Christophe Cuny

J Dedecker

F Merlevède

Strong approximations for a class of dependent random variables with semi exponential tails

Keywords: AMS 2020 subject classifications: 60F17, 60G10, 60J05, 60B15 Almost sure invariance principle, Coupling, Markov chains, products of random matrices

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Introduction

Let (Ω, A, P) be a probability space, and let (ε i ) i≥1 be independent and identically distributed (iid) random variables defined on Ω, with values in a measurable space G and with common distribution µ. Let W 0 be a random variable defined on Ω with values in a measurable space X, independent of (ε i ) i≥1 , and let F be a measurable function from G × X to X. For any n ≥ 1, define W n = F (ε n , W n-1 ) , and assume that (W n , n ≥ 1) has a stationary distribution ν. Let now h be a measurable function from G × X to R and define, for any n ≥ 1,

X n = h(ε n , W n-1 ) . (1)
Then (X n ) n≥1 forms a stationary sequence with stationary distribution, say π. Let (G i ) i∈Z be the non-decreasing filtration defined as follows: for any i < 0, G i = {∅, Ω}, G 0 = σ(W 0 ) and for any i ≥ 1, G i = σ(ε i , . . . , ε 1 , W 0 ). It follows that for any n ≥ 1, X n is G n -measurable. We can also consider the following model

X n = h(ε n , ε n-1 , . . .) , (2)
which is in fact included in the preceding situation, by taking W 0 = (ε 0 , ε -1 , . .

.).

In what follows we assume that h is such that E(X n ) = 0. For any n ≥ 0, let us define the sequence (δ(n)) n≥0 by

δ(0) = E(|X 1 |) and δ(n) = X n -X * n 1 , n ≥ 1 ,
where X * n is defined as follows : let W 0 and W * 0 be random variables with law ν, and such that W * 0 is independent of (W 0 , (ε i ) i≥1 ). For any n ≥ 1, let

X * n = h(ε n , W * n-1 ) with W * n = F (ε n , W * n-1 ) . (3)
Note that the coefficients δ(n) are well defined if π has a moment of order 1.

In the paper [START_REF] Cuny | On the Komlós, Major and Tusnády strong approximation for some classes of random iterates[END_REF], we assumed that the coupling coefficients δ n decrease at a polynomial rate, and we used a variant of the Berkes-Liu-Wu method (see [START_REF] Berkes | Komlós-Major-Tusnády approximation under dependence[END_REF]) to establish strong approximation results for partial sums. Another more restrictive coefficient is also considered in [START_REF] Cuny | On the Komlós, Major and Tusnády strong approximation for some classes of random iterates[END_REF] (see also [START_REF] Cuny | An alternative to the coupling of Berkes-Liu-Wu for strong approximations[END_REF] for variants of these coefficients and the resulting conditions). In [START_REF] Cuny | On the Komlós, Major and Tusnády strong approximation for some classes of random iterates[END_REF] we applied our results to various classes of (possibly non irreducible) Markov chains; in particular, we obtained optimal rates in the almost sure invariance principle (ASIP) for some functions of the left random walk on GL d (R) under polynomial moment conditions. In [START_REF] Cuny | Rates in almost sure invariance principle for slowly mixing dynamical systems[END_REF] and [START_REF] Cuny | Rates in almost sure invariance principle for quickly mixing dynamical systems[END_REF] we adapted the proofs of these results to Birkhoff sums of Hölder observables of non-uniformly expanding dynamical systems, using a representation as functions of the trajectory of a particular Markov Chain due to Korepanov [START_REF] Korepanov | Rates in almost sure invariance principle for dynamical systems with some hyperbolicity[END_REF]. In the paper [START_REF] Cuny | Rates in almost sure invariance principle for quickly mixing dynamical systems[END_REF], we consider dynamical systems with exponential or sub-exponential decay of correlations, and we show that the rates in the ASIP are in powers of log n.

In the present paper, we consider the context of [START_REF] Cuny | On the Komlós, Major and Tusnády strong approximation for some classes of random iterates[END_REF], assuming that the coefficients δ n decrease at an exponential, sub-exponential or super-exponential rate (described by an index γ 1 , see ( 4)), and that the variables have an exponential, sub-exponential or super-exponential moments (described by an index γ 2 , see [START_REF] Cuny | Limit theorems for the left random walk on GL d (R)[END_REF]). Using arguments from [START_REF] Cuny | Rates in almost sure invariance principle for quickly mixing dynamical systems[END_REF], we show that the rates in the ASIP are in powers of log n (the exponent of the logarithm depending on γ 1 and γ 2 ). We apply our results to the left random walk on GL d (R), to iid products of positive matrices, and to a class of non-uniformly contracting auto-regressive processes.

Main results

We assume that there exist two positive constants γ 1 and c such that δ(n) ≤ exp(-cn γ 1 ) for any positive integer n ,

and there are constant b ∈]0, ∞[ and γ 2 ∈]0, ∞] such that

P(|X 1 | > t) ≤ exp(1 -(t/b) γ 2 ) for any positive t , (5)
Note that when γ 2 = ∞ (5) means that X n ∞ ≤ b a.s. for any integer n. When X 1 satisfies (5) we say either that it has a semi exponential tail of order γ 2 or that it admits a sub-exponential or super-exponential moment of order γ 2 .

Theorem 1 Let (X n ) n≥1 be defined by (1) or (2). Let S n = n k=1 X k . Assume that (4) and (5) are satisfied. Then n -1 E S 2 n → σ 2 as n → ∞ and one can redefine (X n ) n≥1 without changing its distribution on a (richer) probability space on which there exist iid random variables (N i ) i≥1 with common distribution N (0, σ 2 ), such that, as n → ∞,

S n - n i=1 N i = O((log n) 1+λ ) P-a.s. where λ = 1 γ 1 + 1 γ 2 . ( 6
)
Proof of Theorem 1. Let α = 1 + λ. The proof uses similar arguments as those developed in the proof of Theorem 1.6 in [START_REF] Cuny | Rates in almost sure invariance principle for quickly mixing dynamical systems[END_REF] with the main difference that the random variables X k are no longer bounded. we then have to truncate them. With this aim, let us first define for any positive integer k,

M k = c 1 k 1/γ 2 where c 1 = b(2 log 3) 1/γ 2 , (7)
(recall that b is the constant appearing in condition ( 5)) and

ϕ k (x) = x ∧ M k ∨ -M k .
Then we set

X k,j = ϕ k (X j ) -Eϕ k (X j ) and W k, = +3 k-1 i=1+3 k-1 X k,i .
For n ≥ 2, let h n = (log n)/(log 3) (so that h n is the unique integer such that 3 hn-1 < n ≤ 3 hn ). Note that

S n = hn-1 k=1 3 k i=1+3 k-1 X i + n i=1+3 hn-1 X i ,
and set

S † n = hn-1 k=1 W k,3 k -3 k-1 + n i=1+3 hn-1
X hn,i .

Let also S † 1 = 0. We first prove that max

1≤i≤n S i -S † i = O((log n) α ) P-a.s. (8)
With this aim, it suffices to prove via the Kronecker lemma and stationarity that k≥1

k -α 3 k E(|X 1 | -M k ) + < ∞ . (9)
Obviously this holds when γ 2 = ∞. Assume now that γ 2 < ∞ and note that since

3 k = exp 1 2 M k b γ 2 , 3 k E(|X 1 |-M k ) + = 3 k ∞ M k P(|X 1 | > t)dt ≤ ∞ M k exp(1-2 -1 (t/b) γ 2 )dt ≤ b2 1/γ 2 γ 2 Γ((1-γ 2 )/γ 2 ) < ∞ .
Since α > 1, ( 9) is proved and so (8) also. Now, for any k ≥ 1, let

m k = [c 2 k 1/γ 1 ] + 1 where c 2 = 2c -1 (log 3) 1/γ 1 , (10)
(recall that c is the constant appearing in condition (4)). We then define

Xk,j = E ϕ k (X j )|ε j , ε j-1 , . . . , ε j-m k -E ϕ k (X j ) for any j ≥ m k and W k, = +3 k-1 i=1+3 k-1
Xk,i .

