Lin Han-Mai

On the weak invariance principle for non-adapted stationary random elds under projective criteria

Keywords: 2020 Mathematics Subject Classication. 60F17, 60G60 Invariance principle, Brownian sheet, random eld, orthomartingale approximation

In this paper, we study the central limit theorem (CLT) and its weak invariance principle (WIP) for sums of stationary random elds non necessarily adapted, under dierent normalizations. To do so, we rst state sucient conditions for the validity of a suitable ortho-martingale approximation. Then, with the help of this approximation, we derive projective criteria under which the CLT as well as the WIP holds. These projective criteria are in the spirit of the Hannan's condition and are well adapted to linear random elds with ortho-martingale innovations and which exhibit long memory.

Introduction

Let (X i ) i∈Z d , d ≥ 1, be a stationary random eld with zero mean and nite variance. Let also S n be its associated partial sum with n = (n 1 • • • n d ) ∈ N d , dened as

S n = n 1 i 1 =1 • • • n d i d =1 X i .
In this paper, we are interested in the central limit theorem and the weak invariance principle. In particular the WIP addresses the question of nding a positive sequence s n such that s n → ∞ and

S [n•t] s n t∈[0,1] d ⇒ {σW d (t)} t∈[0,1] d , (1.1)
in D [0, 1] d equipped with the uniform topology, where

[n • t] = ([n 1 t 1 ] , • • • , [n d t d ]
). The notation " ⇒ " means the convergence in distribution. When the X k are functions of an i.i.d random eld, under the so-called Hannan's condition [START_REF] Hannan | Central limit theorems for time series regression[END_REF] generalized to Z d , Wang and Volný [START_REF] Volný | An invariance principle for stationary random elds under Hannan's condition[END_REF] have established that (1.1) holds with s 2 n = n 1 . . . n d . When d = 2, their condition reads as

(i,j)∈Z 2 P 0,0 (X i,j ) 2 < ∞, (1.2)
where P 0,0 is the projection operator dened by (1.4). We also refer to [START_REF] Zhang | On the quenched CLT for stationary random elds under projective criteria[END_REF] for a quenched version of the CLT under (1.2). The result stated in [START_REF] Volný | An invariance principle for stationary random elds under Hannan's condition[END_REF] does not allow to consider models for which the normalization in the WIP does not have a linear growth with respect to n. Our paper is a step in this direction and aims at relaxing the condition (1.2) to still get the WIP. Our results can be viewed as the random eld counterparts of those established in Dedecker et al. [START_REF] Dedecker | On the weak invariance principle for non-adapted sequences under projective criteria[END_REF]. To do so, we start with a new ortho-martingale approximation using the notion of commuting ltration (note that there is no natural ordering of future and past in higher dimension).

Before giving a avor of the results obtained in this paper by considering a simple example of linear random eld with long memory, let us rst mention additional earlier results to [START_REF] Dedecker | On the weak invariance principle for non-adapted sequences under projective criteria[END_REF], involving projective type criteria and using the notion of ortho-martingale. First, in [START_REF] Wang | A new condition for the invariance principle for stationary random elds[END_REF], Wang and Woodroofe proved the WIP for stationary random elds indexed by rectangular sets with the help of an m-dependent random eld approximation. Their condition is in term of conditional expectation and is in the spirit of the Maxwell-Woodroofe's condition in dimension 1 (see [START_REF] Maxwell | Central limit theorems for additive functionals of Markov chains[END_REF]). Then, as an application, they proved the WIP for ortho-martingales when they are functions of an i.i.d random eld. In [START_REF] Machkouri | A central limit theorem for stationary random elds[END_REF], using the same method (m-dependent random eld approximation) and the physical dependence measure, El Machkouri et al. proved a more general result in the sense that their result apply to a wider class of stationary random elds.

Concerning stationary ortho-martingales, a CLT has been obtained by Volný [START_REF] Volný | A central limit theorem for elds of martingale dierences[END_REF]. More precisely, let us consider the case d = 2 and assume that (T i,j ) (i,j)∈Z 2 is a group of commuting probability preserving transformation of (Ω, A, P). Suppose in addition that h is a random variable in L 2 and that (h • T i,j ) (i,j)∈Z 2 is an ortho-martingale (see denition 2.1) with respect to a commuting ltration (F i,j ) (i,j)∈Z 2 (see denition 1.1). In this situation, it has been proved in [START_REF] Volný | A central limit theorem for elds of martingale dierences[END_REF] that, as n 1 ∧ n 2 → ∞, (n 1 n 2 ) -1/2 n 1 i=1 n 2 j=1 h • T i,j converges in distribution to a centered gaussian random variable with variance h 2 2 provided that one of the transformation T 0,1 or T 1,0 is ergodic (here and along the paper the symbol a ∧ b stands for the minimum between a and b). Our strategy of proof is then to derive a new ortho-martingale approximation for S n /s n which combined with the CLT in [START_REF] Volný | A central limit theorem for elds of martingale dierences[END_REF] will lead us to new projective criteria ensuring the convergence in law of S n /s n . To derive the corresponding WIP, we then state a new criterion ensuring the uniform integrability of s -2 n max k≤n S 2 k . To illustrate our results, let us consider the following linear elds in dimension 2 with long memory. Let X k, = i,j∈N 2 1 (i + 1)(j + 1) ξ 0,0 • T k-i, -j , with T the shift transformation dened as in subsection 1.1 and ξ 0,0 a centered random variable in L 2 and such that E (ξ 0,0 |F -1,0 ) = E (ξ 0,0 |F 0,-1 ) = 0 a.s. ,

where F i,j = σ {ξ h,k , h ≤ i, k ≤ j}. As a consequence of our result, we get that for this process (1.1) holds with the normalization s n 1 ,n 2 = √ n 1 n 2 log(n 1 ) log(n 2 ) (see our Corollary 4.1). This kind of limiting behavior seems to be new in the context of linear random elds. Even if all our results hold for any d ≥ 1, for the sake of clarity due to complicated notations, we shall rather state and prove them in case d = 2. However, some indications concerning the statements of the results in case d > 2 are given in Section 5.

Our paper is organized as follows. In Sections 2 and 3, we state our main results in case of random elds with dimension 2. In Section 4, we apply our results to linear elds with orthomartingale innovations. Then, in Section 5, we extend our results to higher dimension. Section 6 is devoted to the proofs of the main results. We end this section by giving some notations and denitions used all along the paper.

Notations and Denitions. Let (Ω, A, P) be a probability space. To dene properly a stationary ltration, as in [START_REF] Zhang | On the quenched CLT for stationary random elds under projective criteria[END_REF], it is convenient to start with an auxiliary stationary process (ξ i,j ) i,j∈Z 2 and then to set

F i,j = σ{ξ h,k , h ≤ i, k ≤ j}. (1.3)
For all i, j, we also dene

F ∞,∞ = (i,j)∈Z 2 F i,j , F i,-∞ = j∈Z F i,j , F -∞,j = i∈Z F i,j , and
F -∞,-∞ = (i,j)∈Z 2 F i,j . Denition 1.1. The ltration (F i,j ) (i,j)∈Z 2 dened as above is said to be commuting if ∀i, j, h, k ∈ Z, E [E [X|F i,j ] |F h,k ] = E [E [X|F h,k ] |F i,j ] = E [X|F i∧h,j∧k ] a.s. ,
provided all the above conditional expectations are well dened.

Note that a ltration dened by an independent and identically distributed (i.i.d) random eld is commuting. This kind of ltrations can also be constructed using stationary random elds with independent rows or columns (see [START_REF] Machkouri | A central limit theorem for stationary random elds[END_REF]). From now on, we assume that the ltration dened in (1.3) is commuting.

Next we introduce the projection operators dened by: for all i, j ∈ Z and X ∈ L 1 (Ω),

P i, j (X) = E [X|F i,j ] -E [X|F i,j-1 ] P i,j (X) = E [X|F i,j ] -E [X|F i-1,j ] .
Note that, if the ltration is commuting then P i,j (X) = P i,j • P i, j (X) = P i, j • P i,j (X).

Therefore P i,j (X) = E [X|F i,j ] -E [X|F i,j-1 ] -E [X|F i-1,j ] + E [X|F i-1,j-1 ] . (1.4)
Let us now introduce the shift operators as follows: on R Z 2 , let

T (1) ((x k, ) (k, )∈Z 2 ) = (x k+1, ) (k, )∈Z 2 , T (2) ((x k, ) (k, )∈Z 2 ) = (x k, +1 ) (k, )∈Z 2 .
(1.5)

We will denote

T i (1) • T j (2) (x k, ) (k, )∈Z 2 by T i,j ((x k, ) (k, )∈Z 2 )
. Note that the ltration dened above can also be rewritten F i,j = T -i,-j (F 0,0 ). Now we introduce a stationary random eld in the following way. For a real-valued measurable function f on R Z 2 , we dene

X 0,0 = f (ξ i,j ) (i,j)∈Z 2 and X k, = f T k, (ξ i,j ) (i,j)∈Z 2 = X 0,0 • T k, . (1.6)
The stationary random eld (X i,j ) (i,j)∈Z 2 will be said to be regular if X 0,0 is F ∞,∞ -measurable, and

E (X 0,0 |F -∞,∞ ) = E (X 0,0 |F ∞,-∞ ) = E (X 0,0 |F -∞,-∞ ) = 0 a.s.
2 Ortho-Martingale Approximation Let (X i,j ) (i,j)∈Z 2 be a stationary random eld dened by (1.6) and let

S n 1 ,n 2 = n 1 i=1 n 2 j=1 X i,j and σ 2 n 1 ,n 2 = S n 1 ,n 2 2 2 .
Denition 2.1. Let m be an integrable F 0,0 -measurable function. We say that (m

• T i,j ) (i,j)∈Z 2
is a eld of ortho-martingale dierences with respect to a commuting ltration

(F i,j ) (i,j)∈Z 2 , if for all i, j, k, ∈ Z, such that either k < i, or < j then E (m • T i,j |F k, ) = 0 a.s. . In addition, M n 1 ,n 2 := n 1 i=1 n 2
j=1 m • T i,j is said to be an ortho-martingale.

Let (s n 1 ,n 2 ) n 1 ,n 2 ≥1 be a double indexed sequence of positive numbers such that s n 1 ,n 2 → ∞ as n 1 ∧ n 2 → ∞. Below we give an ortho-martingale approximation for the normalized partial sum

S n 1 ,n 2 /s n 1 ,n 2 . Theorem 2.2. Let m be a square integrable F 0,0 -measurable function such that E (m|F -1,0 ) = E (m|F 0,-1 ) = 0 a.s. and let R n 1 ,n 2 = E [S n 1 ,n 2 |F n 1 ,0 ] 2 + E [S n 1 ,n 2 |F 0,n 2 ] 2 + E [S n 1 ,n 2 |F 0,0 ] 2 .
Then the following conditions are equivalent

C 0 (s n 1 ,n 2 ): lim n 1 ∧n 2 →∞ S n 1 ,n 2 s n 1 ,n 2 - 1 √ n 1 n 2 n 1 i=1 n 2 j=1 m • T i,j 2 = 0 . C 1 (s n 1 ,n 2 ):                  (a) R n 1 ,n 2 = o(s n 1 ,n 2 ), (b) S n 1 ,n 2 -E [S n 1 ,n 2 |F n 1 ,n 2 ] 2 = o(s n 1 ,n 2 ), (c) lim n 1 ∧n 2 →∞ 1 n 1 n 2 n 1 i=1 n 2 j=1 √ n 1 n 2 s n 1 ,n 2 n 1 h=1 n 2 k=1 P 0,0 (X h-i,k-j ) -m 2 2 = 0 .
If one of the conditions is satised and T (1) or T (2) is ergodic, then s -1 n 1 ,n 2 S n 1 ,n 2 converges in distribution to σN , where N is a standard Gaussian random variable, and

σ 2 = E(m 2 ). Remark 2.3. Note that if E [m 2 ] < ∞ then the convergence in law of M n 1 ,n 2 √ n 1 n 2
always takes place, but to a mixture of normal laws (see [START_REF] Volný | On limit theorems for elds of martingale dierences[END_REF]). However, the additional condition that at least one of the transformation is ergodic ( T (1) or T (2) ) guarantees the convergence towards a normal law. For more detailed results see [START_REF] Volný | A central limit theorem for elds of martingale dierences[END_REF].

Proposition 2.4. If C 1 (σ n 1 ,n 2 )(a) and (b) hold then σ n 1 ,n 2 √ n 1 n 2
is a two-parameter slowly varying function (2p-svf) in the following sense : for any non negative integers k and

lim n 1 ∧n 2 →∞ σ 2 kn 1 , n 2 σ 2 n 1 ,n 2 = k • . (2.1)
In addition, if σ n 1 ,n 2 → ∞ as n 1 ∧ n 2 goes to innity and if

lim n 1 ∧n 2 →∞ σ 1,n 2 σ n 1 ,n 2 = 0, lim n 1 ∧n 2 →∞ σ n 1 ,1 σ n 1 ,n 2 = 0, (2.2)
then,

lim x∧y→∞ σ 2 [kx],[ y] σ 2 [x],[y] = k • . (2.3)
Next we give a similar remark carried from its one dimensional version (see [START_REF] Dedecker | On the weak invariance principle for non-adapted sequences under projective criteria[END_REF]Remark 3] ).

Remark 2.5.

If C 0 (s n 1 ,n 2 ) holds then s -2 n 1 ,n 2 σ 2 n 1 ,n 2 converges to E (m 2 ). Consequently, if E (m 2 ) > 0 then C 0 (σ n 1 ,n 2 ) holds with m = m/ m 2 . Therefore C 1 (σ n 1 ,n 2 )(a)
, and (b) holds, which implies that σ n 1 ,n 2 /

√ n 1 n 2 is a 2p-svf (by using Prop 2.4) and the same is true for

s n 1 ,n 2 / √ n 1 n 2 .
Below we give sucient conditions for C 1 (s n 1 ,n 2 )(c) to hold.

Proposition 2.6. Condition C 1 (s n 1 ,n 2 )(c) is satised as soon as the following conditions hold:

as n 1 ∧ n 2 → ∞, √ n 1 n 2 s n 1 ,n 2 n 1 h=-n 1 n 2 k=-n 2 P 0,0 (X h,k ) -→ m in L 2 , (2.4
)

n 1 n 2 j=1 n 1 h=-n 1 n 2 k=j P 0,0 (X h,k ) 2 2 + n 1 h=-n 1 -j k=-n 2 P 0,0 (X h,k ) 2 2 = o(s 2 n 1 ,n 2 ), (2.5
)

n 1 i=1 n 2 j=1 n 1 h=i n 2 -j k=1-j P 0,0 (X h,k ) 2 2 + -i h=-n 1 n 2 -j k=1-j P 0,0 (X h,k ) 2 = o(s 2 n 1 ,n 2 ). (2.6)
In particular if X 0,0 is F 0,0 -measurable and

s n 1 ,n 2 √ n 1 n 2 = h 1 (n 1 )h 2 (n 2 )
, with h 1 and h 2 two one-parameter slowly varying functions, then C 0 (s n 1 ,n 2 ) is satised as soon as:

R n 1 ,n 2 = o(s n 1 ,n 2 ) and √ n 1 n 2 s n 1 ,n 2 n 1 h=0 n 2 k=0 P 0,0 (X h,k ) → m in L 2 . (2.7)
As a consequence of the previous proposition, we obtain the following corollary : Corollary 2.7. Consider the following conditions:

n 1 i=-n 1 n 2 j=-n 2 P 0,0 (X i,j ) -→ m in L 2 , and S n 1 ,n 2 2 √ n 1 n 2 -→ m 2 , as n 1 ∧ n 2 → ∞, (2.8)
X 0,0 is regular and

(i,j)∈Z 2 P 0,0 (X i,j ) 2 < ∞.
(2.9)

We have the implications (2.9)

⇒ (2.8) ⇒ C 1 ( √ n 1 n 2 ).
Note that in dimension one, (2.9) is the so-called Hannan's condition. The condition (2.8) can be viewed as a non-adapted version of the Theorem 5 of [START_REF] Peligrad | Martingale approximations for random elds[END_REF].

