domain: stat math math.math-pr

text:
Patrice Bertail

Gabriela Ciolek
email: gabrielaciolek@gmail.com

Gabriela Ciolek New Bernstein

Hoeffding 2018 Hal-01440167v2

New Bernstein and Hoeffding type inequalities for regenerative Markov chains

come

New Bernstein and Hoeffding type inequalities for regenerative Markov chains 1

Patrice Bertail * , Gabriela Cio lek * *

Introduction

Exponential inequalities are a powerful tool to control the tail probability that a random variable X exceeds some prescribed value t. They have been extensively investigated by many researchers due to the fact that they are a crucial step in deriving many results in numerous fields such as statistics, learning theory, discrete mathematics, statistical mechanics, information theory or convex geometry. There is a vast literature that provides a comprehensive overview of the theory of exponential inequalities in the i.i.d.

setting. An interested reader is referred to [START_REF] Bai | Probability Inequalities[END_REF], [START_REF] Boucheron | Inequalities. A Nonasymptotic Theory of Independence[END_REF] or van der Vaart and Wellner (1996).

The wealth of possible applications of exponential inequalities has naturally led to development of this theory in the dependent setting. In this paper we are particularly interested in results that establish exponential bounds for the tail probabilities of the additive functional of the regenerative Markov chain of the form

f (X 1 ) + • • • + f (X n ),
where (X n ) n∈N is a regenerative Markov chain. It is noteworthy that when deriving exponential inequalities for Markov chains (or any other process with some dependence structure) one can not expect to recover fully the classical results from the i.i.d. case. The goal is then to get some counterparts of the inequalities for i.i.d. random variables with some extra terms that appear in the bound as a consequence of a Markovian structure of the considered process.

In the recent years such (non-)asymptotic results have been obtained for Markov chains via many approaches: martingale arguments (see [START_REF] Glynn | Hoeffding?s Inequality for Uniformly Ergodic Markov Chains[END_REF], where Hoeffding's inequality for uniformly ergodic Markov chains has been presented), coupling techniques (see [START_REF] Chazottes | Concentration inequalities for Markov processes via coupling[END_REF] and [START_REF] Dedecker | Subgaussian concentration inequalities for geometrically ergodic Markov chains[END_REF]). In fact, [START_REF] Dedecker | Subgaussian concentration inequalities for geometrically ergodic Markov chains[END_REF] have proved that Hoeffding's inequality holds when the Markov chain is geometrically ergodic and thus weakened the assumptions imposed on the Markov chain in [START_REF] Glynn | Hoeffding?s Inequality for Uniformly Ergodic Markov Chains[END_REF]. Winterberger (2016) has generalized the result of [START_REF] Dedecker | Subgaussian concentration inequalities for geometrically ergodic Markov chains[END_REF] by showing that Hoeffding's inequality is valid also for unbounded functions of geometrically ergodic Markov chains provided that the sum is correctly self-normalized. [START_REF] Paulin | Concentration inequalities for Markov chains by Marton couplings and spectral methods[END_REF] has presented McDiarmid inequality for Markov chains using Merton coupling and spectral methods. [START_REF] Clémençon | Moment and probability inequalities for sums of bounded additive functionals of regular Markov chains via the Nummelin splitting technique[END_REF], [START_REF] Adamczak | A tail inequality for suprema of unbounded empirical processes with applications to Markov chains[END_REF], Bertail and Clémençon (2009), and [START_REF] Adamczak | Exponential concentration inequalities for additive functionals of Markov chains[END_REF] have obtained exponential inequalities for ergodic Markov chains via regeneration techniques (see [START_REF] Smith | Regenerative stochastic processes[END_REF]).

Regeneration techniques for Markov chains are particularly appealing to us mainly due to the fact that it requires much fewer restrictions on the ergodicity properties of the chain in comparison to alternative methods. In this paper we establish Hoeffding and Bernstein type of inequalities for statistics of the form

1 n ∑ n i=1 f (X i )
, where (X n ) n∈N is a regenerative Markov chain. We show that under proper control of the size of class of functions F (measured by its uniform entropy number), one can get non-asymptotic bounds on the suprema over the class of F of such empirical process for regenerative Markov chains. It is noteworthy that it is easy to generalize such results from regenerative case to the Harris recurrent one, using Nummelin extension of the initial chain (see [START_REF] Nummelin | A splitting technique for Harris recurrent chains[END_REF]).