Finally, set S1 = 0 and for n > 1,

Sn = hn-1 k=1 W k,3 k -3 k-1 + W hn,n-3 hn-1 .
Note that by Lemma 24 in [START_REF] Cuny | On the Komlós, Major and Tusnády strong approximation for some classes of random iterates[END_REF], for any j ≥ m k ,

X k,j -Xk,j 1 ≤ E(|X m k +1,x -X m k +1,y |)ν(dx)ν(dy) ≤ δ(m k ). (11)
Hence, for k ≥ k 0 where k 0 is such that for any ≥ k 0 , m ≤ 3 -1 , the upper bound [START_REF] Cuny | Limit theorems for iid products of positive matrices[END_REF] implies that max

1≤ ≤3 k -3 k-1 W k, -W k, 1 ≤ 3 k i=1+3 k-1 X k,i -Xk,i 1 ≤ 3 k δ(m k ) 3 k exp(-c × c γ 1 2 k) .
Hence, since c × c γ 1 2 ≥ log 3,

k≥k 0 k -α max 1≤ ≤3 k -3 k-1 W k, -W k, 1 < ∞ ,
which entails by the Kronecker lemma that max 1≤i≤n

S † i -Si = o((log n) α ) P-a.s. (12)
From ( 8) and [START_REF] Cuny | Berry-Esseen type bounds for the matrix coefficients and the spectral radius of the left random walk on GL d (R)[END_REF], the theorem will follow if one can prove that there exists a standard Brownian motion B such that such that, as n → ∞,

max 1≤i≤n Si -B(iσ 2 ) = O((log n) α ) P-a.s. (13)
To prove [START_REF] Dedecker | A new covariance inequality and applications[END_REF], we do exactly as in Steps 3 and 4 of the proof of [START_REF] Cuny | Rates in almost sure invariance principle for quickly mixing dynamical systems[END_REF]Theorem 1.6] by replacing the upper bound ψ ∞ by 2M k (which a bound of the sup norm of the truncated random variables X k,j ), and by noticing that M k m k k 1/γ . This leads to (3.20) in [START_REF] Cuny | Rates in almost sure invariance principle for quickly mixing dynamical systems[END_REF] which combined with (3.9) in [START_REF] Cuny | Rates in almost sure invariance principle for quickly mixing dynamical systems[END_REF] gives the following strong approximation: there exists a standard Brownian motion B such that such that, as n → ∞,

max 1≤i≤n Si -B(σ 2 i ) = O((log n) α ) P-a.s. ( 14
)
where σ 2 i is defined by (3.21) in [START_REF] Cuny | Rates in almost sure invariance principle for quickly mixing dynamical systems[END_REF]. It remains to identify the variance of the Brownian motion and to show that max

1≤i≤n B(σ 2 i ) -B(iσ 2 ) = O((log n) α ) P-a.s. (15)
With this aim, we shall proceed as in step 3.4 in [START_REF] Berkes | Komlós-Major-Tusnády approximation under dependence[END_REF] (see also Step 5 in [7]). We set

ν k = σ 2 + 2 m k i=-m k (γ k,i -γ i ) -2 i≥m k +1 γ i , (16)
where γk,i = cov( Xk,m k , Xk,i+m k ) , γk,i = cov(X k,0 , X k,i ) and γ i = cov(X 0 , X i ) .

To prove [START_REF] Fan | Deviation inequalities for martingales with applications[END_REF], we need to prove that

(log n) max k≤hn (m k ν k ) 1/2 = O((log n) α ) , (17)
and

3 k (ν 1/2 k -σ) 2 = O(k 2α (log k) -1 ) . ( 18
)
By using the upper bound [START_REF] Cuny | Limit theorems for iid products of positive matrices[END_REF], we have, for any i ≥ 0,

γk,i -γk,i = cov( Xk,m k -X k,m k , Xk,i+m k ) + cov(X k,m k , Xk,i+m k -X k,i+m k ) ≤ 2M k Xk,m k -X k,m k 1 + 2M k Xk,i+m k -X k,i+m k 1 k 1/γ 2 exp(-c × c γ 1 2 k) . (19)
Next, by using inequality (1.11a) in [START_REF] Rio | Théorie asymptotique des processus aléatoires faiblement dépendants[END_REF], note that

γk,i -γ i = cov(X k,0 -X 0 , X k,i ) + cov(X 0 , X k,i -X i ) ≤ 2 1/2 0 Q |ϕ k (X)| (u)Q |g k (X)| (u)du+2 1/2 0 Q |X 1 | (u)Q |g k (X)| (u)du ≤ 4 1 0 Q(u)(Q(u)-M k ) + du , where g k (x) = (|x|-M k ) + = x-ϕ k (x). But, for any positive t, P(|X 1 | > t) ≤ exp(1-(t/b) γ 2 ) := G(t). Therefore, for any u ∈]0, 1[, Q(u) ≤ G -1 (u) := b 1 -log u 1/γ 2 . But G -1 (u) > M k ⇐⇒ u -1 > e 2k log 3-1 . So, overall, γk,i -γ i ≤ 4b 2 1 0 log(e/u) 2/γ 2 1 {u -1 >e 2k log 3-1 } du = 4eb 2 ∞ e 2k log 3 log x 2/γ 2 x -2 dx = 4eb 2 ∞ 2k log 3 x 2/γ 2 e -x dx ≤ κ γ 2 b 2 1 + k 2/γ 2 e -2k log 3 ,
where κ γ 2 is a positive constant depending only on γ 2 . Next, by using Proposition 1 in [START_REF] Dedecker | A new covariance inequality and applications[END_REF], we derive

i≥m k |cov(X 0 , X i )| ≤ 2 i≥m k δ(i) 0 Q • H -1 (u)du ≤ 2 i≥m k H(δ(i)) 0 Q 2 (u)du ,
where H is the function defined on [0, 1] by

H(v) := v 0 Q(u)du and H -1 is its inverse. Since Q(u) ≤ b 1 -log u
1/γ 2 , we infer that there exists a positive constant κ depending on c, b, γ 1 and γ 2 , such that, for any i ≥ 1,

H(δ(i)) 0 Q 2 (u)du ≤ κi 2γ 1 /γ 2 exp(-ci γ 1 ) . Therefore, since m k = [c 2 k 1/γ 1 ] + 1, i≥m k |γ i | ≤ κk 2 γ 2 + 1 γ 1 -1 exp(-c × c γ 1 2 k) ,
where κ is a positive constant depending only on c, b, γ 1 , γ 2 and c 2 . So, overall, for any positive integer k,

ν k -σ 2 ≤ K × k 2 γ 2 + 1 γ 1 e -2k log 3 + exp(-c × c γ 1 2 k) . ( 20
)
where K is a positive constant depending only on c, b, γ 1 , γ 2 and c 2 . This shows that 18) since we have selected c 2 such that c × c γ 1 2 = 2 log 3. The proof is complete.

ν k → σ 2 , as k → ∞. Hence (17) is satisfied since (log n) max k≤hn (m k ) 1/2 (log n) 1+1/(2γ 1 ) = O((log n) α ) (indeed α = 1 + 1/γ 1 + 1/γ 2 ). Note now that since |ν 1/2 k -σ| 2 ≤ |ν k -σ 2 |, (20) implies (
3 Applications

Products of iid invertible matrices

Let (ε n ) n≥1 be independent random matrices taking values in G = GL d (R), d ≥ 2, with common distribution µ. Let A 0 = Id and for every n ≥ 1, A n = ε n • • • ε 1 .
Let • be the euclidean norm on R d and for any g ∈ G, let N (g) := max( g , g -1 ) where g = sup x =1 gx . Recall that µ has a moment of order p ≥ 1 if G (log N (g)) p µ(dg) < ∞ and a sub-exponential moment of order γ ∈]0, 1] if there exists c > 0 such that

G e c(log N (g)) γ µ(dg) < ∞ . ( 21
)
Recall also that if µ admits a moment of order 1 then

lim n→∞ 1 n log A n = λ µ P-a.s., (22)
where

λ µ := lim n→+∞ n -1 E(log ε n • • • ε 1 )
is the so-called first Lyapunov exponent (see for instance [START_REF] Furstenberg | Products of Random Matrices[END_REF]).

Let X := P d-1 (R d ) be the projective space of R d -{0} and write x as the projection of

x ∈ R d -{0} to X. G is acting on X as follows: g • x = gx for every (g, x) ∈ G × R d -{0}.
We assume that µ is strongly irreducible (i.e. that no proper finite union of subspaces of R d are invariant by Γ µ , the closed semi-group generated by the support of µ) and proximal (i.e. that there exists a matrix in Γ µ admitting a unique (with multiplicity one) eigenvalue with maximum modulus). Under those assumptions (see e.g. Bougerol-Lacroix [START_REF] Bougerol | Products of random matrices with applications to Schrödinger operators[END_REF] or Benoist-Quint [START_REF] Benoist | Central limit theorem for linear groups[END_REF]) it is well-known that there exists a unique invariant measure ν on B(X), meaning that for any continuous and bounded function f from X to R,

X f (x)ν(dx) = G X f (g • x)µ(dg)ν(dx) .
The left random walk of law µ on X is the process defined as follows. Let W 0 be an X-valued random variable independent of (ε n ) n≥1 , and let

W n = ε n • W n-1 for n ≥ 1. Note that, if W 0 has distribution ν, then (W n ) n≥0 is a strictly stationary Markov chain.
Our aim is to study the partial sums associated with the random sequence (X n ) n≥1 given by

X n := h(ε n , W n-1 ) , n ≥ 1 ,
where for every g ∈ G and every x ∈ X, h(g, x) = log gx x .

As usual, we shall denote by X n,x the random variable for which W 0 = x. We then define S n,x = n k=1 X n,x . Note that, for any x ∈ S d-1 ,

S n,x = n k=1 X k,x = log A n x .
Then, when we keep the notation (X n ) n≥1 , we have in mind that we are in stationary regime, i.e. that W 0 has law ν. In this case we also define

S n = n k=1 X n = log A n V 0 ,
where V 0 is such that V 0 = 1 and W 0 = V0 .

We denote by ρ(g) the spectral radius of a matrix g. Applying Theorem 1, the following strong approximation with rate holds.