Weak invariance principle

We start this section by giving sucient conditions for s -2

n 1 ,n 2 max 1≤k≤n 1 1≤l≤n 2 S 2 k,l (n 1 ,n 2 )∈N 2
to be uniformly integrable. With this aim, let us introduce the following notation :

R n 1 ,n 2 (S k,l ) = E(S k,l |F n 1 ,0 ) + E(S k,l |F 0,n 2 ) -E(S k,l |F 0,0 ). Proposition 3.1. The sequence s -2 n 1 ,n 2 max 1≤k≤n 1 1≤l≤n 2 S 2 k,l (n 1 ,n 2 )∈N 2
is uniformly integrable if the following conditions are satised :

C 2 (s n 1 ,n 2 ):                                  (a) sup 1≤k≤n 1 1≤l≤n 2 |R n 1 ,n 2 (S k,l )| 2 = o(s n 1 ,n 2 ), (b) sup 1≤k≤n 1 1≤l≤n 2 |S k,l -E [S k,l |F n 1 ,n 2 ]| 2 = o(s n 1 ,n 2 ),
(c) for some positive sequence (u i,j ) i,j∈Z 2 such that

√ n 1 n 2 s n 1 ,n 2 n 1 i=-n 1 n 2 j=-n 2 u i,j is bounded, lim λ→∞ lim sup n 1 ∧n 2 →∞ √ n 1 n 2 s n 1 ,n 2 n 1 i=-n 1 n 2 j=-n 2 E P 2 0,0 (X i,j ) u i,j 1 P 2 0,0 (X i,j )>λu 2 i,j = 0.
Remark 3.2. Note that if the rst part of C 2 (s n 1 ,n 2 )(c) holds then its second part does as soon as

P 0,0 (X i,j ) u i,j (i,j)∈Z 2
is a square uniformly integrable family.

We give now sucient conditions for C 2 (s n 1 ,n 2 )(a) and (b) to hold.

Proposition 3.3. C 2 (s n 1 ,n 2 )(a) and (b) hold as soon as X 0,0 is regular and

C 3 (s n 1 ,n 2 ):                  (a) n 1 u=1 n 2 v=1 |i|≥u |j|≥v P 0,0 (X i,j ) 2 2 = o(s n 1 ,n 2 ), (b) √ n 1 n 1 -1 i=1-n 1 n 2 v=1 |j|≥v P 0,0 (X i,j ) 2 2 = o(s n 1 ,n 2 ), (c) √ n 2 n 1 u=1 n 2 -1 j=1-n 2 |i|≥u P 0,0 (X i,j ) 2 2 = o(s n 1 ,n 2 ).
In addition, in the adapted case, C 3 (s n 1 ,n 2 ) holds provided

C 3 (s n 1 ,n 2 ) :          (a) √ n 1 n 1 u=1 n 2 v=1 1 √ u E (X u,v |F 0,0 ) 2 = o(s n 1 ,n 2 ), (b) √ n 2 n 1 u=1 n 2 v=1 1 √ v E (X u,v |F 0,0 ) 2 = o(s n 1 ,n 2 ).
Remark 3.4. If s 2 n 1 ,n 2 is regular enough, so for instance if there exist α and β in ]0, 1[ such that

n 1-α 1 n 1-β 2 (n 1 + n 2 ) s 2 n 1 ,n 2 → ∞ as n 1 ∧ n 2 → ∞, then Hölder's inequality combined with Kro- necker's Lemma for double indexed sequences implies that C 3 (s n 1 ,n 2 ) is satised as soon as u,v≥1 uv s 2 u,v E (X u,v |F 0,0 ) 2 2 < ∞. According to [1, Page 88], the uniform integrability of s -2 n 1 ,n 2 max 1≤k≤n 1 1≤l≤n 2 S 2 k,l (n 1 ,n 2 )∈N 2 implies the tightness of {s -1 n 1 ,n 2 S [n 1 t 1 ],[n 2 t 2 ] , (t 1 , t 2 ) ∈ [0, 1] 2 } in D([0, 1] 2 ).
Hence, Theorem 2.2 together with Proposition 3.1 give the following weak invariance principle. Theorem 3.5. Suppose that s

[n 1 t 1 ],[n 2 t 2 ] /s n 1 ,n 2 is bounded for any t ∈ [0, 1] 2 . If C 1 (s n 1 ,n 2 )(c) and C 2 (s n 1 ,n 2 ) hold and T (1) or T (2) is ergodic, then s -1 n 1 ,n 2 S [n 1 t 1 ],[n 2 t 2 ] , t ∈ [0, 1] 2 converges in distribution in D([0, 1] 2 )
to σW 2 where W 2 is a 2-parameter Brownian sheet and σ = E (m 2 ). Remark 3.6. The proof reveals that if we assume that C 1 (s n 1 ,n 2 )(c) holds with m such that E (m 2 ) > 0, then we do not need to assume the boundedness of (s

[n 1 t 1 ],[n 2 t 2 ] /s n 1 ,n 2 ) t∈[0,1] 2 in
the statement of the previous theorem. [START_REF] Dedecker | On the weak invariance principle for non-adapted sequences under projective criteria[END_REF] Application to linear elds of ortho-martingales Dene X 0,0 = i,j∈Z 2 a i,j ξ 0,0 •T -i,-j where (a i,j ) (i,j)∈Z 2 a double-indexed sequence of real numbers in 2 , and ξ 0,0 is a regular F 0,0 -measurable function in L 2 such that

E (ξ 0,0 |F -1,0 ) = E (ξ 0,0 |F 0,-1 ) = 0 a.s. . Let ξ k, = ξ 0,0 • T k, and S n 1 ,n 2 = n 1 k=1 n 2 =1 X k, = n 1 k=1 n 2 =1 (i,j)∈Z 2 a i,j ξ k-i, -j .
Corollary 4.1. Let ξ 0,0 , (a k, ) k, ∈Z 2 and X 0,0 be dened as above. We then dene the doubleindexed sequence

s n 1 ,n 2 = √ n 1 n 2 n 1 i=-n 1 n 2 j=-n 2 a i,j . Assume the following conditions: lim sup n 1 ∧n 2 →∞ n 1 k=-n 1 n 2 =-n 2 |a k, | n 1 k=-n 1 n 2 =-n 2 a k, < ∞ , (4.1
)

n 1 u=1 n 2 v=1 |i|≥u |j|≥v a 2 i,j = o(s n 1 ,n 2 ) , (4.2)
√ n 1 n 1 -1 i=1-n 1 n 2 v=1 |j|≥v a 2 i,j = o(s n 1 ,n 2 ) , (4.3)
and

√ n 2 n 1 u=1 n 2 -1 j=1-n 2 |i|≥u a 2 i,j = o(s n 1 ,n 2 ) . (4.4) Then s -1 n 1 ,n 2 S [n 1 t 1 ][n 2 t 2 ] , (t 1 , t 2 ) ∈ [0, 1] 2 converges in distribution in D([0, 1] 2 ) to E ξ 2 0,0 W 2 , where W 2 is a 2-parameter Brownian sheet. Remark 4.2. If Condition (4.1) is satised and i,j∈Z |a i,j | < ∞ instead of assuming (4.2)- (4.4)
, then the conclusion of Corollary 4.1 follows from Theorem 5.1 in [START_REF] Volný | An invariance principle for stationary random elds under Hannan's condition[END_REF] since, according to Condition (4.1), as

n 1 , n 2 → ∞, s n 1 ,n 2 /
Comment 4.3. Condition (4.1) of the above corollary does not allow the following possibility:

n 1 k=-n 1 n 2 =-n 2 |a k, | diverges but n 1 k=-n 1 n 2 =-n 2 a k, converges.
For instance let us consider the following sequence: a i,j = (-1) i+j (i + j) -2 for i, j ≥ 1 and 0 otherwise, then Corollary 4.1 does not apply. More generally, this is also the case for double alternating series i,j a i,j (in the sense that each row and column is an alternating simple series) that are in addition monotonic which means that |a i,j | ≤ |a n,m | for i ≥ n and j ≥ m. Indeed such series are not absolutely convergent if i,j |a i,j | = ∞ but convergent as soon as |a i,j + a i+1,j | ≥ |a i,j+1 + a i+1,j+1 | for all i, j (see Meyer [START_REF] Meyer | On the convergence of alternating double series[END_REF] for more details) and then, in this situation, condition (4.1) fails. However, by simple algebra, we infer that, for such series, condition (2.7) holds with

s n 1 ,n 2 ∼ √ n 1 n 2 i,j≥0 a i,j
(and then s -1 n,n S n,n converges in distribution to a Gaussian random variable) as soon as

u,v≥0 n 1 k=1 n 2 =1 a k+u, +v 2 + n 1 u=1 n 1 k=1 n 2 =1 a k+u, +v 2 = o(n 1 n 2 )
and

n 1 u=1 v≥0 n 1 k=u n 2 =1 a k+u, +v 2 + n 2 v=1 u≥0 n 1 k=1 n 2 =v a k+u, +v 2 = o(n 1 n 2 ).
Clearly, both of the above conditions hold if u,v≥0 k≥u ≥v a k,

2

< ∞ which is satised as soon as i,j≥0 a 2 i,j < ∞ (to see this, use an Abel transformation and the monotonicity conditions on the sequence (a i,j )). for k ≥ 0 and ≥ 0, and 0 otherwise, satises the condition of Corollary 4.1 with

s n 1 ,n 2 ∼ √ n 1 n 2 log(n 1 ) log(n 2 ).
Example 4.5. Let us consider now another example in the same spirit but for which the normalizing sequence s n 1 ,n 2 is not a product of a function of n 1 times a function of n 2 . The double indexed sequence (a k, ) k, ∈Z 2 is this time dened by a k, = 1 (k + ) 2 for k and > 0, and 0 otherwise. In this case the conditions of Corollary 4.1 are satised with

s n 1 ,n 2 ∼ √ n 1 n 2 log n 1 n 2 n 1 + n 2 .
Comment 4.6. It is also possible to apply our Theorem 3.5 to non linear random elds. For instance, let us consider the following Volterra process

X k, = i,j≥0 ξ k-i, -j (α i,j ξ k-i-1, -j + β i,j ξ k-i, -j-1 ) ,
where (ξ k, ) k, is a sequence of ortho-martingale dierences. In this case, setting

s n 1 ,n 2 = √ n 1 n 2 n 1 i=0 n 2 j=0 (α i,j + β i,j
) and assuming that there exist reals α and β such that

lim n 1 ∧n 2 →∞ n 1 i=0 n 2 j=0 α i,j n 1 i=0 n 2 j=0 (α i,j + β i,j ) = α and lim n 1 ∧n 2 →∞ n 1 i=0 n 2 j=0 β i,j n 1 i=0 n 2 j=0 (α i,j + β i,j ) = β , (4.5)
one sees that condition (2.4) holds with m = ξ 0,0 αξ -1,0 + βξ 0,-1 . Moreover, following the lines of the proof of Corollary 4.1, we infer that, if in addition to (4.5), we assume that

lim n 1 ∧n 2 →∞ n 1 i=0 n 2 j=0 (|α i,j | + |β i,j |) n 1 i=0 n 2 j=0 (α i,j + β i,j ) < ∞
and conditions (4.2)-(4.4) are satised with α 2 i,j + β 2 i,j replacing a 2 i,j in the numerators, then

s -1 n 1 ,n 2 S [n 1 t 1 ][n 2 t 2 ] , (t 1 , t 2 ) ∈ [0, 1] 2 converges in distribution in D([0, 1] 2 ) to c α,β W 2 , where W 2 is a 2-parameter Brownian sheet and c 2 α,β = E(m 2 ).

Extension to higher dimension d > 2

In this section, we extend our results to dimension d > 2. To do so, we rst introduce some notations in dimension d. Let i, j be two elements of Z d . For i := (i 1 , . . . , i d ), and j := (j 1 , . . . , j d ), we set

• i + j := (i 1 + j 1 , . . . , i d + j d ) • i ≤ j mean that i k ≤ j k for all 1 ≤ k ≤ d • i ∧ j := (i 1 ∧ j 1 , . . . , i d ∧ j d ) • |n| := (n 1 × • • • × n d )
Note also that the integers of Z d will be in bold, for example 1 = (1, . . . , 1) ∈ Z d . Next for n = (n 1 , . . . , n d ), and for every 1 ≤ k ≤ d, we set n k = (0, . . . , n k , . . . , 0), n k 1 k 2 = (0, . . . , 0, n k 1 , 0, . . . , 0, n k 2 , 0, . . . , 0) and so on. Also n -k = (n 1 , . . . , n k-1 , 0, n k+1 , . . . , n d ).

As in dimension 2, let F i = σ(ξ j , j ≤ i) where (ξ i ) i∈Z d is an auxiliary stationary process. We assume that the ltration is commuting. The shift transformation

T (1) , • • • , T (d) is dened as follows: on R Z d , for 1 ≤ j ≤ d, T (j) ((x k ) k∈Z d ) = (x k 1 ,••• ,k j-1 ,k j +1,k j+1 ,••• ,k d ) k∈Z d .
(5.1)

For i = (i 1 , • • • , i d ), we set T i 1 (1) • • • • • T i d (d) ((w k ) k∈Z d ) by T i ((w k ) k∈Z d .
Next, for a real valued measurable function f on R Z d , we dene

X 0 = f (ξ i ) i∈Z d , and X k = f T k (ξ i ) i∈Z d = X 0 • T k .
Let S n be its associated partial sum dened as

S n = n 1 i 1 =1 • • • n d i d =1 X i .
We now clarify the rest R n (S n ) in the decomposition of S n with the help of the projective operators. We have

S n = n 1 k 1 =1 • • • n d k d =1 P k (S n ) + R n (S n ), where R n (S i ) = (-1) d-1 E [S i |F 0 ] + (-1) d-2 d k=1 E S i |F n k + (-1) d-3 d-1 k 1 =1 d k 2 >k 1 E S i |F n k 1 k 2 +(-1) d-4 d-2 k 1 =1 d-1 k 2 >k 1 d k 3 >k 2 E S i |F n k 1 k 2 k 3 + • • • +(-1) d-d d-(d-1)+1 k 1 =1 3 k 2 >k 1 • • • d-1 k d-2 >k d-3 d k d-1 >k d-2 E S i |F n k 1 k 2 ..k d-1 .
To simplify its expression, let us introduce some specic notations. We dene the set D of integers from 1 to d,

(D := (1, ..., d)). Next for any 1 ≤ k ≤ d, Q k will designate a set of the type {i 1 , . . . , i k } with i u = i v for u = v and i u ∈ D for any 1 ≤ u ≤ k.
Notice that for k xed, there exist dierent Q k since there are d k ways to chose a set of k dierent integers among D. Thus we will numerate each combination of

Q k by Q k for 1 ≤ ≤ d k
, and these combinations will be numerated in the natural increasing way. For

example for k = 1, Q 1 1 = {1} , Q 2 1 = {2} . . . Q d 1 = {d} and for k = 2, Q 1 2 = {1, 2} , Q 2 2 = {1, 3} . . . Q d-1 2 = {1, d}, Q (d-1)+1 2 = {2, 3} . . . Q (d-1)+(d-2)+1 2 = {3, 4}. In addition (Q k ) C will denote the complement of Q k in D. For Q k := {e 1 . . . e k }, we denote F Q k by F n e 1 ...e k . Hence, setting u k,d = d k , we have R n (S i ) = (-1) d-1 E [S i |F 0 ] + d-1 k=1 (-1) d-1-k u k,d =1 E S i |F Q k .
(5.2)

5.1

Ortho-Martingale approximation and CLT in dimension d ≥ 2 Theorem 5.1. Let m be a F 0 -measurable function such that, E (m|F -1,0...0 ) = E (m|F 0,-1,0...0 ) = • • • = E (m|F 0...0,-1 ) = 0. Then the two following conditions are equivalent

C d 0 (s n ): lim n→∞ S n s n - 1 √ n n 1 i 1 =1 • • • n d i d =1 m • T i 2 = 0 . C d 1 (s n ):                              (a) for every 0 ≤ k ≤ d -1, and 1 ≤ ≤ d k E S n |F Q k 2 = o(σ n ), (b) S n -E [S n |F n ] 2 = o(s n ), (c) lim n→∞ 1 n n 1 i 1 =1 • • • n d i d =1 √ n s n n 1 k 1 =1 • • • n d k d =1 P 0 (X k-i ) -m 2 2 = 0.
If one of the conditions is satised and one of T (1) , • • • , T (d) is ergodic, then s -1 n S n converges in distribution to σN , where N is a standard gaussian random variable, and σ 2 = E(m 2 ).

lim n→∞ σ 2 k 1 n 1 ,...,k d n d σ 2 n 1 ,...,n d = lim n→∞ E S 2 k 1 n 1 ,...,k d n d E S 2 n 1 ,...,n d = k 1 • • • k d . (5.3) In addition, if σ n 1 ,...,n d → ∞ as n 1 ∧. . .∧n d → ∞ and if for every 1 ≤ k ≤ d-1, and 1 ≤ ≤ u k,d , we have lim n 1 ∧...∧n d →∞ σ Q k σ n 1 ,...,n d = 0, (5.4)
where for example

σ Q 1 1 = σ 1,n 2 ,...,n d . Then, lim x 1 ∧...∧x d →∞ σ 2 [k 1 x 1 ],...,[k d x d ] σ 2 [x 1 ],...,[x d ] = k 1 • • • k d . (5.5)
As in dimension 2, we now give a sucient condition to prove C d 1 (s n )(c).

