown in dotted lines on the graphs, to indicate prediction quality. The GAMs were estimated using the mgcv package in R [START_REF] Woods | Effect of comorbid reading disorder on oculomotor behavior in children with DCD: A study with the Developmental Eye Movement test[END_REF].

Results

Alouette and MABC Scores

Table 1 summarizes the scores of the children in the four groups and the results of the intergroup comparisons. The group effect was significant for both the MABC (ꭕ 2 = 61.5, p < .001, ε² = .477) and Alouette (ꭕ 2 = 67.9, p < .001, ε² = .526) scores. Paired comparisons carried out with the Dwass-Steel-Critchlow-Fligner test showed that the differences between the groups were always significant, except between the DYSDCD and DCD groups for the MABC score, and between the DYSDCD and DYS groups for the Alouette score. Figure 2 shows the children's distribution according to their scores on the MABC and Alouette tests. Correlations between scores were significant (rho = .34, p < .001).

GAM Analyses: Effects of Reading and Motor Skills Levels on Graphomotor Dependent

Variables

In the graphs illustrating the significant GAM models (Figs. 3, 4, 5 and 6) reading level (Alouette scores as percentiles) is on the x-axis, and motor skills level (MABC scores as percentiles) is on the y-axis. The black dots represent the children in the sample. In the heat maps, the color gradient corresponds to the values of the dependent variable predicted by the model. The lighter the color, the higher the predicted value, and the darker the color, the lower the value. For example, in Figure 3, a high number of missed targets is shown in pale yellow, and a low number is shown in red. The green lines are isolines indicating the predictor gradients, where 0 represents the mean. They can be either rectilinear, indicating a linear effect of reading and motor skills scores, or curved, indicating a nonlinear effect of potentiation of one explanatory variable by the other. When the lines are oblique, the effects of reading level and motor skills were additive. When the lines are vertical, reading had a predominant effect, and when they are horizontal, motor skills had a predominant effect. A second type of model presentation illustrates the variability of the GAM estimates. The black lines represent the main estimates, and the red and green dashed lines -1 and +1 standard error respectively.

Contributions of motor skills and reading levels to graphomotor control task.

There was an additive linear effect of reading and motor skills levels on graphomotor performances. The lower children's reading and motor skills levels, the greater the numbers of they missed targets and line crossings, in both normal and fast speed conditions (Fig. 3). This effect was also observed for mean velocity (Fig. 4), as the better the children's reading and motor skills, the faster they completed the task. Finally, in both models, the fast speed condition clearly increased the variance explained by the model. The GAM model applied to the parameters recorded when writing the word Lapin at normal speed did not reveal a significant effect of either motor skills or reading levels. By contrast, in the fast speed condition, several effects appeared (Fig. 5). The graphs show a simple effect of reading level on the number of corrected trials, such that the children with more severe reading difficulties had more corrected trials. In addition, there was an additive effect of reading and motor skills on trace length: the more severe the children's reading and motor difficulties, the longer their trace. Pause duration was sensitive to both reading and motor skills levels: children with a low reading level made longer pauses when they had a good level of motor skills. Good readers with severe motor difficulties spent less time pausing. Writing velocity, by contrast, did not appear to be sensitive to either motor skills or reading levels.

Normal speed condition Fast speed condition

Contributions of motor skills and reading levels to BHK scores.

Regarding BHK scores, the GAM was significant both for quality and for speed (Fig. 6). There was an interaction between the influence of reading and motor skills levels on the quality score. This effect was linear when children had low motor and reading skills, such that the more difficulties they had, the lower the quality. In addition, children with low motor skills had a low score, regardless of their reading level. The effect became nonlinear when children had average-to-high levels of motor and reading skills. In this case, children produced high-quality writing and there was a simple effect of reading.

Concerning the BHK speed score, we observed a simple effect of reading (Fig. 6), such that the lower the children's reading level, the lower their writing speed.

Quality score Speed score

Discussion

The aim of this research was to examine how a dimensional approach to disorders can provide a better understanding of the comorbidity between DYS and DCD and the manifestation of handwriting difficulties in these two disorders. We therefore recruited a (Alouette & Lefavrais, 1965) tests, and we analyzed the links between their scores. We expected children in the CTL, DCD, DYS and DYSDCD groups to lie on a continuum, with children with DYSDCD displaying more impaired performances than children with just one disorder. In addition, we recorded children's graphomotor output in a graphomotor precision task, a word writing task with or without speed constraint, and a handwriting test (BHK). We then ran GAMs to test the contributions of children's reading and motor performances to the variance in measurements recorded during the graphomotor and writing tasks. We expected motor skills level to contribute more than reading level, especially to graphomotor precision.

