Lucile Laulin

Introducing smooth amnesia to the memory of the Elephant Random Walk

Keywords: primary 60G50, secondary 60G42, 60F17 Elephant random walk, Amnesic random walk, Multi-dimensional martingales, Almost sure convergence, Asymptotic normality, Distributional convergence

This paper is devoted to the asymptotic analysis of the amnesic elephant random walk (AERW) using a martingale approach. More precisely, our analysis relies on asymptotic results for multidimensional martingales with matrix normalization. In the diffusive and critical regimes, we establish the almost sure convergence and the quadratic strong law for the position of the AERW. The law of iterated logarithm is given in the critical regime. The distributional convergences of the AERW to Gaussian processes are also provided. In the superdiffusive regime, we prove the distributional convergence as well as the mean square convergence of the AERW.

Introduction

The Elephant Random Walk (ERW) is a discrete-time random walk, introduced by Sch ütz and Trimper [START_REF] Sch Ütz | Elephants can always remember: Exact long-range memory effects in a non-markovian random walk[END_REF] in the early 2000s. At first, the ERW was used in order to see how longrange memory affects the random walk and induces a crossover from a diffusive to superdiffusive behavior. It was referred to as the ERW in allusion to the traditional saying that elephants can always remember anywhere they have been. The elephant starts at the origin at time zero, S 0 = 0. At time n = 1, the elephant moves one step to the right with probability q and to the left with probability 1q for some q in [0, 1]. Afterwards, at time n + 1, the elephant chooses uniformly at random an integer k among the previous times 1, . . . , n. Then, it moves exactly in the same direction as that of time k with probability p or the opposite direction with the probability 1p, where the parameter p stands for the memory parameter of the ERW. The position of the elephant at time n + 1 is given by

S n+1 = S n + X n+1 (1.1)
where X n+1 is the (n + 1)-th increment of the random walk, such that

X n+1 = α n+1 X β n+1 (1.2)
where α n+1 ∼ R(p) and β n+1 ∼ U (1, n) are mutually independent and independant of the past. The ERW shows three differents regimes depending on the location of its memory parameter p with respect to the critical value p = 3/4. On the one hand, a wide literature is now available on the ERW in dimension d = 1 thanks to a variety of approaches. Baur and Bertoin [START_REF] Baur | Elephant random walks and their connection to p ólya-type urns[END_REF] used the connection to P ólya-type urns as well as functional limit theorems for multitype branching processes due to Janson [START_REF] Janson | Functional limit theorems for multitype branching processes and generalized P ólya urns[END_REF]. Bercu [START_REF] Bercu | A martingale approach for the elephant random walk[END_REF] and Coletti et al. [START_REF] Coletti | Central limit theorem and related results for the elephant random walk[END_REF] used martingales to obtain the almost sure convergence and asymptotic normality, among other results. K ürsten [START_REF] Ürsten | Random recursive trees and the elephant random walk[END_REF] and Businger [START_REF] Businger | The shark random swim (Lévy flight with memory)[END_REF] used the construction of random trees with Bernoulli percolation. A strong law of large numbers and a central limit theorem for the position of the ERW, properly normalized, were established in the diffusive regime p < 3/4 and the critical regime p = 3/4, see [START_REF] Baur | Elephant random walks and their connection to p ólya-type urns[END_REF][START_REF] Bercu | A martingale approach for the elephant random walk[END_REF][START_REF] Coletti | Central limit theorem and related results for the elephant random walk[END_REF][START_REF] Coletti | Asymptotic analysis of the elephant random walk[END_REF][START_REF] Guevara | On the almost sure central limit theorem for the elephant random walk[END_REF]. In the superdiffusive regime p > 3/4, Bercu [START_REF] Bercu | A martingale approach for the elephant random walk[END_REF] proved that the limit of the position of the ERW is not Gaussian and Kubota and Takei [START_REF] Kubota | Gaussian fluctuation for superdiffusive elephant random walks[END_REF] showed that the fluctuation of the ERW around this limit is Gaussian.

On the other hand, over the last years, various processes derivated from the ERW have recevied a lot of attention. Bercu and Laulin in [START_REF] Bercu | On the multi-dimensional elephant random walk[END_REF] extended all the results of [START_REF] Bercu | A martingale approach for the elephant random walk[END_REF] to the multidimensional ERW (MERW) where d ≥ 1 and to its center of mass [START_REF] Bercu | On the center of mass of the elephant random walk[END_REF] using a martingale approach, while Bertenghi used the connection [START_REF] Bertenghi | Functional limit theorems for the multi-dimensional elephant random walk[END_REF] to P ólya-type urns for the MERW. The ERW with stops or minimal RW, changing in particular the distribution of α n , has also been investigated [START_REF] Bercu | New insights on the minimal random walk[END_REF][START_REF] Bercu | On the elephant random walk with stops playing hide and seek with the Mittag-Leffler distribution[END_REF][START_REF] Gut | The number of zeros in elephant random walks with delays[END_REF][START_REF] Miyazaki | Limit theorems for the 'laziest' minimal random walk model of elephant type[END_REF]. The ERW with reinforced memory has been studied by Baur [START_REF] Baur | On a class of random walks with reinforced memory[END_REF] via the urn approach, and Laulin [START_REF] Laulin | New Insights on the Reinforced Elephant Random Walk Using a Martingale Approach[END_REF] using martingales.

The idea of this paper is to use the approach developped in [START_REF] Bercu | On the center of mass of the elephant random walk[END_REF] and [START_REF] Laulin | New Insights on the Reinforced Elephant Random Walk Using a Martingale Approach[END_REF] to study how changing the memory allows us to induce amnesia to the ERW. More precisely, the distribution of the memory β n of our new variation of the ERW is such that the probability of choosing a fixed instant k ∈ N * at time n ≥ k decreases approximatly with speed (k/n) β for some amnesia parameter β ≥ 0. The very interesting question of amnesic elephant random walk (AERW) has not been investigated a lot. Gut and Stadm üller [START_REF] Gut | Variations of the elephant random walk[END_REF][START_REF] Gut | The elephant random walk with gradually increasing memory[END_REF] studied variations of the memory for the special cases of ERW with delays or gradually increasing memory. In [START_REF] Gut | Variations of the elephant random walk[END_REF] the elephant could stop and only remember the first (and second) step it tooks. Consequently, it did not induced a phase transition. In [START_REF] Gut | The elephant random walk with gradually increasing memory[END_REF], the elephant only remembered a portion of its past (recent or distant), this portion being fixed or depending on the time n, but was always "small".

The entire study we conduct below can be generalized when β < 0 is not an integer. This can be interpreted as cases where the elephant remembers more vividly the first steps it performed. When β < -1, it appears that the AERW only have one regime that is the diffusive regime. This observation is coherent with the work of Gut and Stadm üller [START_REF] Gut | The elephant random walk with gradually increasing memory[END_REF].

The AERW will appear to be non-Markovian, as the reinfroced ERW. However, unlike the reinforced ERW, the AERW can not be studied using P ólya-type urns. The major change for the AERW is that the distribution of the memory β n in equation (1.2) is no longer uniform but depends on the amnesia parameter β ≥ 0. In this approach, the elephant chooses an instant according to β n+1 as follows,

P(β n+1 = k) = (β + 1)Γ(k + β)Γ(n) Γ(k)Γ(n + β + 1) = (β + 1) n µ k µ n+1 for 1 ≤ k ≤ n, (1.3)
where

µ n = n-1 ∏ k=1 1 + β k = Γ(n + β) Γ(n)Γ(β + 1)
.

(1.4)

The case β = 0 corresponds to the traditionnal ERW. As β grows, the probability of choosing a recent instant gets bigger, see the illustrative Figure 1. We have by definition of the step X n+1 given in (1.2) and the distribution

β = 0 β = 1 β = 2 β = 3 β = 10 β = 100
β n+1 (1.3) that E[X n+1 | F n ] = E[α n+1 ]E[X β n+1 | F n ] = (2p -1)E n ∑ k=1 X k 1 β n+1 =k | F n = (2p -1)(β + 1) nµ n+1 n ∑ k=1 X k µ k . (1.5)
Then, denote a = 2p -1 and

Y n = n ∑ k=1 X k µ k . (1.6)
We deduce from (1.5) that

E[Y n+1 | F n ] = 1 + a(β + 1) n Y n . (1.7)
Hereafter, for any n ≥ 1, let

γ n = 1 + a(β + 1) n (1.8)
and Our strategy for proving asymptotic results for the AERW is as follows. On the one hand, the behavior of the position S n is closely related to the one of the sequences (M n ) and (N n ) defined, for all n ≥ 0, by

a n = n-1 ∏ k=1 γ -1 k = Γ(n)Γ(a(β + 1) + 1) Γ(n + a(β + 1)) . ( 1
M n = a n Y n and N n = S n + a(β + 1) β -a(β + 1) µ -1 n Y n .
(1.12)

We immediatly get from (1.7) and (1.9) that (M n ) is a locally square-integrable martingale adapted to (F n ). Moreover, we have from (1.5) that

E S n+1 + a(β + 1) β -a(β + 1) µ -1 n+1 Y n+1 | F n = S n + a(β + 1) β -a(β + 1) µ -1 n Y n
which also means that (N n ) is also a locally square-integrable martingale adapted to F n . On the other hand, we can rewrite S n as

S n = N n - a(β + 1) β -a(β + 1) (µ n a n ) -1 M n (1.13)
and equation (1.13) allows us to establish the asymptotic behavior of the AERW via an extensive use of the martingale theory.