Proposition 5.3. Condition C d 1 (s n )(c) holds as soon as

|n| s n n 1 k 1 =-n 1 • • • n d k d =-n d P 0 (X k ) -→ m in L 2 , (5.6)
and

d k=1 n 1 u 1 =1 • • • n d u d =1 nq iq=-nq q∈Q 1 k-1 n k i k =u k + -u k i k =-n k np-up ip=1-up p∈(Q 1 k ) C P 0 (X i ) 2 2 = o(s 2 n ), (5.7)
with

Q 1 k = {1, 2, 3, . . . , k} and Q 1 0 = (Q 1 d ) C = ∅.
In particular if X 0 is F 0 -measurable and

s n |n| = h 1 (n 1 ) . . . h d (n d )
with h k a one-parameter slowly varying function for k = 1, ..., d, then

C d 0 (s n )
is satised as soon as:

|n| s n n 1 k 1 =0 • • • n d k d =0 P 0 (X k ) -→ m in L 2 , (5.8)
and

C d 0 (s n )(a) is satised.
As a consequence, we get a generalized version of Corollary 2.7.

Corollary 5.4. Considering the following conditions

i∈Z d P 0 (X i ) -→ L 2 m,
and

S n 2 √ n -→ m 2 , (5.9)
X 0 is regular and

i∈Z d P 0 (X i ) 2 < ∞.
(5.10)

We have the implications (5.10) ⇒ (5.9

) ⇒ C d 1 (s n ).
One can see this as an non-adapted case of Theorem 7 in [START_REF] Peligrad | Martingale approximations for random elds[END_REF] 5.2

Weak invariance principle in dimension d ≥ 2 Proposition 5.5. The sequence s -2 n max 1≤i≤n S 2 i n∈N d is uniformly integrable if and only if the following conditions are satised :

C d 2 (s n ):                          (a) sup 1≤i≤n |R n (S i )| 2 = o(s n ), (b) sup 1≤i≤n |S i -E [S i |F n ]| 2 = o(s n ).
(c) for some positive sequence

(u i ) i∈Z d such that |n| s n n i=-n u i is bounded, lim λ→∞ lim sup n 1 ∧•••∧n d →∞ |n| s n n i=-n E P 2 0 (X n ) u i 1 P 2 0 (X i )>λu 2 i = 0.
Remark 3.3 can also be adapted to dimension d.

Remark 5.6. Condition C d 2 (s n )(a) and (b) hold as soon as, X 0 is regular and

n 1 u 1 =1 • • • n d u d =1 |i 1 |≥u 1 • • • |i d |≥u d P 0 (X i ) 2 2 = o(s n ), (5.11)
and for every

1 ≤ k ≤ d -1, and 1 ≤ ≤ d k , q∈Q k √ n q nq-1 iq=1-nq q∈Q k np ip=1 p∈(Q k ) C +∞ ip=up p∈(Q k ) C P 0 (X i ) 2 2 = o(s n ), (5.12
)

q∈Q k √ n q nq-1 iq=1-nq q∈Q k np ip=1 p∈(Q k ) C -up ip=-∞ p∈(Q k ) C P 0 (X i ) 2 2 = o(s n ), (5.13)
Next, Theorem 5.1 together with Proposition 5.5 give the following weak invariance principle Theorem 5.7. Assume C d 2 (s n ) holds and that one of

T (1) , • • • , T (d) is ergodic. In addition suppose that s [n•t] /s n is bounded for any t ∈ [0, 1] d and E (m 2 ) < ∞. If C d 1 (s n )(c) holds, then s -1 n S [n•t] , t ∈ [0, 1] d converges in distribution in D([0, 1] d ) to σW d where W d is a d-parameter Brownian sheet and σ = E (m 2 ).
The proofs of the results stated in this section are of the same avor as those developed in Section 6 when d = 2 but with more tedious computations due to the higher dimension. They are therefore omitted and left to the reader. Note rst that the following decomposition holds :

S n 1 ,n 2 = S n 1 ,n 2 -E [S n 1 ,n 2 |F n 1 ,n 2 ] + R n 1 ,n 2 (S n 1 ,n 2 ) + E [S n 1 ,n 2 |F n 1 ,n 2 ] -R n 1 ,n 2 (S n 1 ,n 2 ), (6.1)
with

R n 1 ,n 2 (S n 1 ,n 2 ) = E(S n 1 ,n 2 |F n 1 ,0 ) + E(S n 1 ,n 2 |F 0,n 2 ) -E(S n 1 ,n 2 |F 0,0 ). Then we denote n 1 i=1 n 2 j=1 m • T i,j by M n 1 ,n 2 .
Note rst that with the denition of m, we have that

E (M n 1 ,n 2 |F 0,0 ) = E (M n 1 ,n 2 |F n 1 ,0 ) = E (M n 1 ,n 2 |F 0,n 2 ) = 0. Therefore recalling that R n 1 ,n 2 = E [S n 1 ,n 2 |F n 1 ,0 ] 2 + E [S n 1 ,n 2 |F 0,n 2 ] 2 + E [S n 1 ,n 2 |F 0,0 ] 2 , we derive that R n 1 ,n 2 ≤ 3 S n 1 ,n 2 -M n 1 ,n 2 2 .
Hence C 0 (s n 1 ,n 2 ) entails C 1 (s n 1 ,n 2 )(a). Next, using (6.1) and orthogonality, we have :

S n 1 ,n 2 s n 1 ,n 2 - M n 1 ,n 2 √ n 1 n 2 2 2 = S n 1 ,n 2 -E [S n 1 ,n 2 |F n 1 ,n 2 ] s n 1 ,n 2 2 2 + R n 1 ,n 2 (S n 1 ,n 2 ) s n 1 ,n 2 2 2 + E [S n 1 ,n 2 |F n 1 ,n 2 ] -R n 1 ,n 2 (S n 1 ,n 2 ) s n 1 ,n 2 - M n 1 ,n 2 √ n 1 n 2 2 2 := A 2 + B 2 + C 2 . (6.2) Therefore C 0 (s n 1 ,n 2 ) implies C 1 (s n 1 ,n 2 )(b)
and C → 0. Note now that the following decomposition is valid:

E [S n 1 ,n 2 |F n 1 ,n 2 ] -R n 1 ,n 2 (S n 1 ,n 2 ) = n 1 i=1 n 2 j=1 P i,j (S n 1 ,n 2 ). (6.3)
For more details concerning the decomposition we refer to [START_REF] Zhang | On the quenched CLT for stationary random elds under projective criteria[END_REF]. Using (6.3), it follows that

C = 1 √ n 1 n 2 n 1 i=1 n 2 j=1 √ n 1 n 2 n 1 h=1 n 2 k=1 P i,j (X h,k ) s n 1 ,n 2 -m • T i,j 2 .
Hence, by orthogonality, we get

C 2 = 1 n 1 n 2 n 1 i=1 n 2 j=1 √ n 1 n 2 n 1 h=1 n 2 k=1 P i,j (X h,k ) s n 1 ,n 2 -m • T i,j 2 2
.

Using stationarity, we then derive

C 2 = 1 n 1 n 2 n 1 i=1 n 2 j=1 √ n 1 n 2 n 1 h=1 n 2 k=1 P 0,0 (X h-i,k-j ) s n 1 ,n 2 -m 2 2 . (6.4) Therefore, C → 0 is equivalent to C 1 (s n 1 ,n 2 )(c). This ends the proof of C 0 (s n 1 ,n 2 ) ⇒ C 1 (s n 1 ,n 2 ).
For the converse, we use (6.2), (6.4) and the fact that

R n 1 ,n 2 (S n 1 ,n 2 ) 2 ≤ R n 1 ,n 2 . Now if either C 0 (s n 1 ,n 2 ) or C 1 (s n 1 ,n 2 )
holds then, by Remark 2.3, we derive that s -1 n 1 ,n 2 S n 1 ,n 2 converges in distribution to E (m 2 )N , provided that one of the transformations T (1) or T (2) is ergodic. Assume that the conditions C 1 (σ n 1 ,n 2 )(a) and C 1 (σ n 1 ,n 2 )(b) hold. To prove σ n 1 ,n 2 is a 2p-svf, it suces to prove (2.1). Note rst that the following decomposition holds: for every non-negative integers k and ,

σ 2 kn 1 , n 2 = S n 1 , n 2 2 2 + k i=2 S in 1 , n 2 -S (i-1)n 1 , n 2 2 2 + 2B, (6.5)
where

B = E S n 1 , n 2 k i=2 S in 1 , n 2 -S (i-1)n 1 , n 2 .
Hence by stationarity,

σ 2 kn 1 , n 2 = k S n 1 , n 2 2 2 + 2 k-1 i=2 k j=i+1 E S n 1 , n 2 S (j-i+1)n 1 , n 2 -S (j-i)n 1 , n 2 + 2B := k S n 1 , n 2 2 2 + 2A + 2B.
We shall then focus on S n 1 , n 2 2 2 . Proceeding as before, we have

S n 1 , n 2 2 2 = S n 1 ,n 2 2 2 + 2 -1 i=2 j=i+1 E S n 1 ,n 2 S n 1 ,(j-i+1)n 2 -S n 1 ,(j-i)n 2 + 2E S n 1 ,n 2 i=2 S n 1 ,in 2 -S n 1 ,(i-1)n 2 := S n 1 ,n 2 2 2 + 2C + 2D.
Therefore,

σ 2 kn 1 , n 2 = k • S n 1 ,n 2 2
2 + 2A + 2B + 2C + 2D. Hence to prove (2.1), it is sucient to show that A, B, C and D are o(σ 2 n 1 ,n 2 ). We rst handle D and write

|D| ≤ E (S n 1 ,n 2 -E [S n 1 ,n 2 |F n 1 ,n 2 ]) i=2 S n 1 ,in 2 -S n 1 ,(i-1)n 2 + E E [S n 1 ,n 2 |F n 1 ,n 2 ] i=2 S n 1 ,in 2 -S n 1 ,(i-1)n 2 := D 1 + D 2 .
By stationarity and using C 1 (σ n 1 ,n 2 )(b) we infer that

D 1 ≤ S n 1 ,n 2 -E [S n 1 ,n 2 |F n 1 ,n 2 ] 2 i=2 S n 1 ,in 2 -S n 1 ,(i-1)n 2 2 ≤ ( -1) S n 1 ,n 2 2 S n 1 ,n 2 -E [S n 1 ,n 2 |F n 1 ,n 2 ] 2 = o(σ 2 n 1 ,n 2 ).
On another hand, using stationarity and C 1 (σ n 1 ,n 2 )(a) we get

D 2 ≤ S n 1 ,n 2 2 E i=2 S n 1 ,in 2 -S n 1 ,(i-1)n 2 |F n 1 ,n 2 2 ≤ ( -1) S n 1 ,n 2 2 E (S n 1 ,n 2 |F n 1 ,0 ) 2 = o(σ 2 n 1 ,n 2 ).
Thus D = o(σ 2 n 1 ,n 2 ). We handle now the quantity C and write

|C| ≤ -1 i=2 j=i+1 E (S n 1 ,n 2 -E [S n 1 ,n 2 |F n 1 ,n 2 ]) S n 1 ,(j-i+1)n 2 -S n 1 ,(j-i)n 2 + -1 i=2 j=i+1 E E [S n 1 ,n 2 |F n 1 ,n 2 ] S n 1 ,(j-i+1)n 2 -S n 1 ,(j-i)n 2 := C 1 + C 2 .
Using stationarity C 1 (σ n 1 ,n 2 )(b), we get

C 1 ≤ -1 i=2 j=i+1 S n 1 ,n 2 -E [S n 1 ,n 2 |F n 1 ,n 2 ] 2 S n 1 ,(j-i+1)n 2 -S n 1 ,(j-i)n 2 2 ≤ 2 S n 1 ,n 2 2 S n 1 ,n 2 -E [S n 1 ,n 2 |F n 1 ,n 2 ] 2 = o(σ 2 n 1 ,n 2 ).
Next, by stationarity and C 1 (σ n 1 ,n 2 )(a) we infer that,

C 2 ≤ -1 i=2 j=i+1 S n 1 ,n 2 2 E S n 1 ,(j-i+1)n 2 -S n 1 ,(j-i)n 2 |F n 1 ,n 2 ≤ 2 S n 1 ,n 2 2 E (S n 1 ,n 2 |F n 1 ,0 ) 2 = o(σ 2 n 1 ,n 2 ). Recall that, S n 1 , n 2 2 2 = S n 1 ,n 2 2 2 + 2C + 2D. Thereby, lim n 1 ∧n 2 →∞ S n 1 , n 2 2 2 S n 1 ,n 2 2 2
= . (6.6)

We handle now the quantity B. We infer that

|B| ≤ E (S n 1 , n 2 -E [S n 1 , n 2 |F n 1 , n 2 ]) k i=2 S in 1 , n 2 -S (i-1)n 1 , n 2 + E E S n 1 , n 2 |F n1, n 2 k i=2 S in 1 , n 2 -S (i-1)n 1 , n 2 := B 1 + B 2 .
Using stationarity and C 1 (σ n 1 ,n 2 )(b) and (6.6), we get

B 1 ≤ S n 1 , n 2 -E [S in 1 , n 2 |F in 1 , n 2 ] 2 k i=2 S in 1 , n 2 -S (i-1)n 1 , n 2 2 ≤ (k -1) S n 1 , n 2 -E [S in 1 , n 2 |F in 1 , n 2 ] 2 S n 1 , n 2 2 = o(σ 2 n 1 ,n 2 ).
Using stationarity and C 1 (σ n 1 ,n 2 )(a) and (6.6), we derive

B 2 ≤ S n 1 , n 2 2 E k i=2 S in 1 , n 2 -S (i-1)n 1 , n 2 |F n 1 , n 2 2 ≤ (k -1) S n 1 , n 2 2 E (S n 1 , n 2 |F 0, n 2 ) 2 = o(σ 2 n 1 ,n 2 ). Therefore B = o(σ 2 n 1 ,n 2
). Now we handle the quantity A and write

|A| ≤ k-1 i=2 k j=i+1 E (S n 1 , n 2 -E [S n 1 , n 2 |F n 1 , n 2 ]) S (j-i+1)n 1 , n 2 -S (j-i)n 1 , n 2 + k-1 i=2 k j=i+1 E (E [S n 1 , n 2 |F n 1 , n 2 ]) S (j-i+1)n 1 , n 2 -S (j-i)n 1 , n 2 := A 1 + A 2 .
Thereby, with similar arguments as above,

A 1 ≤ k 2 S n 1 , n 2 2 S n 1 , n 2 -E (S n 1 , n 2 |F 0, n 2 ) 2 = o(σ 2 n 1 ,n 2 ), A 2 ≤ k 2 S n 1 , n 2 2 E (S n 1 , n 2 |F 0, n 2 ) 2 = o(σ 2 n 1 ,n 2 ).
Hence A = o(σ 2 n 1 ,n 2 ). This ends the proof of (2.1). Now we prove (2.3). Let introduce the two following conditions :

lim x∧y→∞ k[x]+k+1 i=1 [y]+ +1 j=1 X i,j -[kx] i=1 [ y] j=1 X i,j 2 σ [x],[y]
= 0 (6.7)

and

lim x∧y→∞ k[x]+k+1 i=1 [y]+ +1 j=1 X i,j -k[x] i=1 [y] j=1 X i,j 2 σ [x],[y] = 0. (6.8)
We infer that if (6.7) and (6.8) hold then (2.1) can be extended to (2.3). Next, it is easy to see that (6.7) and (6.8) hold as soon as σ The rst part of the proof is quite direct. Indeed, let us denote the double sums in the norm of the conditions (2.5) and (2.6) by a, b, c, d, and the quantities appearing in the left hand side of the conditions (2.5) and(2.6) by A, B, C, and D. For example:

[x],[y] → ∞ as [x] ∧ [y] → ∞,
a = √ n 1 n 2 s n 1 ,n 2 n 1 h=-n 1 n 2 k=j P 0,0 (X h,k ) and A = n 1 n 2 j=1 n 1 h=-n 1 n 2 k=j P 0,0 (X h,k ) 2 2
.