The paper is organized as follows. In chapter 2 we introduce the notation and preliminary assumptions for Markov chains. We also recall some classical results from the i.i.d. setting which we generalize to the Markovian case. In chapter 3 we present the main results -Bernstein and Hoeffding type inequalities for regenerative Markov chains. The main ingredient to provide a crude exponential bound (with bad constants) is based on Montgomery-Smith which allows to reduce the problem on a random number of blocks to a fixed number of independent blocks. We then proposed a refined inequality by first controlling the the number of blocks in the inequality and then applying again Montgomery-Smith inequality on a remainder term. Next, we generalize these results and obtain Hoeffding and Bernstein type of bounds for suprema of empirical processes over a class of functions F. We also present the inequalities when the chain is Harris recurrent. Some technical parts of the proofs are postponed to the Appendix.

Preliminaries

We begin by introducing some notation and recall the key concepts of the Markov chains theory (see [START_REF] Meyn | Markov chains and stochastic stability[END_REF] for a detailed review and references). Let X = (X n ) n∈N be a positive recurrent, ψ-irreducible Markov chain on a countably generated state space (E, E) with transition probability Π and initial probability ν. We assume further, that X is regenerative (see [START_REF] Smith | Regenerative stochastic processes[END_REF]), i.e. there exists a measurable set A, called an atom, such that ψ(A) > 0 and for all (x, y) ∈ A 2 we have Π(x, •) = Π(y, •). We define the sequence of regeneration times (τ A (j)) j≥1 which is the sequence of successive points of time when the chain visits A and forgets its past. Throughout the paper we write τ A = τ A [START_REF] Adamczak | A tail inequality for suprema of unbounded empirical processes with applications to Markov chains[END_REF]. It is well-known that we can cut the sample path of the process into data segments of the form

B j = (X 1+τ A (j) , • • • , X τ A (j+1) ), j ≥ 1
according to consecutive visits of the chain to the regeneration set A. By the strong Markov property the blocks are i.i.d. random variables taking values in the torus ∪ ∞ k=1 E k . In the following, we assume that the mean inter-renewal time α = E A [τ A ] < ∞ and point out that in this case, the stationary distribution is a Pitman occupation measure given by

∀B ∈ E, µ(B) = 1 E A [τ A ] E A [ τ A ∑ i=1 I {X i ∈B} ]
,

where I B is the indicator function of the event B. Assume that we observe (X 1 , • • • , X n ). We introduce few more pieces of notation: throughout the paper we write l n = ∑ n i=1 I{X i ∈ A} for the total number of consecutive visits of the chain to the atom A, thus we observe l n + 1 data blocks. We make the convention that B (n) ln = ∅ when τ A (l n ) = n. Furthermore, we denote by l(B j ) = τ A (j + 1) -τ A (j), j ≥ 1, the length of regeneration blocks. Let f : E → R be µ-integrable function. In the following, we assume without loss of generality that µ(f ) = E µ [f (X 1 )] = 0. We introduce the following notation for partial sums of the regeneration cycles f

(B i ) = ∑ τ A (j+1) i=1+τ A (j) f (X i ).
Then, the regenerative approach is based on the following decomposition of the sum

∑ n i=1 f (X i ) : n ∑ i=1 f (X i ) = ln ∑ i=1 f (B i ) + ∆ n ,
where

∆ n = τ A ∑ i=1 f (X i ) + n ∑ i=τ A (ln-1) f (X i ).
We denote by

σ 2 (f ) = 1 E A (τ A ) E A ( τ A ∑ i=1 {f (X i ) -µ(f )} ) 2
the asymptotic variance.

For the completeness of the exposition, we recall now well-known classical results concerning some exponential inequalities for independent random variables. Firstly, we present the inequality for the i.i.d. bounded random variables due to [START_REF] Hoeffding | Probability inequalities for sums of bounded random variables[END_REF].

a i ≤ X i ≤ b i (i = 1, • • • , n), then for t > 0 P ( 1 n n ∑ i=1 X i -EX 1 ≥ t ) ≤ exp ( - 2t 2 ∑ n i=1 (b i -a i ) 2
) .