The main results of this paper are given in Section 2. We first investigate the diffusive regime and we establish the strong law of large numbers, the law of iterated logarithm and the quadratic strong law for the AERW. The functional central limit theorem is also provided. Next, we prove similar results in the critical regime. Finally, we establish a strong limit theorem in the superdiffusive regime. Our martingale approach is described in Section 3. Finally, we give some of the technical proofs in Section 4.

Main results

The diffusive regime

Our first result deals with the strong law of large numbers for the AERW in the diffusive regime where p < 4β+3 4(β+1) . The following strong law for the AERW will be deduced from both the strong laws for (N n ) and (M n ). The almost sure rate of convergence for the AERW is as follows, for

σ 2 β = 2β + 1 -a (1 -a)(1 + 2β -2a(β + 1))
.

Theorem 2.2. We have the quadratic strong law

lim n→∞ 1 log n n ∑ k=1 S 2 k k 2 = σ 2 β a.s. (2.2)
Hereafter, we are interested in the distributional convergence of the AERW, which holds in the Skorokhod space D([0, ∞[) of right-continuous functions with left-hand limits.

Theorem 2.3. The following convergence in distribution in D([0, ∞[) holds S nt √ n , t ≥ 0 =⇒ W t , t ≥ 0 (2.3)
where W t , t ≥ 0 is a real-valued centered Gaussian process starting from the origin with covariance

E[W s W t ] = a(1 + β)(1 -a) + aβ (2(β + 1)(1 -a) -1)(a -β(1 -a))(1 -a) s t s a-β(1-a) + β (β(1 -a) -a)(1 -a) s (2.4) for 0 < s ≤ t.
In particular, we have

S n √ n L -→ N 0, σ 2 β .
(2.5)

Remark 2.4. When β = 0 we find again the results from [START_REF] Baur | Elephant random walks and their connection to p ólya-type urns[END_REF] for the ERW

S nt √ n , t ≥ 0 =⇒ W t , t ≥ 0
where W t , t ≥ 0 is a real-valued mean-zero Gaussian process starting from the origin and

E[W s W t ] = 1 1 -2a s t s a .

The critical regime

Hereafter, we investigate the critical regime where p = 4β+3 4(β+1) . It is interesting to notice that, when β is really large (or β → ∞) the critical regime is reached for the memory parameter p = 1. Hence, the greater β is, the more there are values of the memory parameter p for which the AERW stays in the diffusive regime; but whatever the value of β, we still observe a phase transition.

lim n→∞ S n √ n log n = 0 a.s. (2.6)
The almost sure rates of convergence for the AERW are as follows.

Theorem 2.6. We have the quadratic strong law

lim n→∞ 1 log log n n ∑ k=1 S 2 k (k log k) 2 = (2β + 1) 2 a.s.
(2.7)

In addition, we also have the law of iterated logarithm lim sup n→∞ S (2.10)

The superdiffusive regime

Finally, we focus our attention on the superdiffusive regime where p > 4β+3 4(β+1) .

Theorem 2.8. We have the following distributional convergence in D([0, ∞[)

S nt n a(β+1) , t ≥ 0 =⇒ (Λ t , t ≥ 0) (2.11)
where the limiting Λ t = t a(β+1) L β , L β being some non-denegerate random variable. In particular, we have

lim n→∞ S n n a(β+1)-β = L β a.s.
(2.12)

where the limiting L β is a non-degenerate random variable. We also have the mean square convergence

lim n→∞ E S n n a(β+1)-β -L β 2 = 0.
(2.13)

Remark 2.9. The expected value of L β is

E[L β ] = a(β + 1)(2q -1)Γ(β + 1) a(β + 1) -β Γ a(β + 1) + 1 (2.14)
while its second order moment is given by

E L 2 β = a 2 (β + 1) 2 Γ(β + 1) 2 Γ 2(a -1)(β + 1) + 1 a(β + 1) -β 2 Γ (2a -1)(β + 1) + 1 2 .
(2.15)

When β = 0 we find again the expected values for the ERW from [START_REF] Bercu | A martingale approach for the elephant random walk[END_REF] E

[L] = 2q -1 Γ(a + 1) and E[L 2 ] = 1 (2a -1)Γ(2a)
.

A two-dimensional martingale approach

In order to investigate the asymptotic behavior of (S n ), we introduce the two-dimensional martingale (M n ) defined by

M n = N n M n (3.1)
where (M n ) and (N n ) are the two locally square-integrable martingales introduced in (1.12).

As for the CMERW and the RERW, the main difficulty we face is that the predictable quadratic variations of (M n ) and (N n ) increase to infinity with two different speeds. A matrix normalization will again be necessary to establish the asymptotic behavior of the AERW. We will alternatively study

(M n ), (M n ) or (N n ). Denote the martingale increment ε n+1 = Y n+1 -γ n Y n .
We obtain that

∆M n+1 = M n+1 -M n = S n+1 -S n + a(β+1) β-a(β+1) Y n+1 µ n+1 -Y n µ n a n+1 Y n+1 -a n Y n = 1 + a(β+1) β-a(β+1) X n+1 - a(β+1) (β-a(β+1))µ n+1 β n Y n a n+1 ε n+1 = β (β-a(β+1))µ n+1 Y n + X n+1 µ n+1 -(γ n -1)Y n a n+1 ε n+1 .
Consequently

∆M n+1 = β (β-a(β+1))µ n+1 a n+1 ε n+1 .
We also obtain that

E[ε 2 n+1 | F n ] = E[Y 2 n+1 | F n ] -γ 2 n Y 2 n = Y 2 n + 2(γ n -1)Y 2 n + µ 2 n+1 -γ 2 n Y 2 n = µ 2 n+1 -(γ n -1) 2 Y 2 n . (3.2)
Therefore, we deduce that

E (∆M n+1 )(∆M n+1 ) T | F n = (µ 2 n+1 -(γ n -1) 2 Y 2 n ) β (β-a(β+1))µ n+1 2 βa n+1 (β-a(β+1))µ n+1 βa n+1 (β-a(β+1))µ n+1 a 2 n+1 .
We are now able to compute the quadratic variation of M n

M n = n-1 ∑ k=0 β β-a(β+1) 2 βa k+1 µ k+1 β-a(β+1) βa k+1 µ k+1 β-a(β+1) (a k+1 µ k+1 ) 2 -ξ n (3.3)
where

ξ n = n-1 ∑ k=0 (γ k -1) 2 Y 2 k β (β-a(β+1)) 2 βa k+1 µ k+1 (β-a(β+1)) βa k+1 µ k+1 (β-a(β+1)) (a k+1 µ k+1 ) 2 .
Hereafter, we immediatly deduce from (3.3) that

M n = n ∑ k=1 (a k µ k ) 2 -ζ n where ζ n = n ∑ k=1 a 2 k (γ k -1) 2 Y 2 k (3.4)
and

N n = β β -a(β + 1) 2 n.
(3.5)

The asympotic behavior of M n is closely related to the one of

w n = n ∑ k=1 (a k µ k ) 2 (3.6)
as one can observe that we always have M n ≤ w n and that ζ n is negligeable when compared to w n . Consequently, it follows from the definitions of (a n ) and (µ n )that we have three regimes of behavior for (M n ). In the diffusive regime where is p < 4β+3 4(β+1) or a < 1 -1 2(β+1) ,

lim n→∞ w n n 1-2(a(β+1)-β) = where = 1 1 + 2(β -a(β + 1)) Γ(a(β + 1) + 1) Γ(β + 1) 2 .
(3.7)

In the critical regime where p = 4β+3 4(β+1) or a = 1 -1 2(β+1) ,

lim n→∞ w n log n = Γ(β + 1 + 1 2 ) Γ(β + 1) 2 .
(3.8)

In the superdiffusive regime where p > 4β+3 4(β+1) or a > 1 -1 2(β+1) ,

lim n→∞ w n = ∞ ∑ k=1 Γ(a(β + 1) + 1)Γ(k + β) Γ(k + a(β + 1))Γ(β + 1) 2 < +∞. (3.9)
4 Proofs of the main results

The diffusive regime

Lemma 4.1. Let (V n ) be the sequence of positive definite diagonal matrices of order 2 given by

V n = 1 √ n 1 0 0 a(β+1) β-a(β+1) (a n µ n ) -1 . (4.1) Let v = 1 -1 such that v T V n M n = S n √ n . (4.2)
The quadratric variation of M n satisfies in the diffusive regime where is a < 1 -1 2(β+1) ,

lim n→∞ V n M n V T n = V a.s. (4.3)
where the matrix V is given by

V = 1 (β -a(β + 1)) 2 β 2 aβ 1-a aβ 1-a a 2 (β+1) 2 1+2β-2a(β+1) .
(4.4) Remark 4.2. Following the same steps as in the proof of Lemma 4.1, we find that in the critical regime a = 1 -1 2(β+1) , the sequence of normalization matrices (V n ) has to be replaced by

W n = 1 n log n 1 0 0 (2β + 1)(a n µ n ) -1 .
(4.5)

The limit matrix V also need to be replaced by

W = (2β + 1) 2 0 0 0 1 . ( 4
V n M n V T n = lim n→∞ 1 n   ∑ n-1 k=0 β (β-a(β+1)) 2 a(β+1)β (β-a(β+1)) 2 a n µ n ∑ n-1 k=0 a k+1 µ k+1 a(β+1)β (β-a(β+1)) 2 a n µ n ∑ n-1 k=0 a k+1 µ k+1 a(β+1) (β-a(β+1))a n µ n 2 ∑ n-1 k=0 (a k+1 µ k+1 ) 2   = 1 (β -a(β + 1)) 2   β 2 a(β+1)β β+1-a(β+1) a(β+1)β β+1-a(β+1) a(β+1) 2 2(β-a(β+1))+1  
which is exactly what we wanted to prove.