Corollary 2 Let µ be a proximal and strongly irreducible probability measure on B(G). Assume that µ has a sub-exponential moment of order γ ∈]0, 1]. Then n -1 E ν (S n -nλ µ ) 2 → σ 2 as n → ∞ and for every (fixed) x ∈ X, one can redefine (S n,x ) n≥1 without changing its distribution on a (richer) probability space on which there exist iid random variables (N i ) i≥1 with common distribution N (0, σ 2 ), such that,

S n,x -nλ µ - n i=1 N i = O((log n) 1+2/γ ) a.s.
Moreover, the result remains true if we replace the sequence (S n,x ) n≥1 with (S n ) n≥1 , (log

A n ) n≥1 , (log ρ(A n )) n≥1 or (log | A n x, y |) n≥1 , for some x, y ∈ S d-1 .
Remark 3 It follows from item c) of Theorem 4.11 of Benoist-Quint [START_REF] Benoist | Central limit theorem for linear groups[END_REF] that σ > 0 if µ is strongly irreducible and the image of Γ µ in P GL d (R) is unbounded.

Remark 4 If µ has a super-exponential moment meaning that (21) holds with γ > 1, we infer from the proof of Corollary 2 that for every (fixed) x ∈ X, one can redefine (S n,x ) n≥1 without changing its distribution on a (richer) probability space on which there exist iid random variables (N i ) i≥1 with common distribution N (0, σ 2 ), such that,

S n,x -nλ µ - n i=1 N i = O((log n) 2+1/γ ) a.s.
In particular if µ has compact support, the rate in the almost sure invariance principle is of order O((log n) 2 ). Again, this extension holds with (S n ) n≥1 , (log

A n ) n≥1 , (log ρ(A n )) n≥1 or (log | A n x, y |) n≥1 , for some x, y ∈ S d-1 .
Proof of Corollary 2. It follows from Theorem 2 (ii) in [START_REF] Cuny | Limit theorems for the left random walk on GL d (R)[END_REF] that n -1 E ν (S n -nλ µ ) 2 → σ 2 as n → ∞. Then, the strong invariance principle in stationary regime, i.e. for (S n ) n≥1 , is a direct application of Theorem 1 by taking into account the estimate given in Proposition 5 below. The proof of Proposition 5 is given in appendix.

Proposition 5 Assume that µ satisfies (21) with γ > 0. Then, there exists β > 0 such that, for any positive integer k,

sup x,ȳ∈X E(|X k,x -X k,ȳ |) e -βk min(γ,1) . ( 23
)
As in [START_REF] Cuny | Limit theorems for the left random walk on GL d (R)[END_REF] we obtain the result for the matrix norms (log( A n ) n≥1 from the stationary regime, using an argument from [START_REF] Bougerol | Products of random matrices with applications to Schrödinger operators[END_REF]. Indeed, it suffices to apply the estimate given just after equation (66) of [START_REF] Cuny | Limit theorems for the left random walk on GL d (R)[END_REF] to get the ASIP in that case. Similarly, we get the ASIP for (S n,x ) n≥1 , for any x ∈ S d-1 .

It remains to handle the case of the matrix coefficients and of the spectral radius. The proof follows a well-known scheme that has been used in [START_REF] Benoist | Random walks on reductive groups[END_REF] under exponential moments and in [START_REF] Benoist | Central limit theorem for linear groups[END_REF] and [START_REF] Cuny | Berry-Esseen type bounds for the matrix coefficients and the spectral radius of the left random walk on GL d (R)[END_REF] under polynomial moments. Those proofs rely on large deviations inequalities. In the case of sub-exponential moments, the needed large deviations estimates were obtained in [START_REF] Cuny | Large and moderate deviations for the left random walk on GL d (R). ALEA Lat[END_REF].

We start with an auxiliary result of independent interest, see Theorem 2.6 of [START_REF] Grama | Limit theorems on the coefficients of random walks on the general linear group[END_REF], for a related result. We can now prove the result for the matrix coefficients.

Let η be as in Proposition 6. Let n ∈ N and x ∈ X. Using that ν is µ-invariant and Markov's inequality, we have

P -log δ(W n , x) ≥ (2 log n/η) 1/γ = ν ȳ ∈ X : e η|log δ(ȳ,x)| γ ≥ n 2 ≤ K/n 2 .
Hence, by the Borel-Cantelli lemma we see that, P-a.s.

-log δ(W n , x) ≤ (2 log n/η) 1/γ , for all but finitely many n ∈ N. Combining this with the computations after (67) (and page 1864) in [START_REF] Cuny | Limit theorems for the left random walk on GL d (R)[END_REF], we infer that for every x, ȳ ∈ X,

|log δ(A n • x, ȳ)| = O((log n) 1/γ ) ,
and the desired result follows since

log | A n x, y | = log A n x + log y + log δ(A n • x, ȳ) .
It remains to prove the ASIP for the spectral radius. The result follows from the following lemma, with ≥ (2 log n/c) 1/γ , combined with the Borel-Cantelli lemma.

Lemma 7 Under the assumption of Corollary 2, for every ε > 0, there exists C, c > 0 and 0 ∈ N such that for every 0 ≤ ≤ n,

P(log ρ(A n ) -A n ≥ -ε ) ≥ 1 -Ce -c γ .
The lemma may be proved as Lemma 8 in [START_REF] Cuny | Berry-Esseen type bounds for the matrix coefficients and the spectral radius of the left random walk on GL d (R)[END_REF] using our Lemma 12 from the Appendix instead of Lemma 6 in [START_REF] Cuny | Berry-Esseen type bounds for the matrix coefficients and the spectral radius of the left random walk on GL d (R)[END_REF].

Products of iid positive matrices

Let us assume now that G is the semi-group of d-dimensional positive allowable matrices: by positive, we mean that all entries are greater than or equal to 0, by allowable, we mean that any row and any column admits a strictly positive element.

In this case, G is acting on X = S d-1 ∩ (R + ) d as follows: g • x = gx/ gx , and we define N (g) by N (g) = max( g , 1/v(g)), where v(g) := inf x∈X gx . Here, • stands for the 1 norm on R d which is more convenient for the problem, see [START_REF] Cuny | Limit theorems for iid products of positive matrices[END_REF]. This change does not affect the final results since all norms on R d are equivalent.

Then, similarly to the previous section, we define polynomial moments of order p ≥ 1 for µ as well as sub-exponential moments of order γ ∈]0, 1].

We shall also say that µ is strictly contracting if its support contains a matrix whose all entries are strictly positive. When µ is stricty contracting there exists a unique µ-invariant probability ν on B(X), see Section 3 of [START_REF] Cuny | Limit theorems for iid products of positive matrices[END_REF].

We have the following analogue to Corollary 2 with S n,x = log A n x , x ∈ X, where

A n = ε n • • • ε 1 with (ε n ) n≥1 a sequence of independent random matrices taking values in G, d ≥ 2,
with common distribution µ. Let also W 0 be an X-valued random variable with law ν and independent of (ε n ) n≥1 , and S n = log A n W 0 . The following corollary was announced in [START_REF] Cuny | Limit theorems for iid products of positive matrices[END_REF].

Corollary 8 Let µ be a strictly contracting probability measure on B(G). Assume that µ has a sub-exponential moment of order γ ∈]0, 1]. Then n -1 E ν (S n -nλ µ ) 2 → σ 2 as n → ∞ and for every (fixed) x ∈ X, one can redefine (S n,x ) n≥1 without changing its distribution on a (richer) probability space on which there exist iid random variables (N i ) i≥1 with common distribution N (0, σ 2 ), such that,

S n,x -nλ µ - n i=1 N i = O((log n) 2+1/γ ) a.s.
Moreover,the result remains true if we replace the sequence (S n,x ) n≥1 with (S n ) n≥1 , (log

A n ) n≥1 , (log ρ(A n )) n≥1 , (log v(A n )) n≥1 or (log | A n x, y |) n≥1 , for some x, y ∈ X.
Remark 9 When µ is aperiodic, see Definition 5.1 of [START_REF] Cuny | Limit theorems for iid products of positive matrices[END_REF], then, by Proposition 5.2 of [START_REF] Cuny | Limit theorems for iid products of positive matrices[END_REF], σ 2 > 0. When µ admits super-exponential moments, it is possible to improve the rate as in the remark after Corollary 2.

Proof of Corollary 8. Again, we start with the stationary regime. The existence of the asymptotic variance follows from Proposition 5.2 of [START_REF] Cuny | Limit theorems for iid products of positive matrices[END_REF]. By Proposition 3.2 of [START_REF] Cuny | Limit theorems for iid products of positive matrices[END_REF], when µ is strictly contracting and admits a moment of order greater than one, (23) holds. Then, the result (for (S n ) n≥1 ) follows from Theorem 1.

The ASIP for (log A n ) n≥1 and (log v(A n )) n≥1 follows from (4.3) in [START_REF] Cuny | Limit theorems for iid products of positive matrices[END_REF] and the fact that for every x ∈ X, v(A n ) ≤ A n x ≤ A n . Similary, we obtain the ASIP for (S n,x ) n≥1 for any x ∈ X.

Then, the ASIP for (log ρ(A n )) n≥1 follows from (4.3) in [START_REF] Cuny | Limit theorems for iid products of positive matrices[END_REF], combined with the inequalities, for every

n ≥ 1, log v(A n ) ≤ log ρ(A n ) ≤ log A n .
The ASIP for the matrix coefficients is proved in [START_REF] Cuny | Limit theorems for iid products of positive matrices[END_REF], see Theorem 6.4.