Below we recall the generalization of Hoeffding's inequality to unbounded functions. Interested reader, can find different variations of the following inequality (depending on imposed conditions on the random variables) in [START_REF] Boucheron | Inequalities. A Nonasymptotic Theory of Independence[END_REF]. Theorem 2.2 (Bernstein's inequality) Let X 1 , • • • , X n be independent random variables with expectation EX l for X l , l ≥ 1 respectively, such that, for all integers p ≥ 2,

E|X l | p ≤ p!R p-2 σ 2 l /2 for all l ∈ {1, • • • , n}.
Then, for all t > 0,

P ( n ∑ i=1 (X l -EX l ) ≥ t ) ≤ 2 exp ( - t 2 2(σ 2 + Rt)
)

,

where σ 2 = ∑ n i=1 σ 2 l . The purpose of this paper is to derive similar bounds for Markov chains using the nice regenerative structure of Markov chains.

Exponential inequalities for the tail probability for suprema of empirical processes for Markov chains

In the following, we denote f (x) = f (x) -µ(f ). Moreover, we write respec-

tively f (B 1 ) = ∑ τ A i=1 f (X i ) and | f |(B 1 ) = ∑ τ A i=1 | f |(X i
). We will work under following conditions. A1. (Bernstein's block moment condition) There exists a positive constant M 1 such that for any p ≥ 2 and for every f ∈ F

E A f (B 1 ) p ≤ 1 2 p!σ 2 (f )M p-2 1 . ( 1
)
A2. (Non-regenerative block exponential moment assumption) There exists

λ 0 > 0 such that for every f ∈ F we have E ν [ exp [ λ 0 ∑ τ A i=1 f (X i ) ]] < ∞.

A3. (Exponential block moment assumption) There exists λ

1 > 0 such that for every f ∈ F we have E A [ exp [ λ 1 f (B 1 ) ]] < ∞.
Remark 3.1 It is noteworthy to mention that assumption A1 implies the existence of an exponential moment of f (B 1 ) :

E A exp(λ f (B 1 )) ≤ exp ( λ 2 /2 1 -M 1 |λ| ) for all λ < 1 M 1 .
In this section, we formulate two Bernstein type inequalities for Markov chains, one is established via simple use of Montgomery-Smith inequality (see Montgomery-Smith (1993) and de la Peña, and Giné (1999)) which results in much larger constants (comparing to the i.i.d. setting) in the dominating parts of the bound. The second Bernstein's bound contains small constants in the main counterparts of the bound, however at a cost of having an extra term in the bound. Before we state the theorems, we will give a short discussion on already existing results for exponential inequalities for Markov chains.

Remarks 3.1 Since there is plenty of results concerning exponential inequal-

ities for Markov chains under many assumptions, it may be difficult to compare their strength (measured by assumptions imposed on the chain) and applicability. Thus, before we present the proofs of Theorem 3.2 and Theorem 3.3 , we make a short comparison of our result to already existing inequalities for Markov chains. We also strongly recommend seeing an exhaustive overview on the recent results of that type in [START_REF] Adamczak | Exponential concentration inequalities for additive functionals of Markov chains[END_REF].

The bounds obtained in this paper are related to the Fuk and Nagaev sharp bound inequality obtained in Bertail and Clémençon (2010). It is also based on the regeneration properties and decomposition of the chain. However, our techniques of proof differ and allow us to obtain a better rate in the main subgaussian part of the inequality under the hy-

potheses. The proofs of the inequalities are simplified and do not require the partitioning arguments which was used in [START_REF] Bertail | Sharp bounds for the tail of functionals of Markov chains[END_REF]. [START_REF] Dedecker | Subgaussian concentration inequalities for geometrically ergodic Markov chains[END_REF] and [START_REF] Chazottes | Concentration inequalities for Markov processes via coupling[END_REF] or any restrictions on the starting point of the chain as in [START_REF] Dedecker | Subgaussian concentration inequalities for geometrically ergodic Markov chains[END_REF]. Moreover, [START_REF] Adamczak | Exponential concentration inequalities for additive functionals of Markov chains[END_REF] use the assumption of strong aperiodicity for Harris Markov chain. We state a remark that this condition can be relaxed and we can only assume that Harris Markov chain is aperiodic (see Remark 3.9). [START_REF] Adamczak | A tail inequality for suprema of unbounded empirical processes with applications to Markov chains[END_REF], [START_REF] Clémençon | Moment and probability inequalities for sums of bounded additive functionals of regular Markov chains via the Nummelin splitting technique[END_REF], [START_REF] Dedecker | Subgaussian concentration inequalities for geometrically ergodic Markov chains[END_REF]). Our inequalities work for unbounded functions satisfying Bernstein's block moment condition. Moreover, all terms involved in our inequalities are given by explicit formulas. Thus, the results can be directly used in practical considerations. Note also that all the constants are given in simple, easy to interpret form and they do not depend on other underlying parameters.