We have

1 n 1 n 2 n 1 i=1 n 2 j=1 √ n 1 n 2 s n 1 ,n 2 n 1 -i h=1-i n 2 -j k=1-j P 0,0 (X h,k ) -m 2 2 = 1 n 1 n 2 n 1 i=1 n 2 j=1 √ n 1 n 2 s n 1 ,n 2 n 1 h=-n 1 n 2 k=-n 2 P 0,0 (X h,k ) -m -a -b -c -d 2 2 ≤ √ n 1 n 2 s n 1 ,n 2 n 1 h=-n 1 n 2 k=-n 2 P 0,0 (X h,k ) -m 2 2 + A + B + C + D s 2 n 1 ,n 2 . Hence if (2.4)-(2.6) are satised then C 1 (s n 1 ,n 2 )(c) holds.
Now assume that X 0,0 is F 0,0 -measurable and s n 1 ,n 2 √ n 1 n 2 is a composition of two one-parameter slowly varying functions. In this case, conditions (2.4)-(2.6) reduce to

√ n 1 n 2 s n 1 ,n 2 n 1 h=0 n 2 k=0 P 0,0 (X h,k ) -→ m in L 2 , (6.9
)

n 1 n 2 j=1 n 1 h=0 n 2 k=n 2 -j+1 P 0,0 (X h,k ) 2 2 = o(s 2 n 1 ,n 2 ), (6.10
)

n 1 i=1 n 2 j=1 n 1 h=n 1 -i+1 n 2 -j k=0 P 0,0 (X h,k ) 2 2 = o(s 2 n 1 ,n 2 ). (6.11)
Note that the following decomposition holds: for every

n 1 ≥ 1, n 2 > j > 0, √ n 1 n 2 s n 1 ,n 2 n 1 h=0 n 2 k=n 2 -j+1 P 0,0 (X h,k ) = √ n 1 n 2 s n 1 ,n 2 n 1 h=0 n 2 k=0 P 0,0 (X h,k ) - n 1 (n 2 -j) s n 1 ,n 2 -j n 1 h=0 n 2 -j k=0 P 0,0 (X h,k ) + n 1 (n 2 -j) s n 1 ,n 2 -j n 1 h=0 n 2 -j k=0 P 0,0 (X h,k ) 1 - √ n 1 n 2 n 1 (n 2 -j) s n 1 ,n 2 -j s n 1 ,n 2 . Since s n 1 ,n 2 = √ n 1 n 2 h 1 (n 1 )h 2 (n 2 )
with h 1 and h 2 two slowly varying functions, we get

√ n 1 n 2 s n 1 ,n 2 n 1 h=0 n 2 k=n 2 -j+1 P 0,0 (X h,k ) = √ n 1 n 2 s n 1 ,n 2 n 1 h=0 n 2 k=0 P 0,0 (X h,k ) - n 1 (n 2 -j) s n 1 ,n 2 -j n 1 h=0 n 2 -j k=0 P 0,0 (X h,k ) + n 1 (n 2 -j) s n 1 ,n 2 -j n 1 h=0 n 2 -j k=0 P 0,0 (X h,k ) 1 - h 2 (n 2 -j) h 2 (n 2 ) .
It follows that condition (6.9) implies condition (6.10) provided

1 n 2 n 2 j=1 1 - h 2 (j -1) h 2 (n 2 ) 2 -→ 0.
This holds since h 2 is a one-parameter slowly varying function. With similar arguments and using that h 1 is a one-parameter slowly varying function, we get that (6.9) implies (6.11).

Proof of Corollary 2.7

This proof is quite similar to its one dimensional version [4, Corollary 1]. We give it for completeness. Let rst prove (2.8) ⇒ C 1 ( √ n 1 n 2 ). Taking s n 1 ,n 2 = √ n 1 n 2 and m = i,j∈Z 2 P 0,0 (X i,j ), it follows that if the rst part of (2.8) holds then the conditions (2.4)-(2.6) of Proposition 2.6 are clearly satised. Therefore C 1 ( √ n 1 n 2 )(c) holds. Now starting from the decomposition (6.1) and using orthogonality, we derive

S n 1 ,n 2 2 2 n 1 n 2 = 1 n 1 n 2 S n 1 ,n 2 -E [S n 1 ,n 2 |F n 1 ,n 2 ] 2 2 + 1 n 1 n 2 R n 1 ,n 2 (S n 1 ,n 2 ) 2 2 + 1 n 1 n 2 E [S n 1 ,n 2 |F n 1 ,n 2 ] -R n 1 ,n 2 (S n 1 ,n 2 ) 2 2 := A 2 + B 2 + C 2 , (6.12)
where

R n 1 ,n 2 (S n 1 ,n 2 ) = E [S n 1 ,n 2 |F n 1 ,0 ] + E [S n 1 ,n 2 |F 0,n 2 ] -E [S n 1 ,n 2 |F 0,0 ]. According to the proof of Theorem 2.2, since C 1 ( √ n 1 n 2 )(c) holds, C 2 converges to m 2 2 . Since by assumption, (n 1 n 2 ) -1 S n 1 ,n 2 2 2 converges to m 2 2
, we get overall that A and B converge to zero. Note that

A → 0 is exactly C 1 ( √ n 1 n 2 )(b). It remains to prove that C 1 ( √ n 1 n 2 )(a) is satised. With this aim, note that since M n 1 ,n 2 := n 1 i=1 n 2
j=1 m • T i,j is an ortho-martingale with respect to ltration (F i,j ) i,j∈Z , we have

E (S n 1 ,n 2 |F n 1 ,0 ) = E [E (S n 1 ,n 2 |F n 1 ,n 2 ) -R n 1 ,n 2 (S n 1 ,n 2 ) + R n 1 ,n 2 (S n 1 ,n 2 ) -M n 1 ,n 2 |F n 1 ,0 ] . Therefore E (S n 1 ,n 2 |F n 1 ,0 ) 2 √ n 1 n 2 ≤ B + 1 √ n 1 n 2 E (S n 1 ,n 2 |F n 1 ,n 2 ) -R n 1 ,n 2 (S n 1 ,n 2 ) -M n 1 ,n 2 2 ,
which converges to zero by C 1 ( √ n 1 n 2 )(c) and the fact that B → 0. Similarly, we get that

(n 1 n 2 ) -1/2 E (S n 1 ,n 2 |F 0,n 2 ) 2 → 0 and (n 1 n 2 ) -1/2 E (S n 1 ,n 2 |F 0,0 ) 2 → 0. Hence C 1 ( √ n 1 n 2 )(a)
holds.

It remains to prove that (2.9) ⇒ (2.8). Clearly if (2.9) holds so does the rst part of (2.8). Next we shall prove that

1 n 1 n 2 E S 2 n 1 ,n 2 → (k 1 ,k 2 )∈Z 2 E [X 0,0 X k 1 ,k 2 ] a.s. and E m 2 = (k 1 ,k 2 )∈Z 2 E [X 0,0 X k 1 ,k 2 ] , (6.13)
which implies the second part of (2.8). Since X 0,0 is regular, the following decomposition holds

E [X 0,0 X k 1 ,k 2 ] = (i 1 ,i 2 )∈Z 2 (j 1 ,j 2 )∈Z 2 E [P i 1 ,i 2 (X 0,0 )P j 1 ,j 2 (X k 1 ,k 2 )] .
By orthogonality (which comes from the fact that the ltration (F i,j ) is commuting) and stationarity, we derive

E [X 0,0 X k 1 ,k 2 ] = (i 1 ,i 2 )∈Z 2 E [P 0,0 (X i 1 ,i 2 )P 0,0 (X k 1 +i 1 ,k 2 +i 2 )] . (6.14)
Hence

|E [X 0,0 X k 1 ,k 2 ]| ≤ (i 1 ,i 2 )∈Z 2 P 0,0 (X i 1 ,i 2 ) 2 P 0,0 (X k 1 +i 1 ,k 2 +i 2 ) 2 , so that (k 1 ,k 2 )∈Z 2 |E [X 0,0 X k 1 ,k 2 ]| ≤   (i 1 ,i 2 )∈Z 2 P 0,0 (X i 1 ,i 2 ) 2   2 ,
which is nite under condition (2.9). Therefore, the series

(k 1 ,k 2 )∈Z 2 |E [X 0,0 X k 1 ,k 2 ]| converges.
In addition, note that with (6.14), we have

(k 1 ,k 2 )∈Z 2 E [X 0,0 X k 1 ,k 2 ] = (i 1 ,i 2 )∈Z 2 (j 1 ,j 2 )∈Z 2 E [P 0,0 (X i 1 ,i 2 )P 0,0 (X j 1 ,j 2 )] .
In addition,

E m 2 = (i 1 ,i 2 )∈Z 2 (j 1 ,j 2 )∈Z 2 E [P 0,0 (X i 1 ,i 2 )P 0,0 (X j 1 ,j 2 )] .
So,

E m 2 = (k 1 ,k 2 )∈Z 2 E [X 0,0 X k 1 ,k 2 ] .
On another hand,

1 n 1 n 2 E S 2 n 1 ,n 2 = n 1 -1 i 1 =-n 1 +1 n 2 -1 i 2 =-n 2 +1 1 - |i 1 | n 1 1 - |i 2 | n 2 E (X 0,0 X i 1 ,i 2 ) , which then converges to (k 1 ,k 2 )∈Z 2 E [X 0,0 X k 1 ,k 2 ] = E (m 2
). This ends the proof of (2.8).

6.5

Proof of Proposition 3.1

By simplicity in the next proof, we denote max 1≤k≤n 1 1≤l≤n 2 by max k,l , and max 1≤k≤n 1 by max k respectively for l. All along the proof, C will be a universal positive constant (depending on the dimension) which may vary from line to line. Our main aim is to prove that lim λ→∞ lim sup

n 1 ∧n 2 →∞ 1 s 2 n 1 ,n 2 E max k,l S 2 k,l 1 max k,l| S k,l| >λsn 1 ,n 2 = 0.
Note that the following decomposition holds: for every

1 ≤ k ≤ n 1 , 1 ≤ l ≤ n 2 , S k,l = S k,l -E [S k,l |F n 1 ,n 2 ] + R n 1 ,n 2 (S k,l ) + E [S k,l |F n 1 ,n 2 ] -R n 1 ,n 2 (S k,l ), (6.15)
where

R n 1 ,n 2 (S k,l ) = E [S k,l |F n 1 ,0 ] + E [S k,l |F 0,n 2 ] -E [S k,l |F 0,0 ].
Starting from (6.15) and using the fact that for any A ≥ 0,

x 2 1 |x|>A ≤ 4 |x| - A 2 2 +
, where (x) + = x1 x≥0 , we get by convexity that, for any λ ≥ 0,

E max k,l S 2 k,l 1 max k,l| S k,l| >2λsn 1 ,n 2 ≤ 4E max k,l |S k,l | -λs n 1 ,n 2 2 + ≤ 12E max k,l |S k,l -E [S k,l |F n 1 ,n 2 ]| 2 + 12E max k,l |R n 1 ,n 2 (S k,l )| 2 + 12E max k,l |E [S k,l |F n 1 ,n 2 ] -R n 1 ,n 2 (S k,l )| -λs n 1 ,n 2 2 + .
Therefore by the condition C 2 (s n 1 ,n 2 )(a) and (b), it is sucient to show that,

lim λ→∞ lim n 1 ∧n 2 →∞ 1 s 2 n 1 ,n 2 E max k,l |E [S k,l |F n 1 ,n 2 ] -R n 1 ,n 2 (S k,l )| -λs n 1 ,n 2 2 + = 0. (6.16)
For the sake of simplicity, we set S k,l Note that using the projective operators, we have

:= E [S k,l |F n 1 ,n 2 ]-R n 1 ,n 2 (S k,l ), S + n 1 ,n 2 := max(0, S
S k,l = k u=1 l v=1 u-1 i=u-n 1 v-1 j=v-n 2 P u-i,v-j (X u,v ) = k-1 i=1-n 1 l-1 j=1-n 2 k∧(n 1 +i) u=1∨(i+1) l∧(n 2 +j) v=1∨(j+1) P u-i,v-j (X u,v ).
For any xed positive integer i and j, we introduce the double indexed sequence (Y i,j,k,l ) k,l≥1 dened by

Y i,j,k,l = k∧(n 1 +i) u=1∨(i+1) l∧(n 2 +j) v=1∨(j+1) P u-i,v-j (X u,v ).
Notice that, (Y i,j,k,l ) k,l≥1 is an ortho-martingale w.r.t the ltration (F i,j ) i,j∈Z .

With these notations,

S k,l = k-1 i=1-n 1 l-1 j=1-n 2 Y i,j,k,l . Setting b i,j = u i,j ( n 1 k=-n 1 n 2 =-n 2 u k, ) -1 , with (u i,j ) i,j∈Z a positive sequence such that √ n 1 n 2 s n 1 ,n 2 n 1 -1 i=1-n 1 n 2 -1 j=1-n 2 u i,j < ∞, we get S k,l -λs n 1 ,n 2 + ≤ k-1 i=1-n 1 l-1 j=1-n 2 (Y i,j,k,l -λb i,j s n 1 ,n 2 ) + . (6.19)
By Hölder's inequality and taking the maximum over (k, l) on both sides, we derive that

S + n 1 ,n 2 -λs n 1 ,n 2 2 + ≤ n 1 -1 i=1-n 1 n 2 -1 j=1-n 2 u i,j n 1 -1 i=1-n 1 n 2 -1 j=1-n 2 1 u i,j Y + i,j,n 1 ,n 2 -λb i,j s n 1 ,n 2 2 + ,
where Y + i,j,n 1 ,n 2 = max k,l (max(0, Y i,j,k,l )). Hence to prove (6.17), it is sucient to show that,

lim λ→∞ lim sup n 1 ∧n 2 →∞ 1 √ n 1 n 2 s n 1 ,n 2 n 1 -1 i=1-n 1 n 2 -1 j=1-n 2 1 u i,j E Y + i,j,n 1 ,n 2 -λb i,j s n 1 ,n 2 2 + = 0. (6.20)
To prove this, we will need the two following lemmas, whose proofs are postponed to Section 7.

E max k,l |M k,l | p 1 max k,l| M k,l| >λ ≤ 2 2p p p -1 2 E |M n 1 ,n 2 | p 1 |Mn 1 ,n 2 |> λ 4 . Lemma 6.3. Let M n 1 ,n 2 ≡ n 1 i=1 n 2
j=1 d i,j be an ortho-martingale in L 2 . Then for all λ > 0

E (M n 1 ,n 2 -λ) 2 + ≤ n 1 i=1 E   n 2 j=1 d i,j 2
1 max k |Mk,n 2 |>λ   , (6.21)
and

E |M n 1 ,n 2 | 2 1 |Mn 1 ,n 2 |>λ ≤ 8 n 1 i=1 E   n 2 j=1 d i,j 2
1 max k |Mk,n 2 |> λ 2   . (6.22)
In addition, for any xed i (the same goes for j), we have

E   n 2 j=1 d i,j -λ 2 +   ≤ n 2 j=1 E d 2 i,j 1 max l≤n 2 | l j=1 d i,j |>λ . (6.23)
To reduce the complexity of the notation, we write n 1 -1 i=1-n 1 n 2 -1 j=1-n 2 for i,j , and

k∧(n 1 +i) u=1∨(i+1)
by k u , the same goes for l v . Now we denote

U (n 1 , n 2 , λ) := 1 √ n 1 n 2 s n 1 ,n 2 i,j 1 u i,j E Y + i,j,n 1 ,n 2 -λb i,j s n 1 ,n 2 2 + .
Since (|x| -λ) 2 + ≤ x 2 1 |x|>λ , we have

U (n 1 , n 2 , λ) ≤ 1 √ n 1 n 2 s n 1 ,n 2 i,j 1 u i,j E Y + i,j,n 1 ,n 2 2 1 Y + i,j,n 1 ,n 2 >λb i,j sn 1 ,n 2
.

Next applying Lemma 6.2, we get

U (n 1 , n 2 , λ) ≤ C √ n 1 n 2 s n 1 ,n 2 i,j 1 u i,j E Y 2 i,j,n 1 ,n 2 1 |Yi,j,n 1 ,n 2 |> λ 4 b i,j sn 1 ,n 2
.

Then using (6.22) of Lemma 6.3, we derive

U (n 1 , n 2 , λ) ≤ C √ n 1 n 2 s n 1 ,n 2 i,j 1 u i,j n 1 u E   n 2 v P u-i,v-j (X u,v ) 2 1 maxu|Y i,j,u,n 2 |> λ 8 b i,j sn 1 ,n 2   .
Let A > 0 and set A i,j,n 2 = Au 2 i,j n 2 , we have that U (n 1 , n 2 , λ) is less than or equal to

C √ n 1 n 2 s n 1 ,n 2 i,j 1 u i,j n 1 u E   n 2 v P u-i,v-j (X u,v ) 2 1 ( n 2 v P u-i,v-j (Xu,v)) 2 >A i,j,n 2   + C √ n 1 n 2 s n 1 ,n 2 i,j 1 u i,j n 1 A i,j,n 2 P max u |Y i,j,u,n 2 | > λ 8 b i,j s n 1 ,n 2 := C(I + II).
Using (6.23) of Lemma 6.3, we get

I ≤ C 1 √ n 1 n 2 s n 1 ,n 2 i,j 1 u i,j n 1 u n 2 v E P 2 u-i,v-j (X u,v )1 max l| l v P u-i,v-j (Xu,v)|> √ A i,j,n 2 2
.