Lipschitz autoregressive models

We consider the autoregressive Lipschitz model introduced by Dedecker and Rio [START_REF] Dedecker | On the functional central limit theorem for stationary processes[END_REF]. Let τ ∈ [0, 1), C ∈ (0, 1] and f : R → R a 1-Lipschitz function such that

f (0) = 0 and |f (t)| ≤ 1 - C (1 + |t|) τ
for almost every t .

Let (ε i ) i≥1 be iid real-valued random valued with common law µ and define for any n ≥ 1

W n = f (W n-1 ) + ε n , with W 0 independent of (ε i ) i≥1 . (24)
Let S n (g) = n k=1 g(W i ) for any measurable function g. Assume that µ has a sub-exponential moment of order η ∈]0, 1] (hence E(e c|ε 0 | η ) < ∞, for some c > 0). For this model, there exists an unique invariant probability measure ν (see Proposition 2 of [START_REF] Dedecker | On the functional central limit theorem for stationary processes[END_REF]) and the following result holds:

Corollary 10 Assume that µ has a sub-exponential moment of order η ∈]0, 1]. Let (W n ) n≥0 be defined by (24) with L(W 0 ) = ν. Assume furthermore that g is Lipschitz and that, for some ζ ∈ [0, 1] and some positive constant κ, |g(x)| ≤ κ(1 + |x| ζ ). Suppose that τ + ζ > 0. Then, n -1 Var(S n (g)) → σ 2 (g) as n → ∞ and one can redefine (W n ) n≥0 without changing its distribution on a (richer) probability space on which there exist iid random variables (N i ) i≥1 with common distribution N (0, σ 2 (g)), such that,

S n (g) -nν(g) - n i=1 N i = O((log n) 1+1/γ 1 +1/γ 2 ) P-a.s.
where

γ 1 = η(1 -τ )(η(1 -τ ) + τ ) -1 and γ 2 = η(1 -τ )/ζ.
Proof. The result comes from an application of Theorem 1 by taking into account Proposition 3 of [START_REF] Merlevède | A Bernstein type inequality and moderate deviations for weakly dependent sequences[END_REF] and its proof. More precisely, [20, (4.3)] and [START_REF] Merlevède | A Bernstein type inequality and moderate deviations for weakly dependent sequences[END_REF]Remark 5] ensure that g(W 1 ) -ν(g) satisfies ( 5) with index γ 2 = η(1 -τ )/ζ. In addition, as it appears clearly in the proof of [20, (4.4)], the control of the τ -coefficients is done via a control of the coupling coefficients δ(n) (see [20, (4.6)]).

Appendix

Proof of Proposition 5

The proof of the proposition is based on the following lemma whose statement needs the following notation: For x, ȳ ∈ X, define d(x, ȳ) :=

x ∧ y x y

where ∧ stands for the exterior product, see e.g. [4, page 61] for the definition and some properties. Then d is a metric on X.

Lemma 11 Assume that µ satisfies (21) with γ > 0. Then, for any positive integer k and any κ > 0, there exists > 0, not depending on k, such that 1) .

max k≤j≤2k sup x,ȳ∈X,x =ȳ P log(d(A j-1 • x, A j-1 • ȳ)) ≥ -k e -κk min(γ,
Proof of Lemma 11. Without loss of generality, let us assume that c > 4 in condition [START_REF] Rio | Théorie asymptotique des processus aléatoires faiblement dépendants[END_REF]. The general case can be deduced by normalizing suitably the quantities of interest. We proceed with similar arguments as in the proof of Lemma 6 in [START_REF] Cuny | Limit theorems for the left random walk on GL d (R)[END_REF]; namely, we use a martingale decomposition. Since G log N (g)µ(dg) < ∞, we can define the following bounded function F 1 :

F 1 (x, ȳ) = G log(d(g • x, g • ȳ)/d(x, ȳ))µ(dg) , ∀x, ȳ ∈ X, x = ȳ .
Then we define the following centered cocycle:

σ 1 (g, (x, ȳ)) = log(d(g • x, d(g • ȳ)/d(x, ȳ)) -F 1 (x, ȳ) .
Setting

M n = M n (x, ȳ) := n k=1 σ 1 (ε k , (A k-1 • x, A k-1 • ȳ)) and R n = R n (x, ȳ) := n k=1 F 1 (A k-1 • x, A k-1 • ȳ) , we have log(d(A n • x, A n • ȳ)/d(x, ȳ)) = M n + R n .
Hence, since d(x, ȳ) ≤ 1, the lemma will be proved if one can show that for any κ > 0, there exists a > 0 such that

max k≤j≤2k sup x,ȳ∈X,x =ȳ P R j (x, ȳ) ≥ -2 k e -κk min(γ,1) (25)
and max k≤j≤2k sup

x,ȳ∈X,x =ȳ

P |M j (x, ȳ)| ≥ k e -κk min(γ,1) . (26)
The estimate (25) comes from the proof of (53) in [START_REF] Cuny | Limit theorems for the left random walk on GL d (R)[END_REF]. Indeed from the last inequality in the proof of [5, (53), page 1857], there exist ρ ∈]0, 1[, n 0 ∈ N and α > 0 such that

max k≤j≤2k sup x,ȳ∈X,x =ȳ P R j (x, ȳ) ≥ -αk ≤ Cρ k/(2n 0 ) .
Hence (25) follows by taking ≥ αn 0 /(| log ρ|).

We turn to the proof of (26). Let

D k = σ 1 (ε k , (A k-1 • x, A k-1 • ȳ)) and F k = σ(W 0 , ε 1 , . . . , ε k ). Note that (D k , F k ) k>0
is a sequence of martingale differences. Moreover setting X = 4 log(N (g)), Lemma 4.2 page 56 in [START_REF] Bougerol | Products of random matrices with applications to Schrödinger operators[END_REF] (see also their Proposition 5.3 page 62) ensures that

P(D k ≥ x|F k-1 ) ∞ ≤ P(X + E(X) ≥ x) , implying that n k=1 E(D 2 k exp{|D k | γ }|F k-1 ) ∞ ≤ nE((X + E(X)) 2 exp{(X + E(X)) γ }) := nK .
Note that since µ satisfies [START_REF] Rio | Théorie asymptotique des processus aléatoires faiblement dépendants[END_REF]

P |M j (x, ȳ)| ≥ k exp(-c 1 2 k/K) + exp(-c 2 ( k) min(γ,1) ) ,
which proves (26) and ends the proof of the lemma.

We turn now to the proof of Proposition 5.

Proof of Proposition 5. Let x, ȳ ∈ X and κ > 0. Let > 0 be as in Lemma 11. We start from inequality (18) in [START_REF] Cuny | Limit theorems for the left random walk on GL d (R)[END_REF], namely: setting

A = {log N (ε k ) ≥ k/2} and B = {log(d(A k-1 • x, d(A k-1 • ȳ)) ≥ -k}, we write |X k,x -X k,ȳ | ≤ |σ(ε k , A k-1 • x) -σ(ε k , A k-1 • ȳ)|1 A + |σ(ε k , A k-1 • x) -σ(ε k , A k-1 • ȳ)|1 B + |σ(ε k , A k-1 • x) -σ(ε k , A k-1 • ȳ)|1 {A c ∩B c } .
Recall the estimates ( 13) and ( 14) in [START_REF] Cuny | Limit theorems for the left random walk on GL d (R)[END_REF]: there exists C > 0 such that for every x, ȳ ∈ X, |σ(g, x) -σ(g, ȳ)| ≤ CN (g)d(x, ȳ) , and |σ(g, x)| ≤ log N (g) .

We then infer that

E(|X k,x -X k,ȳ |) ≤ 2E((log N (ε k ))1 A ) + 2E((log N (ε k ))1 B ) + CE N (ε k )d(A k-1 • x, A k-1 • ȳ)|1 {A c ∩B c } . (27) Now E((log N (ε k ))1 A ) ≤ e -( k/2) γ E log N (g)e (log N (g)) γ . ( 28
)
On another hand, by independence between ε k and B, and Lemma 11, E((log N (ε k ))1 B ) = P(B)E(log N (g)) ≤ e -κk γ E(log N (g)) .

(29)

Moreover,

E N (ε k )d(A k-1 • x, A k-1 • ȳ)|1 {A c ∩B c } ≤ e -k E N (g)1 {log N (g)≤ k/2} ≤ e -k/2 . ( 30
)
Starting from ( 27) and taking into account the upper bounds (28)-(30), the lemma follows.

Proof of Proposition 6

It suffices to find η > 0 such that (notice that δ(•, •) ≤ 1) The fact that one may find such a η > 0 may be proved as Lemma 4.8 (and more particularly (4.13)) of [START_REF] Benoist | Central limit theorem for linear groups[END_REF], using the next lemma.

Lemma 12 Under the assumption of Corollary 2, for every ε > 0, there exist C, c > 0, such that

sup x∈S d-1 P max 1≤k≤n | log A k x -kλ µ | > εn ≤ Ce -cn γ , (31)
P max 1≤k≤n | log A k -nλ µ | > εn ≤ Ce -cn γ ; ( 32
) P max 1≤k≤n | log Λ 2 (A k ) -k(λ µ + γ µ )| > εn ≤ Ce -cn γ . ( 33
)
Remark 13 Let us recall that, for any A ∈ GL d (R), Λ 2 (A) is the matrix on Λ 2 (R d ) defined by Λ 2 (A)(x ∧ y) = Ax ∧ Ay, where ∧ is the exterior product. In addition, in (33) , γ µ is the second Lyapunov exponent of µ. With the notations of [2, Section 14], λ µ is denoted either λ 1,µ or λ 1 , while γ µ is either denoted λ 2,µ or λ 2 .