Proof of Theorem 2.1. We shall make extensive use of the strong law of large numbers for martingales given, e.g. by theorem 1.3.24 of [START_REF] Duflo | Random iterative models[END_REF]. First, we have for (M n ) that for any γ > 0,

M 2 n = O (log w n ) 1+γ w n a.s.
which by definition of M n and as a n is asymptotically equivalent to n -a(β+1) and w n is asymptotically equivalent to n 1+2(β-a(β+1)) ensures that

Y 2 n n 2 = O (log n) 1+γ n 1+2(β-a(β+1)) n 2(1-a(β+1)) a.s.
Finally as µ n is asymptotically equivalent to n β , we obtain that

Y 2 n (µ n n) 2 = O (log n) 1+γ n a.s. which reduces to lim n→∞ Y n µ n n = 0 a.s. (4.7)
We now focus our attention on (N n ). By the same token as before, we have that for any γ > 0, (H.1) We have from (4.3) and the fact that a nt is asymtotically equivalent to t -a(β+1) a n that

N 2 n = O (log n) 1+γ n a.s. which by definition of (N n ) gives us S n -a(β+1) β-a(β+1) µ -1 n Y n 2 n 2 = O (log n) 1+γ n a.
V n M nt V T n -→ n→∞ V t a.s.
where

V t = 1 (β -a(β + 1)) 2    β 2 t aβ 1 -a t 1+β-a(β+1) aβ 1 -a t 1+β-a(β+1) a 2 (β + 1) 2 1 + 2β -2a(β + 1) t 1+2β-2a(β+1)    .
(H.2) In order to verify that Lindeberg's condition is satisfied, we start by deducing from (1.12) together with (3.1) and V n given by (4.1) that for all 1 ≤ k ≤ n

V n ∆M k = 1 (β -a(β + 1)) √ nµ n β µ n µ k a a k a n ε k which implies that V n ∆M k 2 = 1 (β -a(β + 1)) 2 n β 2 µ 2 k + a 2 a 2 k (a n µ n ) 2 ε 2 k . ( 4.9)
Consequently, we obtain that for all ε > 0,

n ∑ k=1 E V n ∆M k 2 1 { V n ∆M k >ε} | F k-1 ≤ 1 ε 2 n ∑ k=1 E V n ∆M k 4 | F k-1 . (4.10)
It follows from (1.10) that

a -2 n n ∑ k=1 a 2 k = O(n) and a -4 n n ∑ k=1 a 4 k = O(n).
Hence, using that the sequence

(ε n ) is bounded sup 1≤k≤n |ε k | ≤ sup 1≤k≤n (β + 2)µ k ≤ (β + 2)µ n a.s. (4.11)
we find that

n ∑ k=1 E V n ∆M k 4 | F k-1 = O 1 n a.s.
which ensures that Lindeberg's condition (H.2) holds almost surely, that is for all ε > 0,

lim n→∞ n ∑ k=1 E V n ∆M k 2 1 { V n ∆M k >ε} | F k-1 = 0 a.s. (4.12) Since V n V -1 nt converges, we immediatly obtain that lim n→∞ nt ∑ k=1 E V n ∆M k 2 1 { V n ∆M k >ε} | F k-1 ≤ lim n→∞ nt ∑ k=1 E V n ∆M k 4 ≤ lim n→∞ nt ∑ k=1 E (V n V -1 nt )V nt ∆M k 4 = 0 a.s.
(H.3) In this particular case, we have

V t = tK 1 + t α 2 K 2 + t α 3 K 3 where α 2 = 1 -a(β + 1) > 0 and α 3 = 1 -2a(β + 1) > 0 as a < 1 -1 2(β+1)
, and the matrix are symmetric

K 1 = β 2 (β -a(β + 1)) 2 1 0 0 0 , K 2 = aβ (1 -a)(β -a(β + 1)) 2 0 1 1 0 , K 3 = a 2 (β + 1) 2 (1 + 2β -2a(β + 1))(β -a(β + 1)) 2 0 0 0 1 .
Consequently, we obtain that

V n M nt , t ≥ 0 =⇒ B t , t ≥ 0
where B is defined as in Theorem A.2 from [START_REF] Laulin | New Insights on the Reinforced Elephant Random Walk Using a Martingale Approach[END_REF]. Finally, using the fact that S nt is asymp-

totically equivalent to N nt + t β-a(β+1) a(β+1) β-a(β+1) (µ n a n ) -1 M nt , and multiplying by u t = 1 t a(β+1)-β , we conclude 1 √ n S nt , t ≥ 0 =⇒ W t , t ≥ 0 (4.13)
where

W t = u T t B t . It only remains to compute the covariance function of (W t ) that is for 0 ≤ s ≤ t E W s W t = u T s E B s B T t u t = u T s V s u t = u T s sK 1 + s 1+β-a(β+1) K 2 + s 1+2β-2a(β+1) K 3 )u t = β 2 (β -a(β + 1)) 2 s + aβs 1+β-a(β+1) (1 -a)(β -a(β + 1)) 2 (s a(β+1)-β + t a(β+1)-β ) + a 2 (β + 1) 2 (1 + 2β -2a(β + 1))(β -a(β + 1)) 2 s 1+2β-2a(β+1) (st) a(β+1)-β = a(1 + β)(1 -a) + aβ (2(β + 1)(1 -a) -1)(a -β(1 -a))(1 -a) s t s a-β(1-a) + β (β(1 -a) -a)(1 -a)
s.

Proof of Theorem 2.2. We need to check that all the hypotheses of Theorem A. Therefore, we can replace log(det V -1 n ) 2 by log n in (4.1). Hereafter, we obtain from (4.9) and (4.11) that

∞ ∑ n=2 1 (log n) 2 E V n ∆M n 4 F n-1 = O ∞ ∑ n=1 1 (n log n) 2 .
(4.16)

Thus, (4.16) guarentees that (H.4) is verified. We are now going to apply the quadratic strong law given by Theorem A.3 in [START_REF] Laulin | New Insights on the Reinforced Elephant Random Walk Using a Martingale Approach[END_REF]. We get from equation (4.15) that

lim n→∞ 1 log n n ∑ k=1 (det V k ) 2 -(det V k+1 ) 2 (det V k ) 2 V k M k M T k V T k = 1 + 2β -2a(β + 1) V a.s. ( 4
(det V n ) 2 -(det V n+1 ) 2 (det V n ) 2 = 1 + 2β -2a(β + 1). (4.18)
Finally, we can deduce from (4.2), (4.17 which, together with

v T Vv = 2β + 1 -a (1 -a)(1 + 2β -2a(β + 1)) (4.20)
completes the proof of Theorem 2.2.

The critical regime

The proofs of Theorems 2.5 and 2.7 follows essentially the same lines as the ones in the diffusive regimes, provided one exchange V n with W n . Hence, they shall not be explicited here.

Proof of Theorem 2.6. The proof of the quadratic strong law (2.7) is left to the reader as it follows essentially the same lines as that of (2.2). The only minor change is that the matrix V n has to be replaced by the matrix W n defined in (4.5). We shall now proceed to the proof of the law of iterated logarithm given by (2.8). On the one hand, it follows from (1. Consequently, we deduce from the law of iterated logarithm for martingales due to Stout [START_REF] Stout | Maximal inequalities and the law of the iterated logarithm[END_REF], see also Corollary 6.4.25 in [START_REF] Duflo | Random iterative models[END_REF], that (M n ) satisfies when a = 1 -1/2(β + 1)

lim sup n→∞ M n (2w n log log w n ) 1/2 = -lim inf n→∞ M n (2w n log log w n ) 1/2 = 1 a.s. However, as a n w -1/2 n is asymptotically equivalent to (n 2β+1 log n) -1/2 , we immediately ob- tain from (3.8) that lim sup n→∞ Y n (2n 2β+1 log n log log log n) 1/2 = -lim inf n→∞ Y n (2n 2β+1 log n log log log n) 1/2 lim sup n→∞ n -β Y n (2n log n log log log n) 1/2 = -lim inf n→∞ n -β Y n (2n log n log log log n) 1/2 = 1 a.s. (4.22)
The law of iterated logarithm for martingales also allow us to find that (

N n ) satisfies lim sup n→∞ N n (2n log log n) 1/2 = -lim inf n→∞ N n (2n log log n) 1/2
N n + (2β + 1)(µ n a n ) -1 M n (2n log n log log log n) 1/2 = lim sup n→∞ (2β + 1) Y n (2n 2β+1 log n log log log n) 1/2 = -lim inf n→∞ (2β + 1) Y n (2n 2β+1 log n log log log n) 1/2 = -lim inf n→∞ S n (2n log n log log log n) 1/2 .
Hence, we obtain that lim sup which immediately leads to (2.8), thus completing the proof of Theorem 2.6.