It follows that for any B > 0

I ≤ C 1 √ n 1 n 2 s n 1 ,n 2 i,j 1 u i,j n 1 u n 2 v E P 2 u-i,v-j (X u,v )1 P 2 u-i,v-j (Xu,v)>Bu 2 i,j + C 1 √ n 1 n 2 s n 1 ,n 2 i,j 1 u i,j n 1 u n 2 v Bu 2 i,j P max l l v P u-i,v-j (X u,v ) > A i,j,n 2 2 .
By stationarity, it follows that,

I ≤ C 1 √ n 1 n 2 s n 1 ,n 2 i,j 1 u i,j E P 2 0,0 (X i,j )1 P 2 0,0 (X i,j )>Bu 2 i,j + C 1 √ n 1 n 2 s n 1 ,n 2 i,j 1 u i,j n 1 u n 2 v Bu 2 i,j P max l l v P u-i,v-j (X u,v ) > A i,j,n 2 2 . := C 1 (I 1 + I 2 ).
The quantity I 1 converges to zero by C 2 (s n 1 ,n 2 )(c), if we rst let n 1 ,n 2 tend to innity and then B to innity. To handle I 2 , we use Doob's inequality that leads to

I 2 ≤ 4 √ n 1 n 2 s n 1 ,n 2 i,j 1 u i,j n 1 u n 2 v Bu 2 i,j n 2 ∧(n 2 +j) v=1∨(j+1) E [P u-i,v-j (X u,v )] 2 A i,j,n 2 .
Using stationarity again, and recalling that A i,j,n 2 = Au 2 i,j n 2 , we derive

I 2 ≤ 4 √ n 1 n 2 s n 1 ,n 2 i,j 1 u i,j n 1 u n 2 v Bu 2 i,j n 2 E P 2 0,0 (X i,j ) Au 2 i,j n 2 ≤ 4 B A √ n 1 n 2 s n 1 ,n 2 i,j E P 2 0,0 (X i,j ) u i,j . Now by C 2 (s n 1 ,n 2 )(c), we have sup n 1 ≥1 n 2 ≥1 √ n 1 n 2 s n 1 ,n 2 n 1 -1 i=1-n 1 n 2 -1 j=1-n 2 E P 2 0,0 (X i,j ) u i,j < ∞. (6.24)
Hence I 2 converges to zero if we rst let A goes to innity and then B to innity. So overall I converges to zero by letting rst n 1 ∧ n 2 tend to innity and then A. Now to deal with II, we proceed as for I 2 . Indeed, Doob's maximal inequality leads to

II = A √ n 1 n 2 s n 1 ,n 2 i,j u i,j P max u |Y i,j,u,n 2 | > λ 8 b i,j s n 1 ,n 2 , ≤ C 2 A √ n 1 n 2 s n 1 ,n 2 i,j u i,j E Y 2 i,j,n 1 ,n 2 λ 2 b 2 i,j s 2 n 1 ,n 2 .
Hence, by stationarity,

II ≤ C 2 A λ 2 √ n 1 n 2 s n 1 ,n 2 i,j E P 2 0,0 (X i,j ) u i,j √ n 1 n 2 s n 1 ,n 2 i,j u i,j 2
, which converges to zero by letting rst n 1 ∧ n 2 tend to innity and then λ. This ends the proof of (6.17). The proof of (6.18) can be done similarly. Notice rst that the condition C 2 (s n 1 ,n 2 )(a) holds as soon as,

sup 1≤k≤n 1 1≤l≤n 2 |E [S k,l |F n 1 ,0 ] -E [S k,l |F 0,0 ]| 2 = o(s n 1 ,n 2 ), (6.25) sup 1≤k≤n 1 1≤l≤n 2 |E [S k,l |F 0,n 2 ] -E [S k,l |F 0,0 ]| 2 = o(s n 1 ,n 2 ), (6.26)
and sup

1≤k≤n 1 1≤l≤n 2 |E [S k,l |F 0,0 ]| 2 = o(s n 1 ,n 2 ). (6.27) Note that sup 1≤k≤n 1 1≤l≤n 2 |E [S k,l |F 0,0 ]| 2 ≤ n 1 k=1 n 2 =1 E (X k, |F 0,0 ) 2 ,
and since

X k, = +∞ i=-∞ +∞ j=-∞ P 0,0 (X k, ), E (X k, |F 0,0 ) = 0 i=-∞ 0 j=-∞ P i,j (X k, ).
Therefore, the condition (6.27) is implied by C 3 (s n 1 ,n 2 )(a). Now, we focus on (6.25). Notice rst that

E [S k,l |F n 1 ,0 ] -E [S k,l |F 0,0 ] = k u=1 v=1 n 1 i=1 0 j=-∞ P i,j (X u,v ).
Thus (6.25) is equivalent to

sup 1≤k≤n 1 1≤l≤n 2 k u=1 v=1 n 1 i=1 0 j=-∞ P i,j (X u,v ) 2 = o(s n 1 ,n 2 ). (6.28) But sup 1≤k≤n 1 1≤l≤n 2 k u=1 v=1 n 1 i=1 0 j=-∞ P i,j (X u,v ) 2 ≤ n 2 v=1 sup 1≤k≤n 1 k u=1 n 1 i=1 0 j=-∞ P i,j (X u,v ) 2 ≤ n 2 v=1 sup 1≤k≤n 1 k u=1 u-1 i=u-n 1 0 j=-∞ P u-i,j (X u,v ) 2 ≤ n 2 v=1 sup 1≤k≤n 1 k-1 i=1-n 1 k∧(i+n 1 ) u=(i+1)∨1 0 j=-∞ P u-i,j (X u,v ) 2 .
We denote 0 j=-∞ P u-i,j (X u,v ) by D i,u (v), and notice that D i,u (v) is orthogonal with respect to u. Next denote k∧(i+n 1 ) u=(i+1)∨1 D i,u (v) by Y i,k (v). Let (U i (v)) i≥1-n 1 be a sequence of positive real numbers that we will dene later. By Cauchy-Schwarz's inequality,

k-1 i=1-n 1 Y i,k (v) 2 = k-1 i=1-n 1 U i (v) 1 U i (v) Y i,k (v) 2 ≤ k-1 i=1-n 1 U i (v) k-1 i=1-n 1 1 U i (v) (Y i,k (v)) 2 .
Therefore,

sup 1≤k≤n 1 k-1 i=1-n 1 Y i,k (v) 2 2 ≤ n 1 -1 i=1-n 1 U i (v) n 1 -1 i=1-n 1 1 U i (v) sup 1≤k≤n 1 |Y i,k (v)| 2 2
.

Note now that (Y i,k (v)) k≥1 is a martingale. Using Doob's inequality and stationarity, it follows that sup So overall taking U i (v) = +∞ j=v P 0,0 (X i,j ) 2 2 , we have

sup 1≤k≤n 1 k-1 i=1-n 1 Y i,k (v) 2 2 ≤ 4n 1   n 1 -1 i=1-n 1 +∞ j=v P 0,0 (X i,j ) 2 2   2 .
Therefore

n 2 v=1 sup 1≤k≤n 1 k-1 i=1-n 1 Y i,k (v) 2 ≤ 2 √ n 1 n 1 -1 i=1-n 1 n 2 v=1 +∞ j=v P 0,0 (X i,j ) 2 2 .
Thus (6.25) holds as soon as C 3 (s n 1 ,n 2 )(b) holds. Since the ltration is commuting, arguments applied to (6.25) can still be applied to show that (6.26) holds. Thus ( We now prove that in adapted case C 3 (s n 1 ,n 2 ) holds as soon as C 3 (s n 1 ,n 2 ) does. In adapted case C 3 (s n 1 ,n 2 )(a) reads as where P ĩ,j (X) = E (X|F i,j ) -E (X|F i-1,j ). We then apply [START_REF] Dedecker | Invariance principles for linear processes with application to isotonic regression[END_REF]Lemma 6.1] to the partial sum with b i = 1 and u i = P -ĩ,-v (X 0,0 ) 2 . Hence we get that

n 1 i=1 n 2 v=1 P -ĩ,-v (X 0,0 ) 2 ≤ C n 1 i=1 1 √ i n 2 v=1 n 1 k=i P -k,-v (X 0,0 ) 2 2 1/2
, Therefore, we deduce that

n 1 i=1 n 2 v=1 P -ĩ,-v (X 0,0 ) 2 ≤ C n 1 i=1 n 2 v=1 1 √ i E (X i,v |F 0,0 ) 2 .
Hence

4 .

 4 As quoted in the introduction, the double indexed sequence (a k, ) (k, )∈Z 2 dened by a k,

Remark 2 .

 2 4 writes in dimension d > 2 as follows: Proposition 5.2. If C d 1 (s n )(a) and (b) hold, then

2 v=1E 1 n 1 i=1 n 2 v=1P

 2112 (X u,v |F 0,0 ) 2 = o(s n 1 ,n 2 ), which is implied by C 3 (s n 1 ,n 2 )(a). Next, note that C 3 (s n 1 ,n 2 )(b) can be rewritten as √ n -ĩ,-v (X 0,0 ) 2 = o(s n 1 ,n 2 ),

  and (2.2) is satised.

	6.3	Proof of Proposition 2.6

  1,1 , . . . , S n 1 ,n 2 ), and S - n 1 ,n 2 := max(0, -S 1,1 , . . . , -S n 1 ,n 2 ).

	and	lim λ→∞	lim sup n 1 ∧n 2 →∞	1 n 1 ,n 2 s 2	E S -n 1 ,n 2 -λs n 1 ,n 2	2 +	= 0.	(6.18)
	Therefore if we assume that C 2 (s n 1 ,n 2 )(a) and C 2 (s n 1 ,n 2 )(b) hold, then max k,l	S 2 k,l s 2 n 1 ,n 2 (n 1 ,n 2 )∈(N * ) 2
	will be uniformly integrable as soon as
		lim λ→∞	lim sup n 1 ∧n 2 →∞	1 n 1 ,n 2 s 2	E S + n 1 ,n 2 -λs n 1 ,n 2	2 +	= 0	(6.17)

  Denition 6.1. We say that a random eld(M n 1 ,n 2 ) (n1,n 2 )∈N 2 is an ortho-submartingale w.r.t the commuting ltration (F n 1 ,n 2 ) (n 1 ,n 2 )∈Z 2 if M n 1 ,n 2 is integrable, F n 1 ,n 2 -measurable,and for all integers i, j, k, , we have E (M i,j |F k, ) ≥ M min(i,k),min(j, ) .

	Lemma 6.2. Let M n 1 ,n 2 ≡ n 1 i=1 and λ > 0	n 2 j=1 d i,j be an ortho-submartingale. Then for all p ∈]1, +∞[,

  6.26) holds as soon as C 3 (s n 1 ,n 2 )(c) does. Now to give sucient condition for C 2 (s n 1 ,n 2 )(b) to hold, note rst that,S k, -E (S k, |F n 1 ,n 2 ) = Hence C 2 (s n 1 ,n 2 )(b)holds as soon as, s n 1 ,n 2 ). (6.31) Clearly (6.31) holds as soon as C 3 (s n 1 ,n 2 )(a) holds. For (6.29), notice now s n 1 ,n 2 ). (6.33) Clearly (6.33) holds as soon as C 3 (s n 1 ,n 2 )(a) is satised. With similar argument as before, we infer that (6.32) holds as soon as C 3 (s n 1 ,n 2 )(b) does. The proof of (6.30) and (6.31) can be done arguing as before.

					k		+∞	+∞
				sup					P i,j (X u,v )
				1≤k≤n 1 1≤l≤n 2	u=1 v=1	i=n 1 +1	j=n 2 +1
		n 1	+∞			n 1		+∞	0	+∞
				P i,j (X u,v ) =				P i,j (X u,v ) +	P i,j (X u,v ).
	i=-∞	j=n 2 +1			i=1	j=n 2 +1	i=-∞	j=n 2 +1
	Therefore, (6.29) holds as soon as
					k		n 1		+∞
				sup 1≤k≤n 1 1≤l≤n 2	u=1 v=1	i=1	j=n 2 +1	P i,j (X u,v )	2	= o(s n 1 ,n 2 ),	(6.32)
	and				k		0		+∞
				sup					P i,j (X u,v )
				1≤k≤n 1 1≤l≤n 2	u=1 v=1	i=-∞	j=n 2 +1
	k	n 1	+∞				+∞	n 2	+∞	+∞
				P i,j (X u,v ) +		P i,j (X u,v ) +
	u=1 v=1	i=-∞	j=n 2 +1			i=n 1 +1	j=-∞	i=n 1 +1	j=n 2 +1
					k		n 1		+∞
				sup 1≤k≤n 1 1≤l≤n 2	u=1 v=1	i=-∞	j=n 2 +1	P i,j (X u,v )	2	= o(s n 1 ,n 2 ),	(6.29)
					k		+∞	n 2
				sup					(6.30)
				1≤k≤n 1 1≤l≤n 2	u=1 v=1	i=n 1 +1	j=-∞

P i,j (X u,v ) . P i,j (X u,v ) 2 = o(s n 1 ,n 2 ), 2 = o(2 = o(

.

  , C 3 (s n 1 ,n 2 )(b) is implied by C 3 (s n 1 ,n 2 )(a). Similar arguments are applied to C 3 (s n 1 ,n 2 )(c) (implied by C 3 (s n 1 ,n 2 )(b)).2.10] to the submartingale (max1≤k≤n 1 M k,n 2 ) n 2 ∈N we have |M k,l | p 1 max 1≤l≤n 2 max 1≤k≤n 1 |M k,l |>λ |M k,n 2 | p 1 max 1≤k≤n 1 |Mk,n 2 |> λ 2The desired result follows by applying [10, Corollary 2.10] again to (M k,n 2 ) k∈N .The inequality (6.21) comes from inequality (3.6) in[START_REF] Dedecker | On the functional central limit theorem for stationary processes[END_REF] withX i = n 2j=1 d i,j by taking the expectation and by noticing that (X i ) i≥1 is a sequence of martingale dierences. Next (6.22) comes from (6.21) by taking account that x 2 1 |x|>λ ≤ 4 |x| -λ 2

	E max 1≤l≤n 2	max 1≤k≤n 1
			≤ 2 p p p -1	E max 1≤k≤n 1
	7.2	Proof of Lemma 6.3
			2
			. Finally (6.23) is again an
			+
	application of [5, inequality (3.6)].

√ n 1 n 2 → i,j∈Z a i,j > 0.

Acknowledgements

The author is indebted to the two referees for carefully reading the manuscript and for helpful comments that improved the presentation of the paper. He is also grateful to his advisors, F. Merlevède and D. Volný, for their advices.

6.7

Proof of Theorem 3.5 Since C 2 (s n 1 ,n 2 ) holds, the sequence s -2 n 1 ,n 2 max 1≤k≤n 1

is uniformly integrable, and the process s -1

It remains to prove that the nite-dimensional distributions converge, that is, for any t (1)

This follows from the invariance principle for elds of stationary ortho-martingale dierences (see [2, Theorem 1]), provided that for any We shall apply Theorem 3.5, and then we only need to show that conditions C 1 (s n 1 ,n 2 )(c) and C 2 (s n 1 ,n 2 ) hold. We rst prove C 2 (s n 1 ,n 2 )(a), and C 2 (s n 1 ,n 2 )(b). Using Remark 3.3, they hold as soon as C 3 (s n 1 ,n 2 ) is satised. Note that P 0,0 (X i,j ) = a i,j ξ 0,0 . Therefore C 3 (s n 1 ,n 2 ) is implied by (4.2), (4.3), and (4.4).

We prove now that C 2 (s n 1 ,n 2 )(c) holds with u i,j = |a i,j |. SinceP 0,0 (X i,j ) = a i,j ξ 0,0 and


Mathieu Achard

Alphonse Tenaglia

Gérard Buono
email: gerard.buono@univ.u-3mrs.fr

First Cobalt(I)-catalyzed [6+2] Cycloadditions of Cycloheptatriene with Alkynes

The CoI2(dppe)/Zn/ZnI2 system effectively catalyzes the [6+2] cycloaddition of cycloheptatriene with terminal alkynes to afford 7-alkyl-bicyclo[4.2.1]nona-2,4,7-trienes in fair to excellent yields. The catalyst proved tolerant towards functional groups such as ketone, sulfone, ester, ketal, ether, alcohol, imide, and nitrile. The enantioselective cycloaddition of CHT and phenylacetylene with BINOL phosphoramidite ligand 4 afforded the cycloadduct 3a (R= Ph) in 91% yield and 74% ee.