We predicted that in the writing tasks, children with a low reading level would make longer pauses. Finally, we predicted that performance on lengthy tasks or tasks with a speed constraint would depend more clearly on the children's motor skills and reading levels. In sum, the variables pertaining to the writing product or writing process would not be sensitive to motor and reading scores in the same way. Our results did not validate all these hypotheses, and even invalidated some of them.

The first interesting result of our analyses concerned children's scores on the Alouette and MABC tests. The clinical profiles of the control children and children with DCD, DYS, or DYSDCD we recruited were more similar than expected. For instance, some of the children who had no reading or coordination problems, according to their parents, had considerable difficulty performing the Alouette or MABC tests. This observation has two possible non-mutually exclusive explanations. First, some children probably have difficulties that go unnoticed by their parents or teachers. Some children also find ways of compensating for their difficulties in the course of everyday life, but these difficulties are still revealed by specific standardized tests. More children presented worrying test scores for motor difficulties, doubtless reflecting the fact that schools are far more vigilant for reading difficulties than for physical ones, which are socially better tolerated [START_REF] Blank | International clinical practice recommendations on the definition, diagnosis, assessment, intervention, and psychosocial aspects of developmental coordination disorder[END_REF].

Children with reading or motor skills test scores below the 5 th percentile are likely to encounter substantial difficulties at home and in school if no help is offered. This observation reinforces the need for elementary and middle schools to have qualified staff who can identify children at risk of neurodevelopmental disorders and refer them to specialists, who can then make appropriate recommendations in terms of assessment and follow-up (Haute Autorité de Santé, 2017, 2020; see also [START_REF] Camden | Understanding parental concerns related to their child's development and factors influencing their decisions to seek help from health care professionals: Results of a qualitative study[END_REF].

Analysis of MABC and Alouette test scores across the whole sample showed that the children were distributed almost over the entire graph. Some had a very high level of motor skills and a very low or very high reading level, while others had a very high reading level and either very good or very poor motor skills. There is therefore considerable heterogeneity: not all children with DYS have a motor disturbance [START_REF] Ramus | The relationship between motor control and phonology in dyslexic children[END_REF], and not all children with DCD have impaired reading ability [START_REF] Zwicker | Developmental coordination disorder: A review and update[END_REF]. However, in the present study, MABC and Alouette scores were significantly correlated. In addition, the group comparison showed that children with DYS performed more poorly on average than the CTL group on the MABC test. By contrast, children with DCD performed more poorly on average than the CTL children on the Alouette test. Children with isolated DCD or DYS therefore generally exhibit a low level of functioning in the domain not concerned by the neurodevelopmental disorder. These results confirm the severity continuum hypothesis whereby in its severe form, one disorder is more frequently associated with another one [START_REF] Jongmans | Consequences of comorbidity of developmental coordination disorders and learning disabilities for severity and pattern of perceptual-motor dysfunction[END_REF][START_REF] Kaplan | Comorbidity, cooccurrence, continuum: What's in a name? Child: Care[END_REF][START_REF] Schoemaker | Is severity of motor coordination difficulties related to co-morbidity in children at risk for developmental coordination disorder[END_REF]. These elements reinforce the relevance of adopting a dimensional approach to neurodevelopmental disorders: the transitions from typical to atypical functioning, and from isolated disorder to comorbid situation occur on a continuum [START_REF] Dewey | What is comorbidity and why does it matter in neurodevelopmental disorders?[END_REF]Licari et al., 2018;[START_REF] Pennington | From single to multiple deficit models of developmental disorders[END_REF]Thapar et al. 2017).

GAMs provided an opportunity to analyze the respective effects of motor skills and reading on graphomotor and writing tasks. Results showed that reading and motor skills levels, respectively assessed with the Alouette (Lefavrais, 2005) and MABC (Henderson & Sugden, 1992) tests, influenced children's performance on the graphomotor control task in an additive way. Velocity and precision were higher in children with high levels of both reading and motor skills, and lower in children who displayed difficulties in both domains at the same time. Performances were average in children with poor motor skills and good reading skills, and in children with poor reading skills and good motor skills. Even though this graphomotor task did not involve language, the contribution of reading level was equivalent to that of motor skills. Insofar as earlier research had suggested that children with DYS have reduced motor abilities, this result was unexpected in a study that considered levels of both motor skills and reading, and in which we controlled for comorbidity. However, it corroborates recent observations showing that writing difficulties are correlated with children's reading difficulties [START_REF] Arfé | The effects of script and orthographic complexity on the handwriting and spelling performance of children with dyslexia[END_REF][START_REF] Martínez-García | Handwriting production in Spanish children with dyslexia: Spelling or motor difficulties?[END_REF][START_REF] Afonso | Writing impairments in Spanish children with developmental dyslexia[END_REF]Sumner et al., 2014). In our opinion, it shows that this simple graphomotor task requires skills related to both reading and motor control, and which can be compensated for. However, further research needs to be undertaken to understand this result better.