The superdiffusive regime

Proof of Theorem 2.8. Hereafter, we shall again make extensive use of the strong law of large numbers for martingales given, e.g. by theorem 1.3.24 of [START_REF] Duflo | Random iterative models[END_REF] in order to prove (2.12). When a > 1 -1 2(β+1) , we have from (3.9) that w n converges. Hence, as M n ≤ w n , we clealy have that M ∞ < ∞ almost surely and we can conclude that Moreover, we still have that for any γ > 0,

N 2 n = O (log n) 1+γ n a.s.
which by definition of N n gives us for all t ≥ 0

S n + a(β+1) β-a(β+1) (µ n ) -1 Y n ) 2 n 2a(β+1)-2β = O (log n) 1+γ n 2a(β+1)-2β-1
a.s.

We know that a > 1 -1 2(β+1) in the superdiffusive regime, which ensures that 2a(β + 1) -2β -1 > 0. Then, we obtain thanks to (1.11) and (4. Finally, the fact that (4.25) holds almost surely ensures that it also holds for the finite-dimensional distributions, and we obtain (2.11) with Λ t = t a(β+1) L β and L β = a(β+1) a(β+1)-β Y. We shall now proceed to the proof of the mean square convergence (2.13). On the one hand, as M 0 = 0 we have from (3.4) that

E M 2 n = E M n ≤ w n .
Hence, we obtain from (3.9) that sup Finally, we obtain the mean square convergence (2.13) from (4.26) and (4.27) and we achieve the proof of Theorem 2.8.

Figure 1 :

 1 Figure 1: Mass function of the memory depending on the value of β.

Theorem 2 . 1 .

 21 We have the almost sure convergence lim

Theorem 2 . 5 .

 25 We have the almost sure convergence

2

 2 = v T Vv a.s. (4.[START_REF] Laulin | New Insights on the Reinforced Elephant Random Walk Using a Martingale Approach[END_REF]

2 n

 2 2n log n log log log n = (2β + 1)2

  lim n→∞ M n = M a.s. where M = ∞ ∑ k=1 a k ε k which by definition of M n , and as a n is asymptotically equivalent to Γ(a(β + 1) + 1)n -a(β+1) , ensures that lim n→∞ Y n n a(β+1) = Y a.s. where Y = 1 Γ(a(β + 1) + 1) M. (4.23)

  7) that for all t ≥ 0 lim n→∞ S nt nt a(β+1)-β + a(β + 1) βa(β + 1) Y nt nt a(β+1) = 0 a.s. (4.24) The convergences (4.23) and (4.24) hold almost surely and nt is asymptotically equivalent to nt which implies lim n→∞ S nt n a (β + 1) = t a(β+1) L β a.s. (4.25)

n≥1 E M 2 2 n

 22 n < ∞ which ensures that the martingale (M n ) is bounded in L 2 . Therefore, we have the mean square convergence limn→∞ E M n -M 2 = 0 which implies that lim n→∞ E Y n n a(β+1) -Y 2 = 0. (4.26)On the other hand, for any n ≥ 0, the martingale (N n ) satisfiesE N 2 n = E N n ≤ β βa(β + 1)and since a(β + 1)β > 1

  Once again, our next result concerns the asymptotic normality of the AERW.

	Theorem 2.7. The following convergence in distribution in D([0, ∞[) holds
	S n t n t log n	, t ≥ 0 =⇒ (2β + 1) B t , t ≥ 0	(2.9)
	where (B t , t ≥ 0) is a one-dimensional standard Brownian motion. In particular, we have the
	asymptotic normality
	S n n log n	L -→ N 0, (2β + 1) 2 .

2 n 2n log n log log log n = (2β + 1) 2 a.s. (2.8)

  .6)

	Proof of Lemma 4.1. We obtain from Theorem 2.1, equations (1.10) and (3.7) that
	lim n→∞

  In order to apply Theorem A.2 from[START_REF] Laulin | New Insights on the Reinforced Elephant Random Walk Using a Martingale Approach[END_REF], we must verify that (H.1), (H.2) and (H.3) are satisfied.

							s.
	and we conclude that	lim n→∞	S n n	-	a(β + 1) β -a(β + 1)	Y n µ n n	= 0 a.s.	(4.8)

This achieves the proof of Theorem 2.1 as the convergences (4.7) and (4.8) hold almost surely.

Proof of Theorem 2.3.

  3 in[START_REF] Laulin | New Insights on the Reinforced Elephant Random Walk Using a Martingale Approach[END_REF] are satisfied. Thanks to Lemma 4.1, hypothesis (H.1) holds almost surely. We also immediately obtain from (4.12) that (H.2) is verified almost surely when t = 1. Hereafter, we need to verify (H.4) is satisfied in the special case β = 2 that is

	We immediately have from (4.1)
			det V -1 n =	β -a(β + 1) a(β + 1)	a n µ n	√	n.	(4.14)
	Hence, we obtain from (1.10) and (4.14) that
		lim n→∞	log(det V -1 n ) 2 log n	= 1 + 2β -2a(β + 1).	(4.15)
	∞ ∑ n=1	1 log(det V -1 n ) 2 2 E V n ∆M n

F n-1 < ∞ a.s.


Nicolas Mascret
email: nicolas.mascret@univ-amu.fr

Olivier Vors
email: olivier.vors@univ-amu.fr

Tanguy Marqueste
email: tanguy.marqueste@univ-amu.fr

François Cury
email: francois.cury@univ-amu.fr

Vors Olivier

Marqueste Tanguy

Cury François

Free-throw performance following acute stress

Keywords: cortisol, sports, anxiety, psychology Free-throw performance following acute stress 3

Experimental stress paradigms have been little used in the sport psychology literature because they are unrelated to the specific sport task. The Trier Social Stress Test (TSST) was used in the present study to investigate its influence on the free-throw performance of skilled basketball players. We also investigated the influence of adopting other-approach goals (i.e., doing well relative to others) on free-throw performance when basketball players were placed in a competition immediately after the TSST, comparatively with a control condition without competition. The results showed that free-throw performance decreased in the control condition, while free-throw performance remained stable when the TSST was followed by a competition. They also highlighted that other-approach goals were significant positive predictors of post-TSST performance in the competition condition but not in the control condition. The TSST may now be used with skilled players as an experimental stress paradigm to induce acute stress in the sport domain.

Introduction

Understanding competitive sport performance is a recurrent concern for sport scientists, coaches, and athletes [START_REF] Nevill | Twenty-five years of sport performance research in the Journal of Sport Sciences[END_REF], based on several theoretical frameworks. In this perspective, stress and achievement goals are major topics of interest in the sport psychology literature. But they have generally been studied separately to understand competitive sport performance, principally with correlational designs. Moreover, the use of experimental stress paradigms in the sport domain has been underrepresented because they are unrelated to the sport task itself. For example, the most robust experimental stress paradigm (i.e., the Trier Social Stress Test (TSST); [START_REF] Kirschbaum | The "Trier Social Stress Test"-a tool for investigating psychobiological stress responses in a laboratory setting[END_REF] has been used rarely in the sport performance domain. This being so, the purposes of this study were to investigate through an experimental design (a) the consequences of the TSST on the free-throw performance of skilled basketball players, (b) the consequences of competition on the free-throw performance of previously stressed basketball players and the predicting role of other-based goals to better explain performance.

Stress -sport performance relationships in experimental studies

Stress appears when a discrepancy arises between the perceived demands of the task and the coping resources available to face this task (Lazarus & Folkman, 1994). Increased stress may impair sport performance (Frenkel et al., 2019a[START_REF] Frenkel | Mindful Sensation Seeking: An examination of the protective influence of selected personality traits on risk sport-specific stress[END_REF] and may also lead to choking, i.e., acute performance decrements due to high pressure situations [START_REF] Baumeister | Choking under pressure: self-consciousness and paradoxical effects of incentives on skillful performance[END_REF]. Stress induces psychological (e.g., state anxiety), physiological (e.g., the secretion of glucocorticoids such as cortisol), and behavioral responses [START_REF] Allen | The Trier Social Stress test: principles and practice[END_REF]. These stress responses have been shown to often negatively influence motor performance [START_REF] Nieuwenhuys | Anxiety and perceptual-motor performance: toward an integrated model of concepts, mechanisms, and processes[END_REF]. But the fact that psychological stress responses (e.g., anxiety) are always detrimental to performance and need to be reduced whenever possible should be considered a myth [START_REF] Mccarthy | Three myths about applied consultancy work[END_REF]. Indeed, some athletes maintain high levels of performance and others even tend to perform better than usual ("clutch performance", [START_REF] Otten | Choking vs. clutch performance: A study of sport performance under pressure[END_REF] while they experience stressful situations. But the number of experimental studies conducted in the laboratory and focusing on stress -sports performance relationships is limited [START_REF] Frenkel | Mindful Sensation Seeking: An examination of the protective influence of selected personality traits on risk sport-specific stress[END_REF]. This kind of study uses experimental stress paradigms which are not directly related to sport tasks.