Proof of Lemma 12. (31) follows directly from Theorem 2.2 of [START_REF] Cuny | Large and moderate deviations for the left random walk on GL d (R). ALEA Lat[END_REF]. Then (32) follows from (31) and the fact that there exists C > 0 such that for every g ∈ G and every x ∈ S d-1 , gx ≤ g ≤ C max 1≤i≤d ge i , with {e i : 1 ≤ i ≤ d} the canonical basis of R d .

The proof of (33) may be done similarly as above. The group G acts on P (∧ 2 (R d )), the projective space of ∧ 2 (R d ), by g • ξ := ∧ 2 (g)ξ/ ξ , for every g ∈ G and ξ ∈ ∧ 2 (R d ), where ξ is the projection of ξ on P (∧ 2 (R d )). Then, (g, ξ) → ∧ 2 (g)(ξ) / ξ defines a cocycle and one may prove, similarly to (31) that (see Section 6 of [START_REF] Cuny | Large and moderate deviations for the left random walk on GL d (R). ALEA Lat[END_REF] where the situation of general cocycles is considered)

sup ξ∈∧ 2 (R d ) : ξ =1 P max 1≤k≤n | log ∧ 2 (A k )(ξ) -k(λ µ + γ µ )| > εn ≤ Ce -cn γ .
Then, we deduce (33) as we deduced (32) from (31).

Proposition 6

 6 Under the condition of Corollary 2, there exists η > 0 such that sup x∈X X e η|log δ(x,ȳ | γ dν(ȳ) < ∞ , where δ(x, ȳ) = | x, y |/( x y ).

n≥1 e n γ sup

  x∈X ν({ȳ ∈ X : -log δ(x, ȳ) ≥ ηn}) < ∞ . Now, using that ν is µ-invariant, for every n ∈ N and x ∈ X,ν({ȳ ∈ X : -log δ(x, ȳ) ≥ ηn}) = ν ⊗ µ * n ({(ȳ, g) ∈ X × G : -log δ(x, g • ȳ) ≥ ηn}) ≤ sup x,ȳ∈X P(log δ(x, A n • ȳ) ≤ -ηn) .


MSc Claire Olivier

PhD, * † ‡ Jean-Charles C Lamy

MD Zuzana Kosutzka

Angèle Van Hamme

MSc Saoussen Cherif

PhD Brian Lau

MD, * ‡ Marie Vidailhet

MD Carine Karachi

MD, PhD Marie-Laure Welter
email: marielaure.welter@icm-institute.org

‡ Angèle

MSc Van Hamme

¶ # Affiliations

Cerebellar transcranial alternating current stimulation in essential tremor patients with thalamic stimulation: a proof-of-concept study

Keywords: 53 References Essential tremor, transcranial alternating current stimulation, cerebellum, deep brain stimulation, motion capture

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Essential tremor (ET) is a frequent and disabling disorder with progressive worsening postural tremor of the arms, which seriously affects quality of life [START_REF] Louis | How common is the most common adult movement disorder? estimates of the prevalence of essential tremor throughout the world[END_REF]. ET has been associated with pathological changes in the cerebellum [START_REF] Louis | Neuropathological changes in essential tremor: 33 cases compared with 21 controls[END_REF], reduced function of the cerebellar cortex and abnormal oscillations within the cerebello-thalamo-cortical (CTC) network [START_REF] Louis | Neuropathological changes in essential tremor: 33 cases compared with 21 controls[END_REF][START_REF] Deuschl | The pathophysiology of essential tremor[END_REF][START_REF] Placantonakis | Fundamental role of inferior olive connexin 36 in muscle coherence during tremor[END_REF][START_REF] Pedrosa | Essential tremor and tremor in Parkinson's disease are associated with distinct "tremor clusters" in the ventral thalamus[END_REF][START_REF] Gallea | Intrinsic signature of essential tremor in the cerebello-frontal network[END_REF][START_REF] Filip | Linking Essential Tremor to the Cerebellum: Physiological Evidence[END_REF][START_REF] Fang | Essential tremor is associated with disruption of functional connectivity in the ventral intermediate Nucleus--Motor Cortex--Cerebellum circuit[END_REF][START_REF] Hanajima | Cerebellar dysfunction in essential tremor[END_REF][START_REF] Pinto | The cerebellothalamocortical pathway in essential tremor[END_REF]. Highfrequency deep brain stimulation (DBS) of the ventral intermediate nucleus of the thalamus (VIM), which receives afferent inputs from the cerebellar deep nuclei, is very effective in improving ET [START_REF] Deuschl | Treatment of patients with essential tremor[END_REF][START_REF] Kumar | Long-term follow-up of thalamic deep brain stimulation for essential and parkinsonian tremor[END_REF][START_REF] Bardinet | Thalamic stimulation for tremor: can target determination be improved?[END_REF]. It can decrease tremor amplitude by up to 40-80%, by disrupting the abnormal rhythmic activity within the CTC network with low-and high-frequency thalamic oscillation reduction [START_REF] Limousin | Multicentre European study of thalamic stimulation in parkinsonian and essential tremor[END_REF][START_REF] Schuurman | A comparison of continuous thalamic stimulation and thalamotomy for suppression of severe tremor[END_REF][START_REF] Rehncrona | Long-term efficacy of thalamic deep brain stimulation for tremor: double-blind assessments[END_REF]. Although effective and well-tolerated [START_REF] Rehncrona | Long-term efficacy of thalamic deep brain stimulation for tremor: double-blind assessments[END_REF], this technique is an invasive neurosurgical procedure, and decreasing responsiveness and the occurrence of sideeffects could impact its benefits in the long-term [START_REF] Gibson | Functional correlates of the therapeutic and adverse effects evoked by thalamic stimulation for essential tremor[END_REF][START_REF] Fasano | Gait ataxia in essential tremor is differentially modulated by thalamic stimulation[END_REF]. Recently, VIM lesioning using focused ultrasound has been reported to drastically improve ET symptoms with good tolerance [START_REF] Iorio-Morin | Bilateral Focused Ultrasound Thalamotomy for Essential Tremor (BEST-FUS Phase 2 Trial)[END_REF][START_REF] Abe | Focused Ultrasound Thalamotomy for Refractory Essential Tremor: A Japanese Multicenter Single-Arm Study[END_REF]. However, both treatments are of high cost with potential adverse events.

Non-invasive brain stimulation (NIBS) techniques targeting the cerebellum, such as repetitive transcranial magnetic stimulation (rTMS), theta burst stimulation (TBS) or transcutaneous direct current stimulation (tDCS), have also been tested in a few studies to manage ET [START_REF] Kang | Does non-invasive brain stimulation reduce essential tremor? A systematic review and meta-analysis[END_REF][START_REF] Latorre | The use of transcranial magnetic stimulation as a treatment for movement disorders: A critical review[END_REF][START_REF] Popa | Cerebellar rTMS stimulation may induce prolonged clinical benefits in essential tremor, and subjacent changes in functional connectivity: an open label trial[END_REF][START_REF] Gironell | Transcranial direct current stimulation of the cerebellum in essential tremor: a controlled study[END_REF][START_REF] Gironell | Transcranial magnetic stimulation of the cerebellum in essential tremor: a controlled study[END_REF][START_REF] Maas | The role of the cerebellum in degenerative ataxias and essential tremor: Insights from noninvasive modulation of cerebellar activity[END_REF]. Open-label studies and a recent meta-analysis suggested positive therapeutic effects of cerebellar NIBS for ET [START_REF] Kang | Does non-invasive brain stimulation reduce essential tremor? A systematic review and meta-analysis[END_REF][START_REF] Popa | Cerebellar rTMS stimulation may induce prolonged clinical benefits in essential tremor, and subjacent changes in functional connectivity: an open label trial[END_REF], although double-blind proof-of-concept studies mainly failed to demonstrate its therapeutic interest of NIBS [START_REF] Gironell | Transcranial direct current stimulation of the cerebellum in essential tremor: a controlled study[END_REF][START_REF] Bologna | Cerebellar continuous theta burst stimulation in essential tremor[END_REF]28]. In two recent controlled trials, low-frequency rTMS applied over the cerebellum induced a significant improvement in tremor severity immediately after [START_REF] Gironell | Transcranial magnetic stimulation of the cerebellum in essential tremor: a controlled study[END_REF][START_REF] Shin | Cerebellar repetitive transcranial magnetic stimulation for patients with essential tremor[END_REF]. Recently, transcranial alternating current stimulation (tACS) has emerged as a NIBS technique to manipulate intrinsic neuronal oscillations with externally applied rhythmic electrical frequencies, with after-effects that can persist over 1 hour after current offset [START_REF] Moliadze | Boosting brain excitability by transcranial high frequency stimulation in the ripple range[END_REF]. When applied over the cerebellum at 5 Hz or 300 Hz in healthy subjects, it led to an increase in cerebellar-brain inhibition (CBI), with no changes in the short-intracortical inhibition and short-afferent inhibition [START_REF] Spampinato | Frequency-dependent modulation of cerebellar excitability during the application of non-invasive alternating current stimulation[END_REF][START_REF] Naro | Does Transcranial Alternating Current Stimulation Induce Cerebellum Plasticity? Feasibility, Safety and Efficacy of a Novel Electrophysiological Approach[END_REF]. Applied at 50 Hz it induced no changes [START_REF] Spampinato | Frequency-dependent modulation of cerebellar excitability during the application of non-invasive alternating current stimulation[END_REF] or a CBI weakening with better sequential tapping adaptation [START_REF] Naro | Does Transcranial Alternating Current Stimulation Induce Cerebellum Plasticity? Feasibility, Safety and Efficacy of a Novel Electrophysiological Approach[END_REF]. Lastly, phase-locked cerebellar tACS has also been shown to decrease tremor amplitude in a small sample of patients with ET [START_REF] Schreglmann | Non-invasive suppression of essential tremor via phase-locked disruption of its temporal coherence[END_REF] or with non-jerky dystonic tremor [START_REF] Nieuwhof | Phase-locked transcranial electrical brain stimulation for tremor suppression in dystonic tremor syndromes[END_REF].