When children had to write the word Lapin with a speed constraint, we observed specific reading level effects. The number of corrected trials depended exclusively on reading level, such that children with a low score on the Alouette (Lefavrais, 2005) made the most corrections. For their part, Sumner and Connelly (2020) found that even at university, students with dyslexia made more crossings out than their peers did. The MABC score did not influence this variable. Trace length and stop duration also depended on both reading and motor skills. However, for the latter variable, the color gradient indicated that children who paused the longest were poor readers with good motor skills. Children with poor motor skills tended to spend less time pausing. This result is consistent with the findings of Downing Finally, analysis of BHK scores showed that in this copy task, motor skills and reading levels both influenced children's performance. Writing quality was most impaired in children with poor motor skills, regardless of their reading level. However, this effect was nonlinear, and when children reached the average motor skills level, reading level became decisive.

Being a proficient reader therefore determined writing quality when children had no motor difficulties. This result sheds light on the many determinants involved during writing, and the complex interactions between linguistic and motor skills in this activity [START_REF] Gosse | Do children with dyslexia present a handwriting deficit? Impact of word orthographic and graphic complexity on handwriting and spelling performance[END_REF] see also Danna, Longcamp, Nalborczyk, Velay, Commengé, & Jover, under review).

Concerning writing speed, a simple reading effect was observed, whereby the lower the children's reading level, the lower their BHK score. Children with poor reading skills never had a high writing speed, whereas children with poor motor skills could write quickly if they could read well. This result should be set against the nature of the BHK test (Hamstra-Bletz, de Bie, & Berry, 1987), which is a text copy test. It is easy to see why children with reading difficulties would perform a copying task more slowly, as this task places particularly heavy demands on working memory (Sumner et al., 2014). Our results also showed that children with a low motor score were not necessarily slower than the others if they did not have to read. This result should be considered in the light of [START_REF] Barnett | Handwriting difficulties in developmental coordination disorder (DCD)[END_REF]'s recent review, which indicated that not all children with DCD are slower.

Ultimately, it is interesting to note that time pressure (or task length) in the graphomotor and writing tasks was linked to GAMs of greater explanatory power. Thus, motor skills and reading levels contributed more to the predicted variable when the situation was constrained. This result is consistent with previous research among children with DCD or DYS, where the gap with controls widened according to the level of task constraint [START_REF] Gosse | Do children with dyslexia present a handwriting deficit? Impact of word orthographic and graphic complexity on handwriting and spelling performance[END_REF]Huau et al., 2015). It confirms the findings of clinicians and of developmental studies highlighting an increase in the difficulties of children with neurodevelopmental disorders when school and/or social constraints become more severe [START_REF] Pennington | Diagnosing learning disorders. From science to practice[END_REF].

The results of the present study provide a better understanding of the discrepancies between research indicating that children have more severe difficulties in the case of comorbidity (e.g., [START_REF] Jongmans | Consequences of comorbidity of developmental coordination disorders and learning disabilities for severity and pattern of perceptual-motor dysfunction[END_REF] and research that does not (e.g., [START_REF] Maziero | Influence of comorbidity on working memory profile in dyslexia and developmental coordination disorder[END_REF]. Depending on the tasks and variables analyzed, the effects of reading and motor skills levels may either be independent, additive, or interact in a nonlinear way.

In conclusion, the indices measured in the graphomotor and writing tasks depended on children's performances on both the MABC motor skills and Alouette reading tests. The effect of reading level was comparable to that of motor skills in the graphomotor tasks, but became more salient in the writing tasks, when the children had to write at a fast speed. This study

shows how important reading level is in writing tasks, and confirms more or less recent results showing a link between basic reading tasks and writing [START_REF] Johannessen | The look of writing in reading. Graphetic empathy in making and perceiving graphic traces[END_REF][START_REF] Longcamp | Visual presentation of single letters activates a premotor area involved in writing[END_REF] or models of human cognition (see Grw factor in Cattell-Horn-Carroll model; Schneider & McGrew, 2018). However, this link is not addressed in diagnostic classifications such as DSM-5 or ICD-11, and writing difficulties are not included among the signs of specific reading impairment [START_REF] Peterson | How specific are learning disabilities?[END_REF].

The results of this research support the multiple factor model explanation for symptom overlaps and comorbidities in neurodevelopmental disorders [START_REF] Mcgrath | The multiple deficit model: Progress, problems, and prospects[END_REF]Pennington, 2009;[START_REF] Pennington | Diagnosing learning disorders. From science to practice[END_REF]. The existence of shared risk and protective factors explains the presence of a continuum of severity (Jongman et al., 2003;[START_REF] Kaplan | Comorbidity, cooccurrence, continuum: What's in a name? Child: Care[END_REF][START_REF] Schoemaker | Is severity of motor coordination difficulties related to co-morbidity in children at risk for developmental coordination disorder[END_REF], where children in the greatest difficulty at the motor level, are generally also in the greatest difficulty at the reading level. GAM models helped to identify the contributions of motor skills and reading levels to task performances, and provided an innovative means of understanding comorbidity and its effects on individuals.