For example, Lautenbach (2017) used a physical stressor (i.e., immersing participants' arm in ice water, up to the elbow, for three minutes) to induce stress. Subsequent golf performance in a putting task was not influenced by this experimental stressor. Surprisingly, the Trier Social Stress Test (TSST, [START_REF] Kirschbaum | The "Trier Social Stress Test"-a tool for investigating psychobiological stress responses in a laboratory setting[END_REF] is strongly underrepresented in experimental studies focusing on stress -motor performance relationships, although this extensively used paradigm has been considered the gold standard for examining the role of acute psychosocial stress on cognitive performance, for example with tasks involving working memory performance [START_REF] Allen | The Trier Social Stress test: principles and practice[END_REF]. Psychosocial stress is induced through two tasks (a mock job interview and a mental arithmetic test) performed in front of a camera and two or three unknown experimenters who have a cold attitude. The TSST induces stress through the combination of social-evaluative threat and uncontrollability [START_REF] Dickerson | Acute stressors and cortisol responses: a theoretical integration and synthesis of laboratory research[END_REF].

Studies conducted with the TSST in the sport domain often focused on the influence of the level of physical activity on stress reactivity following the TSST (e.g., [START_REF] Mizzi | Investigating the role of mindfulness in the relationship between physical activity and mental health[END_REF][START_REF] Rimmele | Trained men show lower cortisol, heart rate and psychological responses to psychosocial stress compared with untrained men[END_REF] for a review see [START_REF] Mücke | Influence of regular physical activity and fitness on stress reactivity as measured with the trier social stress test protocol: A systematic review[END_REF]. But performance in sport tasks following the TSST was not assessed in these studies. But the use of these kinds of experimental stress paradigms in the sport performance domain may be called into question because they are unrelated to the sport task itself [START_REF] Kivlighan | Salivary α-amylase response to competition: relation to gender, previous experience, and attitudes[END_REF] and because performance in competitive sports needs to be delivered within a threatening and stress-inducing environment. [START_REF] Rohleder | Stress on the dance floor: the cortisol stress response to social-evaluative threat in competitive ballroom dancers[END_REF] also showed that the cortisol response of competitive ballroom dancers was higher in real-life competition than in the TSST. However, conducting laboratory investigations focusing on the influence of psychosocial stress induction (through the TSST) on motor performance may be relevant for several reasons: (1) it is undoubtedly impossible to experimentally induce stress during real-life competitions to measure its influence on sport performance, which would certainly be refused by both coaches and athletes; (2) [START_REF] Rimmele | Trained men show lower cortisol, heart rate and psychological responses to psychosocial stress compared with untrained men[END_REF] highlighted that even elite players (national or Olympic levels) showed significant psychological (i.e., anxiety increases) and physiological (i.e., salivary cortisol levels and heart rate increases) stress responses to the TSST, and (3) the influence of cortisol on cognition was recently highlighted as a way to explain physical functioning and sport performance (e.g., Lautenbach, 2017;[START_REF] Robazza | Relationship between biological markers and psychological states in elite basketball players across a competitive season[END_REF].

Two studies used the second part of the TSST only (i.e., the mental arithmetic task) to investigate the influence of this stress induction on sport performance. [START_REF] Lautenbach | Preliminary evidence of salivary cortisol predicting performance in a controlled setting[END_REF] showed a negative correlation between salivary cortisol levels and performance in a second tennis serves task. [START_REF] Lautenbach | Nonautomated pre-performance routine in tennis: An intervention study[END_REF] highlighted that the use of a non-automated pre-performance routine in tennis serves helped to keep participants' performance consistent in the highpressure condition induced by the second part of the TSST. Despite these interesting results, state anxiety significantly increased following the mental arithmetic task in the two previous studies, but this was not the case for the salivary cortisol levels. This can certainly be explained by the fact that only the second part of the TSST was used, and not the protocol as a whole, including the mock job interview and the mental arithmetic test in front of a panel of experimenters [START_REF] Kirschbaum | The "Trier Social Stress Test"-a tool for investigating psychobiological stress responses in a laboratory setting[END_REF]. In the sport performance domain, the whole TSST has been used in one experimental study only, examining the consequences of this paradigm on free-throw performance of beginners in basketball [START_REF] Mascret | The Influence of the 'Trier Social Stress Test' on Free Throw Performance in Basketball: An Interdisciplinary Study[END_REF]. The results showed that no effect was found on free-throw performance and movement execution due to the very low free-throw level of the participants, who were beginners. Because the TSST is the experimental stress paradigm inducing the most elevated levels of cortisol [START_REF] Allen | The Trier Social Stress test: principles and practice[END_REF] and because only one study has used it in the sport domain, studying its influence on sport performance needs to be pursued. Following the null effect of the TSST on free-throw performance of basketball beginners, it seems relevant to examine if similar results may be found with basketball players in clubs who have a more stable free-throw technique and a better initial performance level, since [START_REF] Beilock | On the fragility of skilled performance: what governs choking under pressure[END_REF] showed that one of the psychological responses to stress, namely anxiety, may disrupt the automatic execution of a motor task, impairing performance. Studying the influence of the TSST on free-throw performance of skilled basketball players was the first aim of the present study.

Competition, sport performance, and other-based goals

As previously highlighted by [START_REF] Dickerson | Acute stressors and cortisol responses: a theoretical integration and synthesis of laboratory research[END_REF] for the TSST, the combination of social-evaluative threat and uncontrollability induces the strongest stress responses. The combination of social evaluation of performance (by coaches, teammates, and/or audience) and the uncontrollability of the task (uncertainty of the result) is also inherent parts of competitive sport (Salvador & Costa, 2009). Competition was defined as normative comparison between individuals [START_REF] Murayama | The competition-performance relation: a meta-analytic review and test of the opposing processes model of competition and performance[END_REF]. Competition is naturally an integral part of sport because it implies that one or several individuals try to achieve a goal by confronting another individual or another group who want to reach the same goal. As in the animal and human literatures (e.g., Salvador & Costa, 2009), competition and its outcomes are considered in sport research as a significant form of social stress which results in physiological, psychological, and behavioral consequences (Costa & Salvador, 2012). There is an effect of sport competition itself with an increase in cortisol (e.g., Filaire et al., 2009;[START_REF] Rohleder | Stress on the dance floor: the cortisol stress response to social-evaluative threat in competitive ballroom dancers[END_REF] and anxiety (e.g., [START_REF] Hanton | Recent developments in competitive anxiety direction and competition stress research[END_REF]. Recently, Hogue andcolleagues (2013, 2017) also showed that a coach who promoted competition produced a psychologically (i.e., increases of competitive state anxiety) and physiologically (i.e., increases of salivary cortisol levels) stressful environment in a juggling task. Competition between individuals induces stress responses and increases the likelihood that choking occurs [START_REF] Baumeister | Choking under pressure: self-consciousness and paradoxical effects of incentives on skillful performance[END_REF] because the evaluations of performance and the associated results are emphasized [START_REF] Breske | The effects of goal priming on cortisol responses in an ego-involving climate[END_REF]. While competition between athletes is frequently used by coaches in the sport context to motivate their athletes (Hogue, Fry, & Fry, 2017), previous studies rather showed that competition induces above all a stressful environment. But what happens if athletes are placed in a competitive situation while they are already stressed before the beginning of the competition?

We may hypothesize that it depends on their psychological characteristics, especially on their trait competitiveness, which is a dispositional preference to compete with others in situations in which their performance is evaluated [START_REF] Spence | Achievement-related motives and behaviors[END_REF]. Achievement goal theory specifically focused on behaviors oriented towards demonstrating competence or avoiding demonstrating incompetence when an individual's performance is evaluated (for an historical overview, see Elliot, Murayama, & Pekrun, 2011). Consequently, an individual may pursue different kinds of achievement goals, which represent strivings focused on competence (i.e., how well or poorly an individual is doing at a task or activity) and concrete aims that emerge from personality characteristics such as achievement motives (Elliot et al., 2011).