Finally, the fact that: 1) all these previous data highlight the potential benefits of cerebellar tACS in modulating CTC network functioning; 2) ET patients exhibited various abnormal neuronal oscillations within the CTC network, including very low and high frequency oscillations, which may reflect altered Purkinje cells GABAergic neurotransmission [START_REF] Schreglmann | Non-invasive suppression of essential tremor via phase-locked disruption of its temporal coherence[END_REF][START_REF] Nieuwhof | Phase-locked transcranial electrical brain stimulation for tremor suppression in dystonic tremor syndromes[END_REF][START_REF] Schnitzler | Occurrence of thalamic high frequency oscillations in patients with different tremor syndromes[END_REF];

3) high-frequency thalamic DBS dramatically improves ET severity in correlation with increased activation within the whole CTC network, intra-regional inhibition with decreased neuronal oscillations [START_REF] Gibson | Functional correlates of the therapeutic and adverse effects evoked by thalamic stimulation for essential tremor[END_REF], all strongly support the potential value of cerebellar tACS for ET management.

Here, we hypothesise that high-frequency tACS could improve tremor severity similarly to high-frequency VIM-DBS, by increasing GABAergic inputs to deep cerebellar nuclei, and subsequently modulate the CTC network excitability and neuronal oscillations. To test this hypothesis, we first studied the effects of high-frequency VIM-DBS and cerebellar tACS applied at the same frequency in ET patients previously treated by VIM-DBS. We also tested the effects of high-frequency cerebellar tACS in a group of ET patients not previously operated for VIM-DBS and matched for tremor severity.

Methods

Study design and patients

In this double-blind trial, we recruited patients from the Salpêtrière Hospital, Paris, France.

All patients were assessed at the Paris Brain Institute. Two groups of patients were included.

All patients were eligible for inclusion if they fulfilled ET diagnosis criteria according to the Task Force on tremor of the International Parkinson and Movement Disorders Society [START_REF] Bhatia | Consensus Statement on the classification of tremors. from the task force on tremor of the International Parkinson and Movement Disorder Society[END_REF],

were aged 18 to 75 years, had no other causes of tremor or neurological disorders, were affiliated to the French national health insurance or similar scheme, agreed to participate and provided written informed consent. An additional inclusion criterion was unilateral VIM-DBS for more than 6 months prior to this study for patients in the VIM-DBS group. This study received approval from the local ethics committee of Paris VI University and the French Ministry of Health. All patients gave written informed consent. The study was performed according to the Declaration of Helsinki and Good clinical practice guidelines, and was sponsored by the INSERM (C14-44). Clinical trial.gov registration: NCT02346409.

Procedures

Patients with VIM-DBS were first assessed with unilateral VIM-DBS ON with their usual stimulation parameters, and the DBS was then switched OFF for 3 hours to avoid rebound effects due to sudden neurostimulation cessation. Then, all patients were assessed without treatment, OFF VIM-DBS for patients previously operated (baseline assessment). Then, each patient first received sham-tACS followed by active high-frequency tACS for 10 minutes each, without being informed of the order of application, and at 0, 20 and 40 min after current offset (Figure 1). Anti-tremor drugs were stopped 7 days before the assessment in both groups, except for propranolol which was stopped the day before.

We performed assessments of tremor severity by using the Fahn-Tolosa-Marin (FTM) scale [START_REF] Fahn | Clinical Rating Scale for Tremor[END_REF], and kinetic recordings of arm tremor using a VICON® 3D motion capture system with 29 reflective markers positioned bilaterally on the arms, head and trunk (Figure 1). Cerebellar tACS was administrated to patients at rest, seated in a chair, and positioned in front of a table. tACS was delivered by a battery-driven stimulator (NeuroConn, DC-Stimulator Plus, CE) connected to a pair of saline-soaked square-shaped sponge electrodes (9 cm 2 ). One electrode was positioned over the posterior cerebellum (3 cm lateral to the inion), contralateral to VIM-DBS, whereas the return electrode was positioned over the contralateral cheekbone. Electrodes were fixed by elastic bands. For the control group, we applied cerebellar tACS ipsilaterally to the most severe side with a similar set-up. During active stimulation, tACS was delivered at a frequency of 130 to 180 Hz matched to the chronic VIM-DBS stimulation frequency in the VIM-DBS group; and at 130 Hz in the No-DBS group. The stimulation intensity was set at 1 mA and was applied over a 10-min period resulting in a current density of 0.11 mA/cm 2 [START_REF] Herzog | Cerebellar transcranial current stimulation -An intraindividual comparison of different techniques[END_REF] and a delivered total charge of 0.066 C/cm 2 . Sham-tACS was also applied for a 10 min period starting with delivery of a 1 mA current but for 10 s only. All patients reported a moderate local paraesthesia at the beginning of both sham-and active-tACS and they were unable to differentiate the two conditions.

Randomization and masking

This proof-of-concept trial was double-blind and sham-controlled (Figure 1). All the assessments were recorded, videorecorded and rated in random order by two blinded investigators at the end of the study (ZK performed the clinical scale rating and OC the kinetics analysis), all sequences having been previously coded by a blinded clinical research assistant.

Outcomes

The predefined primary outcome was tremor amplitude, as assessed by the displacements of the markers positioned on the arms during a 5 second holding posture (arms folded, wrists slightly extended, fingers apart). Secondary endpoints included: i) the movement duration for performance of the 'nose-to-target' task with a marker placed on a support on the table at a fixed position with 5 round trips (Figure 1), and the displacement of the finger marker within a 2cm 2 area around the target (reflecting the pointing precision); ii) the clinical severity with the total score of the FTM scale, with part A rating the tremor severity of each body part in different posture and action tasks and part B rating the tremor severity during handwriting, drawing and pouring (we did not assess orthostatic tremor in part A, nor pouring in part B); and iii) the safety and tolerance of the procedure, as assessed by the occurrence of adverse events, serious adverse events or discomfort.

Statistical analysis

In this study, we aimed to examine the changes in tremor severity with active cerebellar high frequency tACS. In line with previous data obtained after cerebellar NIBS [START_REF] Kang | Does non-invasive brain stimulation reduce essential tremor? A systematic review and meta-analysis[END_REF][START_REF] Popa | Cerebellar rTMS stimulation may induce prolonged clinical benefits in essential tremor, and subjacent changes in functional connectivity: an open label trial[END_REF][START_REF] Gironell | Transcranial magnetic stimulation of the cerebellum in essential tremor: a controlled study[END_REF], we expected a decrease of 25% (SD 15%) in the tremor amplitude during active tACS with respect to baseline. With this assumption, our 10 patients should enable us to have a power of 90% with an alpha error rate of 5% (Wilcoxon signed-rank test).

The main outcome criterion was the sum of the displacements of the arm markers during posture holding (postural tremor) at the different timepoints. Secondary outcomes were the time required to perform the 'nose-to-target' task (action tremor) and the precision of targeting (measured by the displacements of the finger markers in the 2 cm 2 area around the target, Figure 1), and the FTM scores. For this purpose, we modelled each outcome using a linear mixed model. We included the treatment condition (baseline or OFF VIM-DBS, ON VIM-DBS, during sham-tACS, during and 0, 20, 40 minutes after active-tACS) and patient intercepts as random effects. We used R (R, version 3.3.1, R Core Development Team, FactoMineR package) for statistical analyses, with the LmerTest package for the linear mixed-effect model computation. We used a significance threshold of 0.05 with Bonferroni correction and p-values were adjusted using the false-discovery-rate method.

Results

Patient characteristics

Between February 2015 and July 2018, we included 11 patients with ET previously operated for VIM-DBS (3F/8M) and 10 patients with ET not previously operated for VIM-DBS (2F/8M) that were not significantly different for age (p=0.24), disease duration (p=0.48) or tremor severity (p=0.95, Table 1).

Effects of cerebellar tACS on ET severity in the VIM-DBS group

Postural tremor amplitude

In the active-tACS vs OFF-VIM-DBS condition, the postural tremor amplitude decreased significantly during and at 0, 20 and 40 min following stimulation (p= 0.02, p=0.03, p=0.04 and p=0.04, respectively, Figure 2, see supplementary video). There was no significant difference in postural tremor amplitude during active-tACS vs ON VIM-DBS condition (p=0.26, Figure 2), nor during sham-tACS vs OFF VIM-DBS (p=0.26). Comparing the effects of active-tACS between arms, we found that the effects of active-tACS were only significant for the ipsilateral arm during and at 0 minutes following active-tACS relative to OFF VIM-DBS (p=0.02 and p=0.02, respectively), and relative to sham-tACS (p=0.03), with no significant difference for the contralateral arm (Figure 2, see supplementary video).