Transition metal-mediated higher order cycloadditions are important synthetic tools for the construction of medium-sized ring compounds. 1 The photoinduced [6+2] cycloaddition of cycloheptatriene (CHT) with internal alkynes was reported first by Pettit 2 using a tricarbonyl( 4 -cycloheptatriene)iron(0) complex to afford bicyclo[4.2.1]nonatriene adducts as tricarbonyl( 4triene)iron(0) complexes. A related chromium(0)mediated

[6+2] cycloaddition with tricarbonyl( 6cycloheptatriene)chromium(0) was demonstrated later by Grevels 3 and thoroughly investigated by Sheridan. 4 In between, Turecek reported the first catalytic procedure with Ziegler catalyst (TiCl4/Et2AlCl) but the cycloaddition suffered from the competitive cyclotrimerization of the alkyne. 5 Recent efforts focused on the development of Cr(CO)nL6-n (n = 2, 3) complexes bearing labile ligands to perform catalytic processes. 6 (1) (a) Schore, N. E. Chem. Rev. 1988, 88, 1081-1119. (b) Lautens, M.; Klute, W.; Tam, W. Chem. Rev. 1996, 96, 49-92. (c) Frühauf, H. W. Chem. Rev. 1997, 97, 523-596. (d) Rigby, J. H. Tetrahedron 1999, 55, 4521-4538. With terminal alkynes the chromium(0)-promoted cycloadditions are difficult to arrest at the 1:1 alkyne:CHT cycloadduct stage; instead tetra-and pentacyclic 2:1 alkyne:CHT adducts are produced (Scheme 1). 7 These multicomponent reactions are believed to occur through a stepwise [6+2] cycloadditionhomo[6+2] cycloaddition sequence. [7][8] To date, the chromium(0)-promoted [6+2] cycloadditions with alkynes remains confined to internal alkynes.

In continuation of studies into metal promoted cycloadditions, 9 we have addressed the possibility of effecting a [6+2] cycloaddition of cycloheptatriene 1 with terminal alkynes 2 and we report in this letter the first cobalt(I)-catalyzed [6+2] cycloaddition which gives 7alkyl-bicyclo[4.2.1]nona-2,4,7-trienes 3. Scheme 1. Cycloadditions of [(η 6 -CHT)Cr(CO)3] and Alkynes

R Cr(CO) 3 R R R R R R Cr(CO) 3
From the outset, we judged that the makeup of Cobased complexes offered an attractive opportunity for the design, synthesis and development of effective cycloaddition catalysts. The following cobalt catalytic systems (A-D) combining CoI2, 1,2bis(diphenylphosphino)ethane (dppe) and tetrabutylammonium or zinc metal as reducing agent with or without ZnI2 in 1,2-dichloroethane (DCE) were selected to test the cycloaddition of 1 with phenylethyne 2a (Table 1):

Catalyst A: CoI2 / dppe / Bu4NBH4 / ZnI2 Catalyst B: CoI2 / dppe / Zn Catalyst C: CoI2 / dppe / ZnI2 Catalyst D: CoI2 / dppe / Zn / ZnI2
These catalysts revealed effective in homo-Diels Alder cycloadditions (A, B and C), [9][10][11][START_REF] Wu | A promoted the [2+2+2] cycloaddition of 1[END_REF][13] [4+2+2] cycloaddition of functionalized norbornadienes and butadiene (catalysts A-D), [START_REF] Chen | metal-catalyzed [4+2+2] cycloaddition of norbornadiene with 1[END_REF] Diels-Alder type reactions of 1,3-dienes with terminal or internal alkynes (catalysts A and D), 10,15 and 1,4-hydrovinylation reactions of 1,3-dienes with functionalized terminal alkenes (catalysts A and D). 16 When CHT and phenylethyne 2a were mixed in the presence of CoI2/dppe (5/5 mol%) or preformed CoI2(dppe) 17 in DCE at 40°C for 20 h no reaction occurred (not shown in Table 1). Catalyst B and C afforded the desired cycloadduct 3a in low yields (entry 1-2). The association of a reducing agent and a Lewis acid, (catalyst D) proved beneficial and improved the yield to 75 % (entry 6). Ligands such as PPh3, P(OPh)3 or dppp were less effective (entries 3, 4, 8). Interestingly, the monodentate phosphoramidite ligand proved satisfactory affording 3a in 61 % yield (entry 5). The cycloaddition was also observed by changing CoI2 to CoBr2 giving the cycloadduct in lower yield (entry 7). The efficiency of the cycloaddition was also demonstrated by using Bu4NBH4 as the reducing agent (catalyst A) (entry 9). Although the (3) Fischler, I.; Grevels, F. W.; Leitich, J.; Özkar, S. Chem. Ber. 1991, 124, 2857-2861 (4) (a) Chaffee, K.; Sheridan, J. B.; Aistars, A. Organometallics 1992, 11, 18-19; (b) Chaffee, K.; Huo, P.; Sheridan, J. B.; Barbieri, A.; Aistars, A.; Lanlancette, R. A.; Ostrander, R. L.; Rheingold, A. L. J. Am. Chem. Soc. 1995, 117, 1900-1907. (5) (a) Mach, K.; Antropiusová, H.; Petrusová, L.; Hanus, V.; Turecek, F. Tetrahedron, 1984, 40, 3295-3302; (b) catalysts A and D were similarly effective to promote the [6+2] cycloaddition, the catalyst D which would be expected to be more tolerant to the functional groups of the trienophile 2 was selected for further studies. The choice of solvent was critical: use of DCE (75%), trifluoroethanol (TFE) (55%) and toluene (53%) are acceptable; MeCN, THF, dioxane or DMF afforded none or trace amount of 3a.

The scope of the [6+2] cycloaddition using catalyst D was further extended to other terminal alkynes (Table 2). This catalytic system tolerates functional groups of 2 such as ketone, sulfone, ester, ketal, ether, alcohol, imide and nitrile which are of interest for further transformations. For instance, cycloadducts featuring a vinyltrimethylsilane (3c, 92%) or allyltrimethylsilane (3d, 65%) (entries 3-4) can be involved in a range of reactions of organosilicon compounds. 18 The yields observed depend on both the nature and the position of the functional group on the alkyne. Alkynes 2h, 2m, 2n and 2o with the functional group remote from the triple bond afforded the cycloadducts in excellent yields (entries 8, 14, 15, 16) compared to alkynes 2d, 2f, or 2j with the functionality at propargylic carbon atom (entries 4, 6, 10). TFE as solvent proved beneficial to promote the cycloaddition of propargyl and homopropargyl alcohols in quite good yields (entries 11-13). Electron-poor alkynes such as methyl propiolate 2i afforded cycloadduct 3i in only 21% yield (entry 9), this is likely due to a Scheme 2. Proposed Mechanism for the Cobalt Catalyzed

[6+2] Cycloaddition R [CoL 2 ] + CoI 2 L 2 + Zn + ZnI 2 I II III IV 3 1 2 Co R L L R L 2 = dppe Co L L R R Co L L R Co L L  +  +  +  +
marked preference of coordination of 2i to the cobalt species triggering the cyclotrimerization. The efficiency of the cycloaddition was improved by using slow addition of the alkyne (syringe pump, 4 h) and lower catalyst loading (2.5 mol %) (entries 1, 2, 5, 7, 9, 10).

A plausible mechanism is depicted in Scheme 2. As suggested by Snyder, ZnI2 accelerates the reduction of a CoL2I2 by zinc metal. 14a The coordination of the alkyne and cycloheptatriene to the cationic [Co(I)L2] + complex 16c is followed by an oxidative cyclometallation to give the cobaltacyclopentene I. 19 The latter undergoes a 1,5migration of the C(sp 3 )-Co bond through consecutive  -allyl complexes (-allyl complexes II and III) to form the bicyclo cobaltacycle IV. Subsequent reductive elimination of IV releases the cycloadduct 3 and regenerates the cationic [Co(I)L2] + species.

On the basis of the above results, a preliminary attempt was made to unravel the catalytic enantioselective [6+2] cycloaddition. 20 By using phosphoramidite ligand 4 21

Table 1 .

 1 Cobalt catalysts tested for the [6+2] cycloaddition of cycloheptatriene 1 with phenylethyne 2a to form 3a a

			Co catalyst
		+ Ph
	1	2a		3a Ph
	entry	catalyst (molar ratio)	yield (%) c
	1	CoI2/dppe/Zn (1/1/5)	39
	2	CoI2/dppe/ZnI2 (1/1/5)	5
	3	CoI2/PPh3/Zn/ZnI2 (1/2/3/2)	35
	4	CoI2/P(OPh)3/Zn/ZnI2 (1/2/3/2)	25
	5	CoI2/L/Zn/ZnI2 (1/2/3/2)	61
			O
		L= Me 2 N	P
			O
	6	CoI2(dppe)/Zn/ZnI2 (1/3/2)	75
	7	CoBr2(dppe)/Zn/ZnI2 (1/3/2)	40
	8	CoI2(dppp)/Zn/ZnI2 (1/3/2) b	45
	9	CoI2(dppe)/Bu4NBH4 /ZnI2 (1/1/2)	74
	a All reactions were carried out at 0.3 M concentration in DCE at 40
	°C for 20 h in a Schlenk tube under nitrogen atmosphere with CHT
	1/2a/[Co] in 1.2/1/0.05 molar ratio. b dppp is for 1,3-
	bis(diphenylphosphino)propane.

Klein, R.; Sedmera, P.; Cejka, J.; Mach, K. J.

Organomet. Chem. 1992, 436, 143-153; (c)

Kaagman, J. F.; Rep, M.; Horacek, M.; Sedmera, P.; Cejka, J.; Varga, V.; Mach, K. Collect.

Czech. Chem. Commun. 1996, 61, 1722-1728

and references cited therein. c Yields of isolated 3a.

Table 2 .

 2 Cobalt-catalyzed [6+2] Cycloaddition of Cycloheptatriene with Terminal Alkynes a Yields of isolated products by column chromatography or distillation. c Yields in parentheses are for optimized conditions (slow addition of the alkyne (4h) using 2.5 mol% of CoI2(dppe)). d TFE as solvent.

			CoI 2 (dppe)/Zn/ZnI 2
		+	R
		1	2	3	R
	Entry	alkyne	R	product	yield (%) b, c
	1	2a	Ph	3a	75 (86)
	2	2b	n-Bu	3b	74 (95)
	3	2c	SiMe3	3c	92
	4	2d	CH2SiMe3	3d	65
	5	2e	CH 2	3e	43 (68)
	6	2f	CH2SO2Ph	3f	50
	7	2g	CH2CH(CO2Me)2	3g	55 (83)
	8	2h	CH2C(Me)2CH2Ac	3h	96
	9	2i	CO2Me	3i	21 (43 d )
	10	2j	CH(OEt)2	3j	43 (59)
	11	2k	CH2OH	3k	19
	12	"	"	"	60 d
	13	2l	CH2CH2OH	3l	67 d
	14	2m	CH2CH2OAc	3m	83
			O
	15	2n	CH 2 (CH 2 ) 3 N	3n	82
			O
	16	2o	(CH2)3CN	3o	90
	a All reactions were carried out with CHT 1/Alkyne 2/
	CoI2(dppe)/Zn/ZnI2 in a 1.2/1/0.05/0.15/0.1 molar ratio in DCE at 40°C
	for 20h.

b

Acknowledgment. The authors would like to thank the CNRS for financial support. M.A. acknowledges MENRT for the award of a doctoral fellowship.

In conclusion, we have developed the first cobaltcatalyzed [6+2] cycloaddition of cycloheptatriene and functional terminal alkynes that proceeds with a noncarbonyl metal catalyst and without irradiation. The reaction proceeded under mild conditions and the catalyst tolerates a wide range of functional groups. Further optimisation of the enantioselectivity with chiral phosphoramidite ligands is underway in our laboratory.

Supporting Information Available: Experimental procedures and characterization data for all unknow compounds (PDF). This material is available free of charge via the Internet at http://pubs.acs.org (21) Phosphoramidites ligands applied in a large number of a catalytic transformations and allow the production of a true ligands library : Duursma, A.; Boiteau, J. G.; Lefort, L.; Boogers, J. A. F.; de Vries, A. H. M.; de Vries, J. G.; Minnaard, A. J.; Feringa, B. L. J. Org. Chem. 2004, 69, 8045-8052.


Esteban Jim Énez-Rodríguez

Aldo Jonathan Mu Ñoz-V Ázquez
email: aldo.munoz.vazquezz@gmail.com

Juan Diego S Ánchez-Torres

Michael Defoort
email: michael.defoort@uphf.fr

Alexander G Loukianov

A Lyapunov-like Characterization of Predefined-Time Stability

Keywords: Nonlinear control systems, Predefinedtime stability, Sliding mode control, Stability of nonlinear systems

This technical note studies Lyapunov-like conditions to ensure a class of dynamical systems to exhibit predefinedtime stability. The origin of a dynamical system is predefinedtime stable if it is fixed-time stable and an upper bound of the settling-time function can be arbitrarily chosen a priori through a suitable selection of the system parameters. We show that the studied Lyapunov-like conditions allow to demonstrate equivalence between previous Lyapunov theorems for predefinedtime stability for autonomous systems. Moreover, the obtained Lyapunov-like theorem is extended for analyzing the property of predefined-time ultimate boundedness with predefined bound, which is useful when analyzing uncertain dynamical systems. Therefore, the proposed results constitute a general framework for analyzing predefined-time stability, and they also unify a broad class of systems which present the predefined-time stability property. On the other hand, the proposed framework is used to design robust controllers for affine control systems, which induce predefined-time stability (predefined-time ultimate boundedness of the solutions) w.r.t. to some desired manifold. A simulation example is presented to show the behavior of a developed controller, especially regarding the settling time estimation.

I. INTRODUCTION

The development of control, observation, and optimization algorithms for many industrial applications requires the fulfillment of certain time-response constraints in order to comply with a certain quality or safety standards. To deal with these requirements, several developments concerning the finite-time stability concept have been carried out in [START_REF] Roxin | On finite stability in control systems[END_REF]- [START_REF] Moulay | Finite time stability conditions for non-autonomous continuous systems[END_REF]. Nevertheless, the finite settling time provided by finite-time convergent algorithms is usually an unbounded function of the system's initial conditions. This concern gives rise to a stronger form of stability called fixed-time stability, where the settling-time function is bounded. The notion of fixed-time stability has been investigated in [START_REF] Andrieu | Homogeneous approximation, recursive observer design, and output feedback[END_REF]- [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF].

Although the concept of fixed-time stability represents a significant advantage over the concept of finite-time stability because of the boundedness of the settling time, it cannot be guaranteed in general that the convergence time can be arbitrarily selected through the system tunable parameters. To overcome this mentioned drawback, it is necessary to consider another class of dynamical systems that exhibit the property of predefined-time stability, which has been studied in [START_REF] Sánchez-Torres | A class of predefined-time stable dynamical systems[END_REF], [START_REF] Jiménez-Rodríguez | On optimal predefined-time stabilization[END_REF]. For these systems, an upper bound of the settling-time function can be arbitrarily chosen through an appropriate selection of the system parameters.

On the other hand, Lyapunov methods have proved to be a handy tool for analyzing and designing nonlinear control systems [START_REF] Bacciotti | Liapunov Functions and Stability in Control Theory[END_REF], [START_REF] Khalil | Nonlinear Systems[END_REF]. In the same manner, they have been highly used for convergence rate estimation in systems exhibiting finite-and fixed-time stability properties [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF], [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF], [START_REF] Defoort | Fixedtime stabilisation and consensus of non-holonomic systems[END_REF], and in particular in systems with sliding modes [START_REF] Polyakov | Stability notions and Lyapunov functions for sliding mode control systems[END_REF]. For systems exhibiting the predefined-time stability property, similar methods have been applied. For instance, different Lyapunov-like theorems for predefined-time stability were proposed in [START_REF] Sánchez-Torres | A class of predefined-time stable dynamical systems[END_REF], [START_REF] Jiménez-Rodrıguez | A note on predefined-time stability[END_REF]- [START_REF] Aldana-López | Enhancing the settling time estimation of a class of fixed-time stable systems[END_REF] allowing the development of several control applications [START_REF] Jiménez-Rodríguez | Variable structure predefined-time stabilization of second-order systems[END_REF]- [START_REF] Sánchez-Torres | Predefined-time stabilisation of a class of nonholonomic systems[END_REF].