Two kinds of goals seemed particularly relevant for our own research because some athletes may be more or less concerned with demonstrating normative competence than others and because trait competitiveness is a positive predictor of these two goals [START_REF] Murayama | The competition-performance relation: a meta-analytic review and test of the opposing processes model of competition and performance[END_REF]. Individuals may pursue other-approach goals (focused on doing well relative to others) and/or other-avoidance goals (focused on not doing poorly relative to others).1 Recent metaanalyses of self-reported achievement goals and objective performance in the sport domain (e.g., [START_REF] Lochbaum | A meta-analytic review of the approach-avoidance achievement goals and performance relationships in the sport psychology literature[END_REF][START_REF] Van Yperen | A meta-analysis of self-reported achievement goals and nonself-report performance across three achievement domains (work, sports, and education)[END_REF] have evidenced that other-approach goals were positively associated with performance attainment, whereas no significant correlations were found between other-avoidance goals and sport performance. Some experimental studies have also evidenced that situationally induced other-approach goals (i.e., promoting competition between participants) may cause better performance in basketball dribbling tasks (e.g., [START_REF] Elliot | Achievement goals, selfhandicapping, and performance attainment: A mediational analysis[END_REF], golf-putting exercises [START_REF] Kavussanu | The effects of achievement goals on performance, enjoyment, and practice of a novel motor task[END_REF], and dart-throwing exercises (e.g., [START_REF] Ntoumanis | Achievement goals, selfhandicapping, and performance: A 2 × 2 achievement goal perspective[END_REF]. In all these studies, competition between participants was experimentally induced while they were not particularly stressed. Consequently, several questions may arise. Would the pattern of results be the same if athletes were already stressed? Would the stress induced by competition itself be added to the stress previously experienced by athletes? Would cumulative stress effects be deleterious or beneficial for sport performance? These questions were theoretically relevant because the compatibility between competition and dispositional other-approach goals may induce a kind of stress-buffering effect, which may in turn positively influence performance. Indeed, dispositional orientations (e.g., dispositional affects, Janicki Deverts, Cohen, & Doyle, 2017) have already been found to influence stress-buffering effects in other domains than sport. Hence, we postulate that being in competition immediately after an acute stress may positively influence the performance of athletes who adopt other-approach goals. Testing this hypothesis was the second aim of the present study.

Free-throw shooting was selected in the present study because (a) the player may easily compare his performance with other players in a competition conducted in an experimental setting; (b) players may pursue other-approach goals (e.g., "make more free throws than others") and/or other-avoidance goals (e.g., "not make fewer free throws than others") through this specific skill; (c) it is one of the most important and stress-provoking game situations in basketball; (d) the player controls his stereotyped movement and performs a standardized performance task; and (e) the cortisol consequences linked to the physical exertion during a sequence of free throws can be neglected [START_REF] Mascret | The Influence of the 'Trier Social Stress Test' on Free Throw Performance in Basketball: An Interdisciplinary Study[END_REF].

The present study

The first aim of the study was to investigate the influence of an experimental stress paradigm unrelated to sport tasks (i.e., the TSST) on free-throw performance of skilled basketball players. Because the TSST induces acute psychosocial stress responses and because skilled basketball players have a stabilized shooting movement, we expected the freethrow performance of skilled basketball players to decrease after the TSST compared with their performance before the TSST (Hypothesis 1). The second aim of the study was to investigate the influence of adopting other-approach goals on free-throw performance when basketball players were placed in a competition immediately after the TSST, by comparing their performance pre-TSST and post-TSST and comparing their performance with a control condition (no competition). Because other-approach goals and attractiveness of competition were theoretically and empirically related [START_REF] Murayama | The competition-performance relation: a meta-analytic review and test of the opposing processes model of competition and performance[END_REF], we expected otherapproach goals to be positive predictors of free-throw performance post-TSST when participants were placed in a competition immediately after the TSST (Hypothesis 2). But we hypothesized that other-approach goals were not significant predictors of free-throw performance post-TSST when no competition was organized immediately after the TSST (Hypothesis 3).

Methods

Participants

Thirty-eight men (18.50 ± 0.92 years with a range of 18-21 years; Body Mass Index of 22.70 ± 2.62 kg/m 2 ) voluntarily participated in the study. The selected participants were students at the Faculty of Sports Science in Marseilles (France). They were all amateur basketball players in clubs at departmental to regional levels. A power analysis using G* Power 3.1 [START_REF] Faul | G* Power 3: A flexible statistical power analysis program for the social, behavioral, and biomedical sciences[END_REF] showed that an n = 17 per condition was needed to produce desirable power of greater than 0.8 (using medium effect size = 0.25).

One month before the beginning of the experiment, all the participants completed a questionnaire collecting general information and a scale measuring their other-based goals.

Exclusion criteria were the following: history of endocrine disorder, chronic stress, cardiovascular diseases, psychological distress, reported medical illness, drugs abuse, medication intake, drinking more than two glasses of alcohol a day, and smoking more than five cigarettes a day. Elite players were excluded from the study, because they showed lower cortisol responses to stress compared with amateur sportsmen [START_REF] Rimmele | Trained men show lower cortisol, heart rate and psychological responses to psychosocial stress compared with untrained men[END_REF].

Participants also had to follow specific instructions: not to take medication, not to use alcohol, and not to practice sport during the twelve hours preceding this experiment. They were also instructed not to smoke, not to eat, and not to drink anything except water in the hour before the experiment. Finally, experimental sessions were conducted between 13:00 and 17:00 h and participants were instructed to wake up at least three hours before the beginning of the experiment to control the cortisol awakening response [START_REF] Stalder | Assessment of the cortisol awakening response: expert consensus guidelines[END_REF].

Experimental design and procedure

Participants were brought into the gymnasium in groups of six, then a short static warm-up session of five free throws began. Each participant was in front of a basketball hoop and a scorer/ball boy was assigned to each player. Communication between players and with the scorer was not allowed and comparing performance with other participants was not possible because they shot simultaneously in different directions. The first sequence (pre-TSST) consisted of 20 consecutive free throws with neutral instructions given by the experimenter, who was specifically trained before the beginning of the experiment with a specific text to memorize. All of the instructions were completely standardized and were given to the participants at the same time from one session to the next.

After the first sequence of 20 free throws, participants left the gymnasium for an adjacent room dedicated to the stress protocol. For approximatively 25 minutes, the six participants of the session were exposed to the Trier Social Stress Test for Groups (TSST-G, Von [START_REF] Dawans | The Trier Social Stress Test for Groups (TSST-G): A new research tool for controlled simultaneous social stress exposure in a group format[END_REF], composed of both the preparation and delivery of a speech task simulating a recruitment interview and a mental arithmetic task in front of a committee of two absolutely neutral experimenters wearing white lab coats, seated behind a table on which a wide-angle video camera, a microphone, and a timer were placed.

Participants were separated by mobile dividing walls that restricted visual and social interaction. After a 3-min preparation period, each participant performed the mock job interview for 2 min. Then the experimenters interrupted the first participant and requested the next one to speak (overall duration per group: 6 participants x 2 min = 12 min). The participants were not aware of the order in which they were being called. In the mental arithmetic task, each participant successively performed serial subtraction for 80 sec (duration: 6 participants x 80 sec = 8 min). To avoid learning effects, each participant received an individual starting number. The TSST-G was deliberately unrelated to the nature of the following free-throw task to investigate Hypothesis 1.

Once the TSST-G was completed, the six participants returned to the gymnasium to carry out the second sequence of 20 free throws (post-TSST) with the same organization as in the first sequence. The participants were blind to the objectives and the topic of the present study. They were debriefed and thanked at the end of the experimentation. The full procedure is presented in Fig. 1 and the experimental manipulation is presented below. This study was in line with the recommendations of the institutional board of the University and was conducted in accordance with the Declaration of Helsinki.

Figure 1 near here

Materials

Physiological stress response

In the vast majority of studies conducted with the TSST, salivary cortisol level was used as the most relevant biological marker of stress [START_REF] Allen | The Trier Social Stress test: principles and practice[END_REF]. The saliva was collected in plastic saliva collection tubes, five times: at baseline (T1 = -20 min before the TSST-G), after the first sequence of free throws and before the TSST-G (T2 = -1 min), after the TSST-G (T3 = +1 min), after the second free-throw sequence (T4 = +10 min), and finally after a static recovery period (T5 = +15 min). The procedure was exactly the same as precisely described in Mascret et al.'s (2016) study, with ELISA intra-assay variation of 4.7 ±1.7% and inter-assay variation of 6.1 ±3.6% (using Salivary Cortisol Enzyme Immunoassay Kit No. 1-3002, Salimetrics, UK).

Psychological stress response

State anxiety was assessed by two corresponding subscales of the French version of the Competitive State Anxiety Inventory 2 Revised (CSAI-2R, [START_REF] Martinent | Validation of the French version of the Competitive State Anxiety Inventory-2 Revised (CSAI-2R) including frequency and direction scales[END_REF]. Participants rated the intensity of each item on a scale of 1 (not at all) to 4 (very much so) to measure cognitive anxiety (e.g., "I am worried about performing poorly") and somatic anxiety (e.g., "My hands are clammy"). Cognitive and somatic anxiety were assessed, twice: after the first sequence of free throws / before the TSST-G and after the TSST-G. Internal consistency was satisfactory for pre-and post-TSST-G cognitive anxiety (a = .78 and .85 respectively) and for post-TSST-G somatic anxiety (a = .84), but surprisingly somewhat weak and questionable for pre-TSST-G somatic anxiety (a = .51). All the descriptive statistics of cortisol levels and cognitive and somatic anxiety scores are presented in Table 1.