Action tremor amplitude

In the active-tACS vs OFF-VIM-DBS condition, finger displacement within the 2cm 2 area around the target during the 'nose-to-target' task was significantly lower at 0 and 20 min following stimulation (p=0.04 and p=0.008, respectively), with a trend during and at 40 min following stimulation (p=0.05). These effects were only significant for the ipsilateral arm during, at 0, 20 and 40 min following stimulation (p=0.002, p=0.045, p=0.005 and p=0.008, respectively, Figure 2). Finger displacement of the ipsilateral arm was also significantly lower with active-tACS vs sham-tACS (p=0.02, Figure 2). Finger displacement was not significantly different between active tACS vs the ON VIM-DBS condition (p=0.32), and between sham-tACS vs the OFF VIM-DBS condition (p=0.26, Figure 3).

The task duration was also significantly lower in the active-tACS vs OFF-VIM-DBS condition (p= 0.012, p<10 -4 , p<10 -4 , p<10 -4 , respectively for during-stimulation and at 0, 20 and 40 min after, not shown), but only in the ipsilateral arm (p=0.003, p<10 -4 , p<10 -4 , p<10 -4 ipsilaterally; contralaterally the respective values were p=0.88, p=0.05, p=0.20, p=0.20).

Clinical ET severity

The FTM total score (blinded video scoring) was significantly lower during active-tACS and after 0, 20, 40 min vs both OFF VIM-DBS (p=0.006, p=0.010, p =0.007, p=0.02, respectively, Figure 4) and sham tACS (p=0.006, p=0.010, p=0.007, p=0.02, respectively, Figure 4) conditions. There was no significant difference in the FTM total score during active-tACS vs the ON VIM-DBS condition (p=0.10, Figure 4), nor during sham-tACS vs the OFF VIM-DBS condition (p=1.0, Figure 4).

The subscore A (tremor severity) was significantly lower during active tACS and after 0, 20 minutes vs both OFF-VIM-DBS (p=0.003, p=0.005, p=0.045, respectively) and the sham-tACS (p=0.003, Figure 4) conditions. The subscore A did not change significantly during sham-tACS vs the baseline (p=0.90, Figure 4). We found no significant changes in the subscore B (writing, drawing) with sham or active-tACS vs OFF VIM-DBS (all p values > 0.35, not shown).

Effects of cerebellar tACS on ET severity in the No-DBS group

Postural tremor amplitude

The postural tremor amplitude did not change significantly during and after active-tACS relative to both baseline (p=0.15) and sham-tACS conditions (p=0.15), and in the two arms (Figure 2).

Action tremor amplitude

In the active-tACS vs baseline condition, finger displacement within the 2 cm 2 area around the target assessed during the 'nose-to-target' task was significantly lower during and 0, 20 and 40 min after stimulation, but on the ipsilateral side only (p<10 -4 , p=0.02, p=0.01, p=0.04, for during, and 0, 20 and 40 min after active tACS, respectively, Figure 3), with a trend in comparison to the sham-tACS (p=0.10). We found no significant change in finger displacements with sham-tACS vs baseline condition, regardless of the arm considered (Figure 3, p=0.13 and p=0.71 for ipsilateral and contralateral arms, respectively, Figure 3).

The 'nose-to-target' task duration was also significantly lower during active-tACS and at 0, 20, 40 min following stimulation vs baseline (p<10 -4 , p<10 -4 , p<10 -4 , and p<10 -4 respectively). This effect was significant for both sides (not shown), and relative to sham-tACS for the ipsilateral arm (p=0.04, and p=0.11 for the contralateral arm, Figure 3).

Clinical ET severity

The FTM total score was significantly lower during active tACS and at 0, 20, 40 min following stimulation vs both baseline (p<10 -4 , p=0.03, p=0.018 and p=0.003, respectively, Figure 4) and sham-tACS (p=0.045, Figure 4) conditions. During sham-tACS vs baseline, there was no significant difference in the FTM total score (p=0.06, Figure 4). With respect to baseline, the subscore A was significantly lower during and at 0, 20, 40 min following active-tACS (p<10 -4 , p=0.01, p=0.02 and p=0.02, respectively, Figure 4) and also with sham-tACS (p=0.02, Figure 4). We found no significant changes for the subcore B with either sham or active-tACS vs baseline (all p values > 0.35, not shown).

Safety and tolerance of the procedure

No adverse or serious adverse event occurred during the trial. Both sham and active-tACS were generally well-tolerated regardless of the group. Two patients with VIM-DBS described mild transient paraesthesia of the ipsilateral arm occurring at current onset (during both the sham and active tACS conditions), and one patient in the control group described moderate transient paraesthesia under the skull electrode during both sham-and active tACS conditions.

Discussion

In this proof-of-concept double-blind sham-controlled study, we assessed for the first time the effects of high-frequency cerebellar tACS on ET patients treated with VIM-DBS; and in ET patients with similar tremor severity with no VIM-DBS, taken as controls. Using objective measurements (kinetic measures using VICON® 3D motion capture) and blind assessments of the FTM scores by independent examiners, we found a significant improvement in kinetic and clinical measures with active-cerebellar tACS. The effects were comparable to those of VIM-DBS for managing posture tremor amplitude and reducing clinical severity, and only present in the ipsilateral arm. In the control group of ET patients not previously treated with VIM-DBS, we also found a significant improvement in the 'nose-to-target' task parameters in the ipsilateral arm, and in tremor clinical severity after active-cerebellar tACS. Interestingly, we observed variable effects across patients of both groups, with in addition, an absence of decreased tremor severity (total FTM score) with active-tACS in 4 out of our 21 patients (19%), and no change in writing tremor for about one third of patients (score B). Such interindividual variability in cerebellar NIBS has been reported in previous NIBS trials [START_REF] Maas | The role of the cerebellum in degenerative ataxias and essential tremor: Insights from noninvasive modulation of cerebellar activity[END_REF], also with individual phase-locked tACS [START_REF] Schreglmann | Non-invasive suppression of essential tremor via phase-locked disruption of its temporal coherence[END_REF]. These variable effects of active-tACS could be due to differences in clinical characteristics, such as disease duration or age [START_REF] Maas | The role of the cerebellum in degenerative ataxias and essential tremor: Insights from noninvasive modulation of cerebellar activity[END_REF], features of the tremor movement [START_REF] Schreglmann | Non-invasive suppression of essential tremor via phase-locked disruption of its temporal coherence[END_REF], electrode-skin capacitance, or individual cerebellar-cortex anatomical and functional connectivity at baseline [START_REF] Hanajima | Cerebellar dysfunction in essential tremor[END_REF][START_REF] Pinto | The cerebellothalamocortical pathway in essential tremor[END_REF].

The physiological effects of non-invasive brain stimulation of the cerebellar cortex are not fully understood. Here, we hypothesised that high frequency tACS would modify cerebellar cortex excitability leading to subsequent changes in the pathological oscillatory activity within the CTC network. The fact that we found similar effects between VIM-DBS and active cerebellar tACS, applied at the same stimulation frequency, would favour this hypothesis. In ET patients, a possible progressive neurodegeneration of Purkinje cells has been reported with reduced activity of these cells [START_REF] Louis | Neuropathological changes in essential tremor: 33 cases compared with 21 controls[END_REF][START_REF] Louis | From neurons to neuron neighborhoods: the rewiring of the cerebellar cortex in essential tremor[END_REF] leading to a deficit in GABAergic inputs to the deep cerebellar nuclei neurons, and pathological oscillatory activities within the CTC network activity [START_REF] Filip | Linking Essential Tremor to the Cerebellum: Physiological Evidence[END_REF]. Previous studies demonstrated an increased theta-alpha band activity and high-frequency oscillations in the VIM of ET patients [START_REF] Schnitzler | Occurrence of thalamic high frequency oscillations in patients with different tremor syndromes[END_REF][START_REF] Brodkey | Tremor cells in the human thalamus: differences among neurological disorders[END_REF][START_REF] Kane | Enhanced synchronization of thalamic theta band local field potentials in patients with essential tremor[END_REF], with, in addition, an abnormal thalamocortical low-frequency oscillation phase-amplitude coupling [START_REF] Pedrosa | Thalamomuscular coherence in essential tremor: hen or egg in the emergence of tremor?[END_REF][START_REF] Kondylis | Movement-related dynamics of cortical oscillations in Parkinson's disease and essential tremor[END_REF], a mechanism mediating movement execution [START_REF] Opri | The Functional Role of Thalamocortical Coupling in the Human Motor Network[END_REF]. With high-frequency VIM-DBS, tremor suppression was found to be correlated to decreased thalamic neuronal firing and lowfrequency oscillations [START_REF] Opri | Chronic embedded cortico-thalamic closed-loop deep brain stimulation for the treatment of essential tremor[END_REF][START_REF] Milosevic | Physiological mechanisms of thalamic ventral intermediate nucleus stimulation for tremor suppression[END_REF][START_REF] Ferleger | Fully implanted adaptive deep brain stimulation in freely moving essential tremor patients[END_REF], and modulation of both olivocerebellar and thalamocortical circuits with activation of the contralateral cerebellar cortex, deep cerebellar nuclei and motor cortex [START_REF] Gibson | Functional correlates of the therapeutic and adverse effects evoked by thalamic stimulation for essential tremor[END_REF][START_REF] Molnar | Changes in cortical excitability with thalamic deep brain stimulation[END_REF]. These data suggests that VIM-DBS could lead to a partial reestablishment of CTC functioning, and act as a filter to uncouple thalamo-cortical from cortico-spinal reflex loops [START_REF] Milosevic | Physiological mechanisms of thalamic ventral intermediate nucleus stimulation for tremor suppression[END_REF]. Applied at the cerebellar cortex, 5 Hz tACS has been shown to increase cerebellocortex inhibition in healthy subjects [START_REF] Spampinato | Frequency-dependent modulation of cerebellar excitability during the application of non-invasive alternating current stimulation[END_REF], and low-frequency phase-locked cerebellar tACS has also been reported to decrease tremor amplitude in ET patients [START_REF] Schreglmann | Non-invasive suppression of essential tremor via phase-locked disruption of its temporal coherence[END_REF]. Conversely, cathodal cerebellar tDCS, thought to inhibit cerebellar cortical activity, does not significantly reduce tremor [START_REF] Gironell | Transcranial direct current stimulation of the cerebellum in essential tremor: a controlled study[END_REF]. Lastly, when applied over the cerebellum, 300 Hz tACS has been shown to increase cerebellum-brain inhibition [START_REF] Naro | Does Transcranial Alternating Current Stimulation Induce Cerebellum Plasticity? Feasibility, Safety and Efficacy of a Novel Electrophysiological Approach[END_REF], and 140 Hz tACS applied over M1 to increase cortical excitability [START_REF] Moliadze | Boosting brain excitability by transcranial high frequency stimulation in the ripple range[END_REF]. Finally, all these previous studies and our results suggest that high frequency tACS applied over the cerebellum may increase Purkinje cell excitability with, in consequence, a modulation of the CTC network oscillatory activity and tremor reduction.