This paper investigates Lyapunov-like sufficient conditions for predefined-time stability of autonomous systems. The derived Lyapunov-like theorem allows to establish equivalence with several previous Lyapunov-like theorems [START_REF] Sánchez-Torres | A class of predefined-time stable dynamical systems[END_REF], [START_REF] Sánchez-Torres | A second order sliding mode controller with predefined-time convergence[END_REF], [START_REF] Aldana-López | Enhancing the settling time estimation of a class of fixed-time stable systems[END_REF], unifying all the past contributions in predefined-time stability for autonomous systems. Moreover, this framework is extended to the analysis of predefinedtime ultimate boundedness, which is specially usefull when analyzing uncertain systems. To demonstrate the applicability of the proposed framework, the developed results are used to design a family of continuous (respectively, discontinuous) controllers, which ensure predefined-time ultimate boundedness of the solutions to an arbitrarily small vicinity of a desired manifold (respectively, ensure predefinedtime stability to a desired manifold). Finally, all the mentioned properties are validated through a simulation example, in order to show the behavior of the proposed controller, especially regarding the settling time estimation.

II. PRELIMINARIES

A. Notation

Throughout the paper, the following notation is prevalent:

• R is the set of real numbers, R + = {x ∈ R : x > 0}, R ≥0 = {x ∈ R : x ≥ 0} and R+ = R + ∪ {∞}. • For x ∈ R n , x T denotes its transpose, ||x|| = √ x T x and, for r ∈ R + , Br(x) = {y ∈ R n : ||y -x|| < r}. • For any real number h, the functions ⌊•⌉ h : R → R and |⌊•⌉| h : R n → R n are defined as ⌊x⌉ h = |x| h sign(x) for any x ∈ R \ {0} and |⌊x⌉| h = x ||x|| 1-h for any x ∈ R n \ {0}, respectively. Moreover, if h > 0, ⌊0⌉ h = 0 and |⌊0⌉| h = 0. • Whereas ẋ = dx
dt denotes the first derivative of the function x : R → R n with respect to time, θ ′ (z) = dθ dz denotes the first derivative of the function θ : R → R with respect to the variable z, which may represent anything but the time variable t.

• For α, β ∈ R + , Γ(α) = ∞ 0 t α-1 e -t dt is the Gamma Function and B(α, β) = 1 0 t α-1 (1 -t) β-1 dt is the Beta Function; additionally, γ(α, r) = r 0 t α-1 e -t

B. On predefined-time stability

Consider the following autonomous system:

ẋ = f (x; ρ), x(0) = x 0 , (1)
where x : R ≥0 → R n is the system state, the vector ρ ∈ R l stands for the tunable parameters of (1). The function f : R n → R n may be discontinuous, and such that the solutions of (1) exist and are unique in the sense of Filippov (see [START_REF] Filippov | Differential equations with discontinuous righthand sides[END_REF] and [START_REF] Cortes | Discontinuous dynamical systems[END_REF]Proposition 5]). Thus, Φ(t, x 0 ) denotes the solution of (1) starting from x 0 ∈ R n at t = 0. Moreover, the origin x = 0 is the unique equilibrium point of (1). Remark 1: The parameter dependent system (1) is equivalent to the controlled system ẋ = g(x, u),

where g : R n × R m → R n , the control u ∈ R m is a feedback function of x with tunable parameters ρ, i.e., u = φ(x; ρ), with φ : R n → R m . Substituting u = φ(x; ρ) in (2) eliminates u and yields f (x; ρ) := g(x, φ(x; ρ)).

All the notions defined and treated hereafter are global, so we will omit to indicate it.

Definition 1 (Stability notions [START_REF] Lopez-Ramirez | On necessary and sufficient conditions for fixed-time stability of continuous autonomous system[END_REF]): The origin of ( 1) is said to be • Lyapunov stable if for any x 0 ∈ R n , the solution Φ(t, x 0 ) is defined for all t ≥ 0, and for any ǫ > 0, there is δ > 0 such that for any

x 0 ∈ R n , if x 0 ∈ B δ (0) then Φ(t, x 0 ) ∈ Bǫ(0)
for all t ≥ 0; • asymptotically stable if it is Lyapunov stable and Φ(t, x 0 ) → 0 as t → ∞, for any x 0 ∈ R n ; • finite-time stable if it is Lyapunov stable and for any x 0 ∈ R n there exists 0 ≤ τ < ∞ such that Φ(t, x 0 ) = 0 for all t ≥ τ . The function T (x 0 ) = inf {τ ≥ 0 : Φ(t, x 0 ) = 0, ∀t ≥ τ } is called the settling-time function of (1); • fixed-time stable if it is finite-time stable and the settling-time function of (1), T (x 0 ), is bounded on R n , i.e. there exists Tmax such that sup

x 0 ∈R n T (x 0 ) ≤ Tmax < ∞ Example 1: Consider system ẋ = - 1 ρ 1 ⌊x⌉ ρ 2 -ρ 1 ⌊x⌉ 2-ρ 2 , (3)
where x ∈ R is the system state, ρ = [ρ 1 , ρ 2 ] T ∈ R 2 is the vector of tunable parameters of (3), which comply to ρ 1 > 0 and 0 < ρ 2 < 1.

Using [8, Lemma 1], one can easily show that the origin of (3) is fixed-time stable. Moreover, from [17, Theorem 1], the settling-time function of (3) satisfies

sup x 0 ∈R T (x 0 ) = B (1/2, 1/2) 2ρ -1/2 1 ρ 1/2 1 (1 -ρ 2 ) = π 2(1 -ρ 2 ) > π 2 .
This example shows that the convergence time for system (3), whose origin is fixed-time stable, cannot be reduced arbitrarily no matter how the parameters ρ are tuned. The case when the convergence time can be arbitrarily assigned through an appropriate tuning of the system parameters ρ corresponds to the notion of predefined-time stability, which is defined as follows:

Definition 2: The origin of ( 1) is said to be predefined-time stable if it is fixed-time stable and for any Tc ∈ R + , there exists some ρ ∈ R l such that the settling-time function of (1) satisfies

sup x 0 ∈R n T (x 0 ) ≤ Tc.
Example 2: Consider system (see [START_REF] Andrieu | Homogeneous approximation, recursive observer design, and output feedback[END_REF], [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF])

ẋ = -ρ 1 ⌊x⌉ ρ 3 + ρ 2 ⌊x⌉ ρ 4 ρ 5 , (4)
where x ∈ R is the state of the system, ρ = [ρ 1 , ρ 2 , ρ 3 , ρ 4 , ρ 5 ] T ∈ R 5 is the vector of tunable parameters of (4), which comply to ρ 1 , ρ 2 , ρ 5 > 0 and 0 < ρ 5 ρ 3 < 1 < ρ 5 ρ 4 . The origin of ( 4) is fixed-time stable, by [8, Lemma 1]. Moreover, given Tc ∈ R + , there

exist ρ 1 = ρ 2 = Γ(1/4) 4 4πT 2 c
, ρ 3 = 1, ρ 4 = 3 and ρ 5 = 1 2 , such that the settling-time function of ( 4) satisfies (see [START_REF] Aldana-López | Enhancing the settling time estimation of a class of fixed-time stable systems[END_REF]Theorem 1])

sup x 0 ∈R T (x 0 ) = Γ(1/4) 2 Γ(1/4) 4 4πT 2 c 1/2 Γ(1/2)(3 -1) = Tc.
Thus, the origin of system ( 4) is, in fact, predefined-time stable.

The following proposition is an immediate consequence of Definition 2, of predefined-time stability.

Proposition 1: If a system does not have tunable parameters, then its origin is not predefined-time stable.

From Proposition 1, every system with fixed-time stable origin whose parameters are fixed numerical values (they are not tunable), cannot exhibit the predefined-time stability property.

On the other hand, when dealing with systems subject to uncertainties or external perturbations, it is difficult or even impossible to ensure exact convergence to the origin. Instead, it is common to provide convergence not to the origin but to a vicinity of it. In this sense, it would be useful to ensure that, not only the convergence time can be arbitrarily assigned, but also that the radius of the vicinity can be arbitrarily selected through an appropriate tuning of the parameters of the system. This notion is formally defined as follows:

Definition 3: A solution Φ(t, x 0 ) of ( 1) is said to be predefinedtime ultimately bounded with predefined bound if for any Tc, b ∈ R + , there exist some ρ ∈ R l such that for any

x 0 ∈ R n , ||Φ(t, x 0 )|| ≤ b for all t ≥ Tc. C. Class K 1 functions
Inspired in the class K functions in [25, Definition 1] and [12, Definition 4.2], the class K 1 functions are defined as follows:

Definition 4 (K 1 functions): A scalar continuous function κ : R ≥0 → [0, 1) is said to belong to class K 1 , denoted as κ ∈ K 1 , if it is strictly increasing, κ(0) = 0 and κ(r) → 1 as r → ∞.

If κ ∈ K 1 is also differentiable, it is said to be a differentiable class K 1 function. In such a case, there exists a continuous function Φ : R ≥0 → R + such that dκ dr = Φ(r) > 0. The above can be equivalently written in an integral form as

κ(r) = r 0 Φ(z)dz. Since κ(r) → 1 as r → ∞, the function Φ is required to satisfy ∞ 0 Φ(z)dz = 1, i.e.
, functions Φ and κ can be viewed as probability density functions and cumulative distribution functions, respectively, of positive random variables.

Proposition 2: Every class K 1 function is bijective.

Proof: Let κ ∈ K 1 .
It is injective because κ is continuous and strictly increasing. Moreover, since κ(0) = 0, κ is continuous and strictly increasing, and limr→∞ κ(r) = 1, its image is κ(R ≥0 ) = [0, 1). Thus, it is surjective. Hence, it is concluded that κ is bijective.

Since every class K 1 function is bijective, their inverse exist. Proposition 3: Let κ ∈ K 1 . Then:

(i) κ -1 (0) = 0; (ii) κ -1 is continuous (i.e.
, κ is a homeomorphism) and strictly increasing.

(iii) lim r→1 -κ -1 (r) = ∞.
Proof: These properties follow directly from Definition 4, and the fact that the inverse of a strictly increasing function is continuous and strictly increasing.

The next lemma states some useful properties of class K∞ and class K 1 functions, which will be used in the next section.

Lemma 1:

Let α ∈ K∞ (see [25, Definition 1]) and κ 1 , κ 2 ∈ K 1 . Then, κ 1 • α ∈ K 1 , and κ -1 1 • κ 2 ∈ K∞. Proof: The composition of increasing functions is increasing. Moreover, note that (κ 1 • α)(0) = κ 1 (α(0)) = κ 1 (0) = 0 and (κ -1 1 • κ 2 )(0) = κ -1 1 (κ 2 (0)) = κ -1 1 (0) = 0. Finally, since κ 1 is an homeomorphism, lim r→∞ (κ 1 • α)(r) = κ 1 lim r→∞ α(r) = 1,
and lim

r→∞ (κ -1 1 • κ 2 )(r) = κ -1 1 lim r→∞ κ 2 (r) = ∞.
Example 3: Some examples of K 1 functions are:

(i) κ(r) = 1 -exp(-r); (ii) κ(r) = 2 π arctan(r); (iii) κ(r) = r r+α , with α > 0; (iv) κ(r) = P (α, r), with α > 0 (see Subsection II-A). (v) κ(r) = I(α, β, r r+1 ), with α, β > 0 (see Subsection II-A).

III. A LYAPUNOV CHARACTERIZATION OF PREDEFINED-TIME STABILITY

This section states a Lyapunov-like theorem for predefined-time stability. The importance of this theorem is twofold. On the one hand, it establishes equivalence between previous Lyapunov theorems for predefined-time stability, constituting a unifying result. On the other hand, it allows designing predefined-time stable controllers, as shown in Section IV. Moreover, this Lyapunov theorem is extended for analyzing the property of predefined-time ultimate boundedness with predefined bound. Consequently, the results presented in this section constitute the main contribution of this note.

Theorem 1: Let κ ∈ K 1 be differentiable in R \ {0}, and V : R n → R ≥0 be a continuous, positive definite and radially unbounded function. If for any Tc ∈ R + , there exists some ρ ∈ R l , such that the time-derivative of V along the trajectories of (1) satisfies

V (x) ≤ - 1 (1 -p)Tc κ(V (x)) p κ ′ (V (x)) , for x ∈ R n \ {0} , (5)
for some 0 ≤ p < 1, then the origin of ( 1) is predefined-time stable. Moreover, if ( 5) is an equality, then sup x 0 ∈R n T (x 0 ) = Tc. Proof: Let Tc ∈ R + . Then, there exists some ρ ∈ R l such that (5) holds. Moreover, since V : R n → R ≥0 is a continuous, positive definite and radially unbounded function, and its timederivative (5) is negative for x ∈ R n \ {0}, the origin of system (1) is asymptotically stable [START_REF] Khalil | Nonlinear Systems[END_REF]. Now, let Φ(t, x 0 ) be a solution of (1) and let y : R ≥0 → R ≥0 be a function that satisfies

ẏ = - 1 (1 -p)Tc κ(y) p κ ′ (y) ,
and V (x 0 ) ≤ y(0). Hence,

κ(y(t)) =    κ(y(0)) 1-p -t Tc 1 1-p if 0 ≤ t ≤ Tcκ(y(0)) 1-p 0 if t > Tcκ(y(0)) 1-p ,
and V (Φ(t, x 0 )) ≤ y(t) (it is an equality only if [START_REF] Moulay | Finite time stability conditions for non-autonomous continuous systems[END_REF], is an equality) by the comparison lemma [START_REF] Khalil | Nonlinear Systems[END_REF]. Thus, V (Φ(t, x 0 )) = 0 for t ≥ Tcκ(V (x 0 )) 1-p , implying that the trajectories of (1) reach the origin in finite time, and the settling-time function satisfies

sup x 0 ∈R n T (x 0 ) ≤ sup x 0 ∈R n Tcκ(V (x 0 )) 1-p = Tc.
Hence, by Definition 2, the origin of system ( 1) is in fact predefinedtime stable. Moreover, if ( 5) is an equality, then sup x 0 ∈R n T (x 0 ) = sup x 0 ∈R n Tcκ(V (x 0 )) 1-p = Tc. Remark 2: Theorem 1 can be equivalently formulated in terms of a function W : R n → R ≥0 , which satisfies W (x) = 0 if and only if x = 0, 0 ≤ W (x) < 1, and lim ||x||→∞ W (x) = 1. This equivalence is given by the transformation W (x) = κ(V (x)). Moreover, in this equivalent reformulation, inequality ( 5) is replaced by inequality

Ẇ (x) ≤ - 1 (1 -p)Tc W (x) p , for x ∈ R n \ {0} .
In this sense, [15, Theorem 10] is a corollary of Theorem 1, which is obtained fixing p = 0. Remark 3: Although the result in Theorem 1 is independent of the choice of κ ∈ K 1 , the form of the differential inequality (5) strongly depends on the particular selection of this function. Indeed, previous Lyapunov-like theorems for predefined-time stability reported in the literature are, in fact, particular forms of Theorem 1. For instance:

(i) [16, Theorem III.1] is obtained from Theorem 1 with the particular selections of κ(r) = P 1-βq s , αr s = γ 1-βq s ,αr s Γ 1-βq s , with α, β, s, q > 0 and βq < 1, and p = 0.