Table 1 near here

Perceived competition

To verify that the manipulation of competition was successful depending on the condition (competition vs. no-competition), we used an adapted five-item version of the questionnaire assessing the degree to which participants found the experimental setting to be ego-involving [START_REF] Sage | The effects of goal involvement on moral behavior in an experimentally manipulated competitive setting[END_REF]. For the purpose of this study, two of the seven initial items were not relevant and were omitted. Responses were made for each item (e.g., "In today's experiment, winning was emphasized") on a five-point Likert scale from 1 (strongly disagree) to 5 (strongly agree). The subscale had high internal consistency (a = .93).

Free-throw other-based goals

These were assessed one month earlier by the corresponding subscales of the 3x2 Achievement Goal Questionnaire for Sport (AGQ-S; [START_REF] Mascret | Extending the 3× 2 achievement goal model to the sport domain: The 3×2 Achievement Goal Questionnaire for Sport[END_REF]. Participants answered the three items assessing other-approach goals (e.g., "When I shoot free throws, my goal is to do better than others") and the three items assessing other-avoidance goals (e.g., "When I shoot free throws, my goal is to avoid worse results than others") on a seven-point Likert scale (1 = strongly disagree, 7 = strongly agree). Internal consistency was good for other-approach goals (a = .92) and other-avoidance goals (a = .93).

Performance measurement

Participants performed two sequences of 20 free throws. Because performance measurement was carried out simultaneously by the six players on six different basketball backboards, the free-throw scoring of each player was manually documented on a score sheet by an experimenter blinded to condition assignment who did not talk to the participant and who was specifically trained for this task. All the experimenters who collected the scores were men and the instructions were completely standardized. Performance was defined as the number of successful free throws in the pre-and the post-TSST.

Experimental manipulation

The experiment used a two-condition comparison design and participants were randomly assigned to one of two conditions after the TSST-G: 19 participants in a competition condition and 19 participants in a control condition (i.e., no-competition). To induce competition in the second sequence of 20 free throws, the experimenter grouped the participants before the first free throw and told them that from then on they were in competition with one another and with other participants in the next sessions. The aim of the competition was to be the best player, with the highest score, at the end of the 20 free throws.

They were also told that the complete ranking from first to last would be made accessible to the public the next day and displayed in the gymnasium. Then, the players performed the second sequence of 20 free throws. After the third, sixth, ninth, tenth, thirteenth, and seventeenth free throws, the experimenter announced instructions aloud in order to maintain the competition during the second sequence of 20 free throws (e.g., "Ranking will be displayed tomorrow"). In the control condition, the coach announced the same neutral instructions as before the first sequence of free throws. He also announced instructions aloud exactly at the same times as in the competition condition, but these instructions were neutral (e.g., "There are ten free throws left").

Data analysis

Repeated measures ANOVAs were conducted with Statistica 12 for PC on the results of cortisol levels, cognitive anxiety, somatic anxiety, and performance scores to reveal possible effect of Condition and Time of assessment. Following [START_REF] Lautenbach | Preliminary evidence of salivary cortisol predicting performance in a controlled setting[END_REF] procedure, we calculated a baseline for cortisol levels as the mean of the first two measures (Pre-TSST = T1 and T2) and we computed the average of the cortisol response to the stress induction (Post-TSST = T3, T4, and T5). Level of significance was defined as p < .05. Partial eta-squared (ηp²) was used to calculate effect sizes and Newman-Keuls tests were used to evaluate differences post-hoc. Finally, multiple regression analyses were conducted to examine how pre-TSST performance, other-approach goals, and other-avoidance goals predicted post-TSST performance for the two conditions (Competition, Control).

Results

Preliminary results

Physiological stress response

For salivary cortisol levels, a repeated measures ANOVA with the factors Time of assessment (Pre-TSST, Post-TSST) and Condition (Competition, Control) revealed a significant effect of Time (F(4, 144) = 40.65, p < .001, ηp² = .53) and a significant Time X Condition interaction (F(4, 144) = 7.31, p < .001, ηp² = .17). No significant effect was found for Condition (p = .965). Newman-Keuls post-hoc tests showed that cortisol levels were significantly higher (p < .001) after the TSST-G than before for both the competition and control conditions. No significant differences were found between competition and control conditions in pre-and post-TSST.

Psychological response to stress

For state anxiety scores, two consecutive repeated measures ANOVAs conducted respectively on cognitive and somatic anxiety scores with the factors Time of assessment and Condition revealed a significant effect of Time for cognitive anxiety (F(1, 36) = 29.12, p < .001, ηp² = .45) and somatic anxiety (F(1, 36) = 22.27, p < .001, ηp² = .38). No significant effects of Condition and Time X Condition interactions were found for cognitive anxiety (p = .803 and p = .888, respectively) and somatic anxiety (p = .764 and p = .804, respectively).

Newman-Keuls tests showed that the cognitive and somatic anxiety scores were significantly higher after the TSST-G than before for both the competition and control conditions (with ps ranging from .001 to .009). The three previous manipulation checks (cortisol levels, cognitive anxiety scores, somatic anxiety scores) evidenced that the TSST-G had indeed induced stress responses at the physiological and psychological levels. Table 1 presents the descriptive statistics for each variable.

Perceived competition

A one-way ANOVA conducted on the perceived competition scores showed a significant effect of Condition (F(1, 36) = 9.53, p = .004, ηp² = .21). Newman-Keuls tests evidenced that perceived competition was significantly higher (M = 3.99, SD = 1.12) for the competition group compared with the control group (M = 2.89, SD = 1.06, p = .004). The manipulation of competition was also successful.

Other-based goals

Two one-way ANOVAs did not reveal differences between the two experimental groups as regards their other-approach goals (p = .827) and their other-avoidance goals (p = .759).

Main results

Free-throw performance

A repeated measures ANOVA conducted on the free-throw performance scores with the factors Time and Condition revealed a significant effect of Time (F(1, 36) = 5.26, p = .028, ηp² = .13) and a significant Time X Condition interaction (F(1, 36) = 5.26, p = .028, ηp² = .13). Newman-Keuls tests showed that (1) there was no significant difference of performance on pre-TSST between the two experimental conditions (p = .529); (2) free-throw scores in the competition condition remained, surprisingly, exactly the same between the pre-TSST and the post-TSST sequences (p = 1); and (3) free-throw performance in the control condition significantly decreased between the pre-and the post-TSST (p = .013). We were initially surprised by the descriptive statistics of the competition condition, in particular by the fact that the means and the standard deviations were identical before and after the TSST-G.

However, if the means and standard deviations are effectively the same, the individual results are not at all identical. An inspection of the raw data showed that the post-TSST-G performance of each of the 19 participants increased or decreased relative to their pre-SST-G performance.

Detailed results of free-throw performance are presented in Table 1 above. Consequently, Hypothesis 1 was supported: free-throw performance decreased in the control condition following the TSST.

Regression analyses

We conducted regression analyses to examine how condition and other-based goals predicted post-TSST performance in the two conditions. Pre-TSST performance was also entered in the regression analyses, because post-TSST performance may be related with pre-TSST performance due to individual differences on initial free-throw performance between participants during the experimentation. Table 2 shows the results of this analysis for the two experimental conditions. As expected, pre-TSST performance was a significant positive predictor of post-TSST performance for the two conditions. Results mainly highlighted that other-approach goals were significant positive predictors of post-TSST performance in the competition condition but not in the control condition, which supported Hypotheses 2 and 3, respectively.

Table 2 near here

Discussion

This study investigated the influence of competition on free-throw performance after an acute psychosocial stress induction through an experimental stress paradigm, the TSST, unrelated to the specific sport task. The TSST effectively induced significant stress responses on the psychological (increases of cognitive and somatic anxiety) and physiological (increases of salivary cortisol levels) levels. The results first showed that free-throw performance decreased for the control condition (i.e., only the TSST), despite the facts that the TSST is not a specific sport task and that performance in competitive sports needs to be delivered within a threatening and stress-inducing environment. Two main models have been used to study the relationships between one of the stress psychological response, namely anxiety, and perceptual-motor performance [START_REF] Nieuwenhuys | Anxiety and perceptual-motor performance: toward an integrated model of concepts, mechanisms, and processes[END_REF]. Distraction models point out that anxiety draws attention towards task-irrelevant stimuli and consequently attention to movement execution is impaired. Finally, movement may be less accurate and may cause poor performance. Execution focus models point out that anxiety may lead athletes to explicitly control or monitor their movements. This may disrupt movement execution, which is often automatized with experienced athletes, and, ultimately, may result in performance decreases. In the present study, participants performed stereotyped free-throw movements, performed a standardized performance task, and were all basketball players in clubs at departmental to regional levels. Discrepancies in the findings were found between the present study and Mascret et al.'s (2016) study, in which the TSST did not impair free-throw performance and movement execution of beginners. In our study, anxiety may have negatively influenced both attention to relevant stimuli and overall automatization of the freethrow movement of participants who have a more automatized movement than beginners, which may account for the decline in performance following the TSST and may explain this difference with Mascret et al.'s (2016) results.