This study has some limitations. First, we cannot fully exclude the possibility that our findings on VIM-DBS ET patients resulted from an artifact due to the setup used with the cerebellar tACS, since current may have flowed through the DBS electrode. However, the facts that kinetic and functional improvements persisted for up to 40 min after current offset, were also found in non-operated ET patients, and more pronounced for the arm ipsilateral to the stimulated cerebellum do not favour this hypothesis. Second, we had no direct evidence that allowed us to determine which part of the cerebellum could be affected by the tACS nor its physiological effects on cerebellar activity. In our protocol, we positioned stimulating electrodes as recommended in previous studies using DC stimulation [START_REF] Parazzini | Evaluation of the current density in the brainstem during transcranial direct current stimulation with extra-cephalic reference electrode[END_REF][START_REF] Parazzini | Modelling the electric field and the current density generated by cerebellar transcranial DC stimulation in humans[END_REF] and the effects observed across patients led us to suppose that the posterior cerebellum was actively modulated. Third, although a strength of the study was the sham-controlled design, sham-tACS was always applied prior to active-tACS which may have primed the effects of active-tACS, and we cannot totally rule out the possibility that patients felt some differences between sham-tACS and active-tACS without mentioning it. We chose this experimental design for the comfort of the patients as the anti-tremor drugs were stopped 7 days preceding the assessment for all patients and stimulation was turned OFF for several hours for VIM-DBS patients, with a return to the pre-operative tremor disability. In addition, our primary outcome was the comparison of active-tACS vs VIM-DBS, and not per se sham vs active-tACS. We also observed some significant changes in kinetics parameters in our control ET patients group with sham-tACS suggesting a possible placebo effect in these patients, as has been previously reported in ET patients using NIBS [28,[START_REF] Shin | Cerebellar repetitive transcranial magnetic stimulation for patients with essential tremor[END_REF]. However, the fact that we found no significant changes in blinded clinical scores after sham-tACS in these control ET patients and better effects after active-tACS for the ipsilateral arm, and no effect of sham tACS in the VIM-DBS group, may indicate that active tACS is indeed more efficient than sham-tACS to decrease tremor severity [START_REF] Shin | Cerebellar repetitive transcranial magnetic stimulation for patients with essential tremor[END_REF].

Conclusions

Overall, this proof-of-concept study supports the fact that high-frequency cerebellar tACS could modulate the CTC network, with objective changes in the postural and kinetic characteristics of essential tremor. In addition, it demonstrates the safety and potential benefits of high-frequency cerebellar tACS for ET patients. Further studies are needed to assess the value of repeated sessions for long-lasting effects, determine the optimal stimulation settings. This needs to be done in relation to individual tremor characteristics [START_REF] Schreglmann | Non-invasive suppression of essential tremor via phase-locked disruption of its temporal coherence[END_REF] and anatomical and functional cerebellar-cortex connectivity [START_REF] Hanajima | Cerebellar dysfunction in essential tremor[END_REF][START_REF] Popa | Cerebellar rTMS stimulation may induce prolonged clinical benefits in essential tremor, and subjacent changes in functional connectivity: an open label trial[END_REF], before we can propose this technique for patients with tremor insufficiently controlled by medical treatment.

Figure 1 .

 1 Figure 1. Study design

Figure 2 .

 2 Figure 2. Effects of active-and sham-tACS at high-frequency on postural tremor.

Figure 3 .

 3 Figure 3. Effects of active-and sham-tACS at high-frequency on action tremor.

Figure 4 .

 4 Figure 4. Effects of active-and sham-tACS at high frequency on clinical tremor severity.

Table 1 . Baseline demographic and clinical characteristics of patients with Essential tremor

 1 VIM-DBS: ventral intermediate thalamic deep brain stimulation, F: female, M: male, a OFF VIM-DBS for at least 3 hours for patients with VIM-DBS.

		Patient	Sex, age	Disease	Time since	FTM total	Medication	tACS	VIM-DBS settings
			(yrs)	duration (yrs)	surgery for VIM-	score	(dosage/day)	frequency	(contact/pulse
					DBS (months)	(baseline) a		(Hz)	width/amplitude)
	ET with	01	M, 61	10	48	41	_	180	1-2-C+/ 60s/3.8V
	VIM-DBS	02	M, 73	18	26	35	Propanolol 100 mg	130	1-2+/120s/3.5V
		03	M, 67	12	24	26	Topiramate 50 mg,	130	1-C+/60s/3.7V
							Primidone 250 mg
		04	F, 47	27	15	11	Propanolol 80 mg,	130	2-C+/60s/3.0V
							Gabapentin 300 mg
		05	M, 56	40	12	19	_	180	0-1+/ 90s/4V
		06	M, 29	12	10	17	_	130	0-C+/60s us/3.0V
		07	M, 69	49	30	36	_	130	2-C+/60s/2.8V
		08	M, 69	22	93	18	_	180	1-0+/60s/4.0V
		09	F, 41	26	12	28	Propanolol 40 mg,	130	0-C+/60s/2.8V
							Topiramate 25 mg,
							Gabapentin 300 mg
		10	F, 72	16	55	26	_	130	1-2+/60s/3.6V
		11	M, 44	34	36	42	Topiramate 50 mg	180	1-C+/60s/3.9V
		Mean (SD)	57.1 (14.8) 24.2 (12.6)	32.8 (24.9)	27.2 (10.3)
	ET	12	M, 68	46	_	34	Propanolol 120 mg,	130
	without						Gabapentin 600 mg
	VIM-DBS	13	M, 70	12	_	26	Propanolol 100 mg	130
		14	M, 74	46	_	23	Topiramate 50 mg,	130
							Primidone 50 mg
		15	M, 66	52	_	28	_	130
		16	M, 52	12	_	21	Propanolol 60 mg	130
		17	M, 71	17	_	19	Propanolol 40 mg	130
		18	M, 71	42	_	29	Propanolol 40 mg	130
		19	F, 71	15	_	23	Propanolol 100 mg	130
		20	M, 61	53	_	32	Propanolol 80 mg	130
		21	F, 42	12	_	33	_	130
		Mean (SD)	64.6 (10.2) 30.7 (18.3)	_	26.8 (5.2)

Table 2 . Effects of high-frequency VIM-DBS, active-tACS and sham-tACS of the cerebellum on clinical tremor severity in patients with Essential Tremor ON VIM-DBS a Baseline (OFF VIM-DBS a )

 2 Values are mean (SD); FTM=Fahn-Tolosa-Marin score, a for patients of the VIM-DBS group only. * p<0.05 relative to Baseline (OFF VIM-DBS) condition; †p<0.05 between Sham-and Active-tACS condition (linear mixed model).

	Sham-tACS	Active-tACS	Change between	Change between	Change between
			baseline and	baseline and	Active and
			Sham-tACS	Active-tACS	Sham-tACS

Olfati N, Shoeibi A, Abdollahian E, et al. Cerebellar repetitive transcranial magnetic

Acknowledgments

We extend our deepest thanks to the patients who participated in this study. We also thank the nurses and administrative staff of the Clinical Investigation Centre and S Mehdi for their help in the conduct of the study.

Funding sources of the study

This study was supported by a grant from the International Essential tremor foundation and 'Investissements d'avenir' (ANR-10-IAIHU-06). This study was sponsored by INSERM (C14-44).

2014;125:577-84. doi:S1388-2457(13)01105-X [pii] 10.1016/j.clinph.2013.09.039