Thus, inequality (5) then becomes

V (x) ≤ - α βq-1 s Γ 1-βq s sTc exp αV (x) s V (x) βq (6)
for x ∈ R n \ {0}. (ii) At the same time, with α = β = 1, q = 1 -s and 0 < s ≤ 1, inequality (6) reduces to

V (x) ≤ - 1 sTc exp V (x) s V (x) 1-s , for x ∈ R n \ {0} ,
which is equivalent to Theorem 1 with the particular selection of κ(r) = 1 -exp(-r s ). In this form, the result presented in [9, Theorem 2.1] is recovered. (iii) [START_REF] Aldana-López | Enhancing the settling time estimation of a class of fixed-time stable systems[END_REF]Theorem 3] is retrieved from Theorem 1 with the particular selections of κ(r) = I ms, mq, βr q-s βr q-s +α , with ms = 1-ks q-s > 0, mq = kq-1 q-s > 0, α, β, k > 0 and 0 < ks < 1 < kq, and p = 0. Replacing the above picks into inequality [START_REF] Moulay | Finite time stability conditions for non-autonomous continuous systems[END_REF], it yields

V (x) ≤ - ζ Tc αV (x) s + βV (x) q k , for x ∈ R n \ {0} , where ζ = Γ(ms)Γ(mq ) α k Γ(k)(q-s) α β ms .
Thus, in this work, it is shown for the first time that all previous Lyapunov-like theorems for predefined-time stability of autonomous systems are equivalent. Lyapunov analysis can also be extended to show predefined-time ultimate boundedness with predefined bound of the solutions of (1) (see Definition 3), even if there is no equilibrium point at the origin. Sufficient conditions are stated in the following theorem:

Theorem 2: Let κ ∈ K 1 be differentiable in R \ {0}, and V : R n → R ≥0 be a continuous, positive definite and radially unbounded function. If for any Tc, µ ∈ R + , there exists some ρ ∈ R l , such that the time-derivative of V along the trajectories of (1) satisfies

V (x) ≤ - 1 (1 -p)Tc κ(V (x)) p κ ′ (V (x))
, for ||x|| ≥ µ,

then, for any x 0 ∈ R n the solution Φ(t, x 0 ) of (1) satisfies

||Φ(t, x 0 )|| ≤ b = α -1 1 (α 2 (µ)), for all t ≥ Tc,
where α 1 , α 2 ∈ K∞. Moreover, if V (x) = α(||x||), with α ∈ K∞, then b = µ in the above inequality. This is, the solutions of (1) are predefined-time ultimately bounded with predefined bound.

Proof: Let Tc, µ ∈ R + . Then, there exists ρ ∈ R l such that (7) holds. Since V is a continuous, positive definite and radially unbounded function, there exist

α 1 , α 2 ∈ K∞ such that α 1 (||x||) ≤ V (x) ≤ α 2 (||x||) [12, Lemma 4.3]. Note that ||x|| < µ ⇐⇒ α 2 (||x||) < α 2 (µ) ⇒ V (x) < α 2 (µ), i.e. the set {x ∈ R n : ||x|| < µ} ⊆ {x ∈ R n : V (x) < α 2 (µ)}, or equivalently {x ∈ R n : V (x) ≥ α 2 (µ)} ⊆ {x ∈ R n : ||x|| ≥ µ}.
Hence, inequality [START_REF] Cruz-Zavala | Uniform second-order sliding mode observer for mechanical systems[END_REF] holds for V (x) ≥ α 2 (µ).

The above implies that the set {x ∈ R n :

V (x) ≤ α 2 (µ)} is positively invariant, since the derivative V (x) is negative in its boundary {x ∈ R n : V (x) = α 2 (µ)}.
Now, we show that all trajectories starting in the 7) and following similar arguments as

set {x ∈ R n : V (x) ≥ α 2 (µ)}, must enter the set {x ∈ R n : V (x) ≤ α 2 (µ)} within at most Tc time units. Let Φ(t, x 0 ) be a solution of (1), with x 0 ∈ {x ∈ R n : V (x) ≥ α 2 (µ)}, i.e. V (x 0 ) ≥ α 2 (µ). From (
in Theorem 1, κ(V (Φ(t, x 0 ))) ≤ κ(V (x 0 )) 1-p -t Tc 1 1-p , for t ∈ 0, Tc(κ(V (x 0 )) 1-p -κ(α 2 (µ)) 1-p ) . Hence, κ(V (Φ(t, x 0 ))) ≤ κ(α 2 (µ)) ⇐⇒ V (Φ(t, x 0 )) ≤ α 2 (µ) for all t ≥ Tc(κ(V (x 0 )) 1-p -κ 2 (µ) 1-p ),
and consequently for all t ≥ Tc.

Furthermore, note that

V (x) ≤ α 2 (µ) ⇒ α 1 (||x||) ≤ α 2 (µ) ⇐⇒ ||x|| < α -1 1 (α 2 (µ)). Hence, ||Φ(t, x 0 )|| ≤ α -1 1 (α 2 (µ)) for all t ≥ Tc. Moreover, if V (x) = α(||x||), one can select α 1 = α 2 = α
, and the result is obtained.

IV. APPLICATION: LYAPUNOV-BASED PREDEFINED-TIME CONTROLLER DESIGN

A. Problem statement

Consider the following affine control system:

ẋ = f (x) + B(x)v + δ(x, t) (8)
where x : R ≥0 → R n is the system state, v ∈ R m is the control input, δ : R n × R ≥0 → R n is a disturbance vector that includes plant parameter variations and external unknown perturbations, and B : R n → R n×n is continuous and such that rank

B(x) = m for all x ∈ R n .
The objective is to design a feedback control input v such that the trajectories of ( 8) reach (a vicinity of) the manifold

s(x, t) = 0, (9)
where s : R n × R ≥0 → R m is a smooth mapping, in an arbitrarily selected time Tc ∈ R + and remain there for all t ≥ Tc. Remark 4: There are several important control problems which take the form of system (8) subject to [START_REF] Sánchez-Torres | A class of predefined-time stable dynamical systems[END_REF]. For instance, (i) An output tracking problem, where s(x, t) represents the output tracking error vector; the relative degree of each output component with respect to the control input is equal to one, and the system (8) with respect to ( 9) is minimum phase [START_REF] Ley-Rosas | Robust observer-based sliding mode controller for vehicles with roll dynamics[END_REF]. The control objective is to ensure the tracking error be predefined time stable. (ii) A sliding mode (SM) control design problem, where s(x, t) = 0 (9) represents a sliding manofold with a desired SM motion [START_REF] Utkin | Sliding Modes in Control and Optimization[END_REF].

In this case, the objective is to induce the SM on the designed manifold in predefined-time (predefined-time reaching phase).

(iii) An optimization problem solved by dynamic networks, where s(x, t) is a variable which expresses the error in the satisfaction of equality constraints [START_REF] Utkin | Sliding Modes in Control and Optimization[END_REF]. The time derivative of s(x, t) is

ṡ = G(x, t)f (x) + G(x, t)B(x)v + G(x, t)δ(x, t) + ∂s(x, t) ∂t . (10)
Then, assuming that rank[G(x, t)B(x)] = m, for all x ∈ R n and t ≥ 0, the control v is chosen as

v = -[G(x, t)B(x)] -1 G(x, t)f (x) + ∂s(x, t) ∂t + u , (11)
where u ∈ R m is a virtual control input and G(x, t) = ∂s(x,t) ∂x . Substituting [START_REF] Bacciotti | Liapunov Functions and Stability in Control Theory[END_REF] in [START_REF] Jiménez-Rodríguez | On optimal predefined-time stabilization[END_REF] results in

ṡ = u + ∆(x, t), s(x 0 , 0) = s 0 , (12)
where ∆(x, t) = G(x, t)δ(x, t), which is assumed to be globally bounded by sup

(x,t)R n ∈×R ≥0 ||∆(x, t)|| ≤ δ with 0 ≤ δ < ∞ a known constant.
From a control design point of view, the perturbation term ∆(x, t) can only be completely rejected by a discontinuous control term (like the unit-vector controller s ||s|| ), given that it is only restricted to be bounded (no conditions of smoothness, Lipschitz continuity neither continuity are assumed). However, such a discontinuous control term might deteriorate the components of a real physical system due to high-frequency oscillations, or might even be impossible to implement due to limited actuator response.

A solution would be to sacrifice the exact convergence to the manifold s(x, t) = 0 (9) in order to obtain a continuous controller (like the continuous approximation of the unit-vector controller s ||s||+ǫ , with ǫ > 0). In this case, as pointed out in Section II, it can be ensured that the trajectories converge to a vicinity of s(x, t) = 0 (see Eq. ( 9)).

Based on the above, the objective is to design the virtual control input u as a feedback control law u = u(s) that:

• enforces predefined-time stability to the origin of [START_REF] Khalil | Nonlinear Systems[END_REF], obtaining a discontinuous controller; or • enforces the solutions of (12) to be predefined-time ultimately bounded with predefined bound, obtaining a continuous controller.

B. Proposed solution

The proposed solution is a corollary of Theorems 1 and 2.

Corollary 1: Consider system [START_REF] Khalil | Nonlinear Systems[END_REF]. Selecting u as

u = - 1 (1 -ρ 2 )ρ 1 κ(||s||) ρ 2 κ ′ (||s||) |⌊s⌉| 0 -ρ 3 s ||s|| + ρ 4 , (13)
where κ ∈ K 1 is such that κ ′ : R ≥0 → R+ , ρ 1 > 0, 0 ≤ ρ 2 < 1, ρ 3 > δ and ρ 4 ≥ 0, the trajectories of the closed-loop system (12)-( 13) are predefined-time ultimately bounded with predefined bound. In fact, for any Tc, b ∈ R + and s 0 ∈ R m , the solution Φ(t, s 0 ) of ( 12)-( 13) satisfies ||Φ(t, s 0 )|| ≤ b = δρ 4 ρ 3 -δ for all t ≥ Tc = ρ 1 . Moreover, if ρ 4 = 0, then the origin s = 0 of the closed-loop system ( 12)-( 13) is predefined-time stable.

Proof: Consider the the continuous, positive definite and radially unbounded function V (s) = ||s||, and let Tc, b ∈ R + . The time-derivative of V along the trajectories of the closed-loop system (12)-( 13) is V (s) =

s T 0 - 1 (1-ρ 2 )ρ 1 κ(||s||) ρ 2 κ ′ (||s||) |⌊s⌉| 0 -ρ 3 s ||s||+ρ 4 + ∆(x, t) .
Note that: (c) by the Cauchy-Schwarz inequality s T 0 ∆ ≤ δ.

Therefore, considering (a), (b) and (c), the time derivative V (s) results in

V (s) ≤ - 1 (1 -ρ 2 )ρ 1 κ(V (s)) ρ 2 κ ′ (V (s)) -ρ 3 -δ - ρ 3 ρ 4 ||s|| + ρ 4 . Since ρ 3 -δ -ρ 3 ρ 4 ||s||+ρ 4 ≥ 0 ⇐⇒ ||s|| ≥ δρ 4 ρ 3 -δ , then V (s) ≤ - 1 (1 -p)Tc κ(V (s)) p κ ′ (V (s)) , for ||s|| ≥ µ, (14)
with Tc = ρ 1 , µ = δρ 4 ρ 3 -δ and p = ρ 2 . From the above and using Theorem 2, the solutions of the closedloop system ( 12)-( 13) satisfy

||Φ(t, s 0 )|| ≤ b = µ = δρ 4 ρ 3 -δ for all t ≥ Tc = ρ 1 .
Moreover, if ρ 4 = 0 ⇐⇒ µ = 0, inequality ( 14) holds for all s ∈ R m . Hence, by Theorem 1, the origin s = 0 of the closed-loop system ( 12)-( 13) is predefined-time stable.

It is worth to notice that the predefined time Tc = ρ 1 and the predefined bound b = δρ 4 ρ 3 -δ can be selected independently since they depend on different parameters.

Example 4: Let x = [x 1 x 2 ] T ∈ R 2 be a point in the plane whose dynamics is given by ẋ = u, where u = [u 1 u 2 ] T ∈ R 2 is the feedback control signal to be designed so that the point x tracks a desired reference trajectory r(t) = [r 1 (t) r 2 (t)] T : R ≥0 → R 2 . Whereas the reference trajectory signal is assumed to be known, its derivative ṙ(t) is assumed to be unknown but bounded of the form sup t∈R ≥0 || ṙ(t)|| ≤ δ. This model is a classic example in teleoperation tasks, such as haptic interfaces, remote command of manipulators, land, aerial and underwater robots, to name a few, where the position reference is given in real-time by the user, but the desired velocity reference is unknown. The dynamics of the error variable, s(x, t) = xr(t), is then ṡ = u -ṙ(t). Under all the above assumptions, the feedback control signal u can be designed as [START_REF] Defoort | Fixedtime stabilisation and consensus of non-holonomic systems[END_REF] in Corollary 1.

For simulation purposes, the function κ(r) in ( 13) is selected as κ(r) = I mρ 7 , mρ 8 , ρ 6 r ρ 8 -ρ 7 ρ 6 r ρ 8 -ρ 7 +ρ 5 , with mρ 7 = 1-ρ 9 ρ 7 ρ 8 -ρ 7 > 0, mρ 8 = ρ 9 ρ 8 -1 ρ 8 -ρ 7 > 0, ρ 5 , ρ 6 , ρ 9 > 0 and 0 < ρ 9 ρ 7 < 1 < ρ 9 ρ 8 . Furthermore, the reference signal is selected as r(t) = [cos(2πt) sin(2πt)] T (i.e., the point x is required to follow a circumference of radius 1), whose derivative norm is || ṙ(t)|| = 2π = δ. Moreover, the following simulations are conducted using the Euler integration method, with a fundamental step size of 1 × 10 -5 s. Finally, the initial conditions of the system are set as x 0 = [x 10 x 20 ] = x 0 [1 1], with x 0 taking the values of 10 1 , 10 3 and 10 21 .

Part I: assume that the tracking error s is required to reach a vicinity of the origin with a radius of measure b = 0.01 in at most Tc = 1 time units. To this end, the parameters of controller (13) are selected as ρ 1 = 1, ρ 2 = 0, ρ 3 = 4π, ρ 4 = 0.01, ρ 5 = 1, ρ 6 = 1, ρ 7 = 0.9, ρ 8 = 1.1, and ρ 9 = 1. Note that, with this parameter selection, controller (13) is continuous.

Part II: assume that the tracking error s is required to reach the origin in at most Tc = 1 time units. To this end, the parameters of controller (13) are selected as ρ 1 = 1, ρ 2 = 0, ρ 3 = 4π, ρ 4 = 0, ρ 5 = 1, ρ 6 = 1, ρ 7 = 0.9, ρ 8 = 1.1, and ρ 9 = 1. Note that, with this parameter selection, controller (13) is discontinuous. Figs. 1 and3 show the state variables and the reference signal over time, respectively, for the continuous (ρ 4 = 0.01) and the discontinuous (ρ 4 = 0) cases. They also show the tracking error norm ||s|| for a better appreciation of the convergence to the 0.01-vicinity of the origin, in the continuous case, or the convergence to the origin, in the discontinuous case. Figs. 2 and4 show the trajectories of the point (x 1 , x 2 ) in the plane, respectively, for the continuous and the discontinuous cases.

V. CONCLUSION

This technical note was devoted to the study of sufficient Lyapunovlike conditions to ensure a class of dynamic systems to exhibit a predefined time stability property. The introduction of class K 1 allowed to establish equivalence with previous Lyapunov-like theorems for predefined-time stability for autonomous systems [START_REF] Sánchez-Torres | A class of predefined-time stable dynamical systems[END_REF], [START_REF] Jiménez-Rodrıguez | A note on predefined-time stability[END_REF]- [START_REF] Aldana-López | Enhancing the settling time estimation of a class of fixed-time stable systems[END_REF]. Moreover, the derived Lyapunov theorem was extended for analyzing predefined-time ultimate boundedness with predefined bound.

On the other hand, the developed framework was used to design a class of robust controllers for uncertain affine control systems. This class of controllers can be continuous, providing predefined-time ultimate boundedness of the solutions, or discontinuous, providing predefined-time stability to some desired manifold. Finally, the theoretical findings were validated through a numerical simulation, which reveals the effectiveness of the proposed control scheme.

As future work, the Lyapunov-like conditions introduced in Theorems 1-2, require further research to exploit them for the controller design of particular classes of nonlinear systems.

  dt and P (α, r) = γ(α,r) Γ(α) are the Incomplete Gamma Function and the regularized Incomplete Gamma Function, respectively, which are defined for all r ∈ R ≥0 ; finally b(α, β, r) = r 0 t α-1 (1 -t) β-1 dt and I(α, β, r) = b(α,β,r) B(α,β) are the Incomplete Beta Function and the regularized Incomplete Beta Function, respectively, which are defined for all r ∈ [0, 1] [21].

4 = |⌊s⌉| 0 - ρ 4 |⌊s⌉| 0 ||s||+ρ 4 ;

 44 (a) the continuous approximation of the unit-control term satisfies s ||s||+ρ (b) the product s T 0 |⌊s⌉| 0 = 1;

Fig. 1 .Fig. 2 .

 12 Fig. 1. Several trajectories of the state variables x1, x2 vs. t, and ||e|| vs. t. Continuous controller.

1 Fig. 3 .

 13 Fig. 3. Several trajectories of the state variables x1, x2 vs. t, and ||e|| vs. t. Discontinuous controller.

Fig. 4 .

 4 Fig. 4. Several trajectories of the point (x1, x2) in the plane. Discontinuous controller.

ACKNOWLEDGMENT

Esteban Jiménez acknowledges to CONACYT-México for the D.Sc. scholarship number 481467 and the project 252405. Aldo Jonathan acknowledges to CONACYT-México for the Project Cátedras 1086 "Ambientes Inteligentes".