Stress psychological and physiological responses to competition were found in previous studies [START_REF] Hogue | The influence of a motivational climate intervention on participants' salivary cortisol and psychological responses[END_REF][START_REF] Hogue | The differential impact of motivational climate on adolescents' psychological and physiological stress responses[END_REF]. In the present study, free-throw performance remained stable between pre-TSST and post-TSST when competition followed the TSST.

Psychosocial stress may be offset by the fact that situationally induced other-approach goals often cause better performance [START_REF] Van Yperen | A meta-analysis of self-reported achievement goals and nonself-report performance across three achievement domains (work, sports, and education)[END_REF]. Other-approach goals have been shown to be associated with challenge-based affect, cognition, and behavior that tends to facilitate performance [START_REF] Murayama | The competition-performance relation: a meta-analytic review and test of the opposing processes model of competition and performance[END_REF]. Effort and persistence are caused by challenge appraisal when individuals are focused on the possibility of success comparatively to others, and subsequently benefit performance attainment. Indeed, there is a greater tendency to produce an active coping response pattern if the competitive situation is considered a challenge, important for the participant, controllable, and depending on effort [START_REF] Salvador | Steroid hormones and some evolutionary-relevant social interactions[END_REF]. Competition used in the present study may be regarded as explicitly challenging (to be the best free-throw shooter), important (the ranking is displayed in the gymnasium), controllable (the free-throw performance depends on the player), and dependent on effort (if the player is at his best, he can expect to win the competition).

Our results also showed that this is essentially the case for participants with high levels of dispositional other-approach goals when they are placed in a competition after the TSST. While experimental instructions inducing situational achievement goals often override personal goals in affecting outcomes (Ntoumanis & Biddle, 1999), this study evidenced that the combination of competition and dispositional other-approach goals led to a positive outcome concerning free-throw performance following psychosocial stress. Results highlighted that other-approach goals were significant positive predictors of post-TSST performance in the competition condition but not in the control condition. The effect of otherapproach goals on the outcome variable was stronger in the condition that exacerbates these goals than in the control condition [START_REF] Ntoumanis | Achievement goals, selfhandicapping, and performance: A 2 × 2 achievement goal perspective[END_REF]. Participants with a dispositional tendency toward competition (i.e., high levels of other-approach goals) may be more concerned to demonstrate normative competence than others in a competition and may feel comfortable in this specific climate despite their high levels of stress responses previously induced by the TSST. While maintaining a high level of self-confidence is more difficult in competitive conditions [START_REF] Kavussanu | Motivation in physical activity contexts: The relationship of perceived motivational climate to intrinsic motivation and self-efficacy[END_REF], it may be easier for stressed players who have high levels of other-approach goals. This assumption derives from the idea that high levels of self-confidence may have a potential role in protection against the debilitating effects of anxiety [START_REF] Hanton | Recent developments in competitive anxiety direction and competition stress research[END_REF]. This kind of stress-buffering effect may be considered because dispositional orientations have already been found to influence stress-buffering effects (e.g., dispositional affects, Janicki [START_REF] Janicki Deverts | Dispositional affect moderates the stress-buffering effect of social support on risk for developing the common cold[END_REF]. But this assumption remains speculative and needs further investigations.

In the competition condition, other-avoidance goals were not found to be significant predictors of post-TSST performance. The review by [START_REF] Lochbaum | A meta-analytic review of the approach-avoidance achievement goals and performance relationships in the sport psychology literature[END_REF] surprisingly evidenced that other-avoidance goals were mainly beneficial to performance in a laboratory setting and detrimental in a naturalistic setting. Even if the participants in this study effectively performed free throws, they were not placed in the context of a match or an official competition with teammates and opponents, and in front of an audience of supportive or hostile fans. Moreover, the participants were all young adults, and other-avoidance goals showed no effect on objective performance with samples aged 18 or above [START_REF] Lochbaum | A meta-analytic review of the approach-avoidance achievement goals and performance relationships in the sport psychology literature[END_REF]. They are also competitive basketball players. The results would certainly be different in another achievement setting such as Physical Education, in which participation is mandatory and students are not necessarily interested in physical activity. In this specific context, other-avoidance goals may be negative significant predictors of post-TSST performance in such a competitive condition. But this hypothesis needs to be tested.

Two main practical implications of this study may be envisaged. First, the present study showed that a stress-buffering effect may occur in competitive situations for athletes with high other-approach goals, leading to performance improvement despite initial psychosocial stress. If a coach knows that one of his athletes follows other-approach goals (focused on doing well relative to others), he should not hesitate to use competitive interventions (e.g., promoting comparison, emphasize ranking) during training and/or official competition to help the athlete to cope with stress and to expect a beneficial effect on performance. In sum, this may be useful for coaches who can adapt their interventions to the athletes' achievement goals when the environment is stressful. For example, a coach could specifically emphasize competition and comparison for a basketball player with high levels of other-approach goals when he performs two free throws in the final seconds of very close games, in order to increase the likelihood of scoring despite the stressful situation. Second, the results showed that a coach may induce psychosocial stress with experimental stress paradigms which are not directly sport-related. With two unknown tasks, the TSST may bring novelty into the training process when a coach wants to confront athletes with stressful situations, especially with athletes for whom competition during training is no longer sufficient to induce significant stress responses.

Our results must, however, be considered with caution because they concern only other-approach goals and are context-specific through a free-throw task performed in an acute state of stress by trained young men. Moreover, this study is not without limitations. First, it focused on competition because we wanted to examine the relations between competition following an acute state of stress, other-based goals, and performance. Based on the literature review of [START_REF] Harwood | A systematic review of the intrapersonal correlates of motivational climate perceptions in sport and physical activity[END_REF], it could be promising to test the same hypothesis with stressed players faced with a complete performance climate (in which outperforming others, ranking, and rewards are emphasized) and a mastery climate (in which self-improvement and individual effort are valorized). In the absence of a previous stress, a mastery climate has positive relations with sport performance and is less likely to produce psychological and physiological stress responses [START_REF] Hogue | The influence of a motivational climate intervention on participants' salivary cortisol and psychological responses[END_REF][START_REF] Hogue | The differential impact of motivational climate on adolescents' psychological and physiological stress responses[END_REF]. Second, the influence of other achievement goals on post-TSST performance may be investigated. Task-approach goals (satisfying the absolute demands of the task) and self-approach goals (improving relative to own's trajectory) may have a protective role against the debilitative effects of acute stress because these kinds of goals are positively associated with performance attainment in the sport domain [START_REF] Van Yperen | A meta-analysis of self-reported achievement goals and nonself-report performance across three achievement domains (work, sports, and education)[END_REF]. Third, all the participants in the present study were men because women's cortisol levels could have been influenced by menstrual cycle and/or oral contraceptives [START_REF] Allen | The Trier Social Stress test: principles and practice[END_REF]. Because men often show higher competitive orientation than women, it could be interesting to replicate our study to elucidate how women react to competition following an acute stress. The continuation of interdisciplinary studies in the sport context is required to protect athletes from the debilitative effects of stress responses on sport performance, but also on their physical and mental health [START_REF] Rimmele | Trained men show lower cortisol, heart rate and psychological responses to psychosocial stress compared with untrained men[END_REF].

Figure 1 .

 1 Figure 1. Study design including timing of saliva sampling for cortisol measurements, completion of questionnaires, and free-throw performance.

Table 1 .

 1 Descriptive statistics of physiological, psychological, and behavioral measures. M = Mean, SD = Standard deviation.

		Before	After
	Variables	the TSST-G	the TSST-G
		M	SD	M	SD
	Cortisol levels in nmol/L (manipulation check)
	Competition	8.25	3.02 11.00 2.56
	Control	6.71	4.07 9.92	5.17
	Cognitive anxiety (manipulation check)
	Competition	1.68	0.62 2.09	0.87
	Control	1.64	0.46 2.03	0.72
	Somatic anxiety (manipulation check)
	Competition	1.27	0.22 1.65	0.54
	Control	1.29	0.30 1.71	0.68
	Free-throw performance
	Competition	11.42	4.25 11.42 4.25
	Control	12.26	3.75 9.63	4.27
	Notes.

Table 2 .

 2 Regression analyses to examine other-approach goals and other-avoidance goals in predicting post-TSST performance in competition and control conditions, after controlling for pre-TSST performance. *p < .05, **p < .01, ***p < .001

	Post-TSST performance	Post-TSST performance
	(competition condition)	(control condition)
	R 2	b	R 2	b
	.685***		.692***
	Pre-TSST performance	.58**		.78***
	Other-approach goals	.69***		.14
	Other-avoidance goals	-.05		-.04
	Note.

Performance-based goals were recently relabeled other-based goals(Elliot et al.,

2011;[START_REF] Mascret | Extending the 3× 2 achievement goal model to the sport domain: The 3×2 Achievement Goal Questionnaire for Sport[END_REF] but their definition remain the same. The more recent denomination (i.e., other-approach and otheravoidance goals) is used in the present study even for studies employing the denomination performanceapproach and performance-avoidance goals.

Declaration of interest

No authors have competing interests