Our results show that while most of the comorbidity effects were additive, reading and motor difficulties had distinct effects on writing pauses, attesting to the influence of specific factors [START_REF] Mcgrath | The multiple deficit model: Progress, problems, and prospects[END_REF][START_REF] Pennington | From single to multiple deficit models of developmental disorders[END_REF].

Nevertheless, the present study had several limitations that restrict the generalization of its results. First, we used children's composite scores on a single test as an indicator of their level in reading or motor skills. This methodological bias constitutes a limitation, as no single test can capture such complex skills. For example, the Alouette test focuses on the decoding process, does not consider text comprehension, and includes drawings meant to mislead children (Lefavrais, 2005). Likewise, the MABC test is not the gold standard for motor assessment (Henderson & Sugden, 1992). These results must therefore be replicated before they can be generalized. It would also have been interesting to compare the dimensional analysis with a categorical analysis, in order to better understand the value of each approach [START_REF] Visser | Learning difficulties in children with symptoms of DCD and/or ADHD: Analyses from a categorical and a continuous approach[END_REF]. Furthermore, a latent profile analysis (e.g., Spurk, Hirschi, Wang, Valero, & Kauffeld, 2020) would have allowed us to look for different profiles, based on the children's test performances. Finally, we only described a small number of indices and graphomotor and writing tasks here. Handwriting analysis with digital tools is a useful way of identifying the handwriting parameters that are typical of a particular difficulty [START_REF] Gargot | Acquisition of handwriting in children with and without dysgraphia: A computational approach[END_REF][START_REF] Palmis | Motor control of handwriting in the developing brain: A review[END_REF]. More research is needed to pinpoint the differences between the dysgraphic handwriting of children with DYS and the dysgraphic handwriting of children with DCD (Jolly, Danna, & Jover, in preparation).

In conclusion, we showed that the dimensional analysis of neurodevelopmental disorders provides a means of documenting their comorbidity and etiology in a complementary way. This article is part of a more general epistemological movement to redefine developmental disorders [START_REF] Habib | The neurological basis of developmental dyslexia and related disorders: A reappraisal of the temporal hypothesis, twenty years on[END_REF][START_REF] Pennington | Diagnosing learning disorders. From science to practice[END_REF][START_REF] Sonuga-Barke | People get ready': Are mental disorder diagnostics ripe for a Kuhnian revolution[END_REF]. In our opinion, future studies should more carefully consider children's comorbidity and the variability in their clinical profiles, in order to better contribute to the understanding of the risk and protective factors involved.

Figure 1 .

 1 Figure 1. Model of the graphomotor control task.

  Note. DYS: dyslexia; DCD: developmental coordination disorder; DYSDCD: DCD + DYS; CTL: controls; SD: standard deviation; ns: nonsignificant. * p < 0.05. ** p < 0.01. *** p < 0.01.

Figure 2 .

 2 Figure 2. Participants' MABC and Alouette scores.

Figure 3 .

 3 Figure 3. Precision in the graphomotor control task: statistical values and graphical

Figure 4 .

 4 Figure 4. Mean velocity in the graphomotor control task: statistical values and graphical

Figure 5 .

 5 Figure 5. Parameters recorded for the word Lapin in the fast speed condition: statistical values

Figure 6 .

 6 Figure 6. BHK test scores. Statistical values and graphical representations of the GAM

(

  2018) andJolly, Danna, and Jover (in preparation) for children with DYS, DCD and DYSDCD. It supports the hypothesis that children with DYS have specific writing difficulties, compared with children with DCD.

Table 1

 1 Group Comparisons on MABC and Alouette Scores

				MABC	Alouette
			Mean age
	Group	n (girls)		mean percentile	mean percentile
			(SD)
				(SD)	(SD)
	CTL	65 (28)	115 (13)	45.6 (27.7)	61.9 (18.5)
	DYS	33 (10)	119 (14)	30.7 (25.7)	16 (18)
	DYSDCD 12 (3)	118 (14)	3.57 (2.63)	12.6 (16.4)
	DCD	20 (4)	115 (13)	2.86 (4.09)	39.7 (28.4)

Acknowledgments

The present research was supported by a grant from the French National Research Agency (ANR-13-APPR-0010-04). The authors would like to thank Jean-Luc Velay for his help with the recordings, and Bruno Dauvier for his assistance with the analyses. This study would not have been possible without the participation of the children, the consent of their parents, and the support of local teachers, psychomotor therapists, and speech therapists. We are most grateful to them all.

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