It is noteworthy that we do not impose condition of stationarity of the considered Markov chain as in

Many results concerning exponential inequalities for Markov chains are established for bounded functions f (see for instance

Winterberger (2016) has established exponential inequalities in unbounded

case extending the result of [START_REF] Dedecker | Subgaussian concentration inequalities for geometrically ergodic Markov chains[END_REF] to the case when the chain can start from any x ∈ E. However, the constant involved in the bound of the Theorem 2.1 (obtained for bounded and unbounded functions) is very large. [START_REF] Boucheron | Inequalities. A Nonasymptotic Theory of Independence[END_REF]. As mentioned in the paper of Adamczak, there is many exponential inequalities that satisfy spectral gaps (see for instance Gao and Guillin, [START_REF] Lezaud | Chernoff and Berry-Esseen inequalities for Markov processes[END_REF]). Spectral gap inequalities allow to recover the Bernstein type inequality at its full strength. We need to mention that the geometric ergodicity assumption does not ensure in the non-reversible case that considered Markov chains admit a spectral gap (see Theorem 1.4 in [START_REF] Kontoyiannis | Geometric ergodicity and the spectral gap of non-reversible Markov chains[END_REF]).

We formulate a Bernstein type inequality for Markov chains below. Theorem 3.2 Assume that X = (X n ) n∈N is a regenerative positive recurrent Markov chain. Then, under assumptions A1 -A3, we have

P ν [ n ∑ i=1 f (X i ) -µ(f ) ≥ x ] ≤ 18 exp [ - x 2 2 × 90 2 (nσ 2 (f ) + M 1 x/90) ] + C 1 exp [ - λ 0 x 3 ] + C 2 exp [ - λ 1 x 3 ] ,
where

C 1 = E ν [ exp λ 0 τ A ∑ i=1 f (X i ) ] , C 2 = E A [ exp[λ 1 f (B 1 )]
] .

Remark 3.2 Observe that we do not impose a moment condition on E

A [τ A ] p < ∞ for p ≥ 2.
At the first glance, this might be surprising since one usually assumes the existence of E A [τ A ] 2 < ∞ when proving central limit theorem for regenerative Markov chains. A simple analysis of the proof of the central limit theorem in a Markovian case (see for instance [START_REF] Meyn | Markov chains and stochastic stability[END_REF])

reveals that it is sufficient to require only E A [τ A ] < ∞ when we consider cen- tered function f instead of f.
Proof. Firstly, we consider the sum of random variables of the following form

Z n ( f ) = ln ∑ i=1 f (B j ).
(2)

Furthermore, we have that

S n ( f ) = Z n ( f ) + ∆ n ( f ).
We recall, that l n is random and correlated with blocks itself. In order to apply Bernstein's inequality for i.i.d. random variables we apply the Montgomery-Smith inequality (see [START_REF] Montgomery-Smith | Comparison of sums of independent identically distributed random vectors[END_REF]) . It follows easily that

P A [ ln ∑ i=1 f (B i ) ≥ x/3 ] ≤ P A [ max 1≤k≤n k ∑ i=1 f (B i ) ≥ x/3 ] ≤ 9P A [ n ∑ i=1 f (B i ) ≥ x/90 ] (3)
and under Bernstein's condition A1 we obtain

P A [ n ∑ i=1 f (B i ) ≥ x/90 ] ≤ 2 exp [ - x 2 2 × 90 2 (M 1 x/90 + nσ 2 (f )) ] . ( 4
)
Next, we want to control the remainder term ∆ n .

∆ n = τ A ∑ i=1 f (X i ) + n ∑ i=τ A (ln-1) f (X i ). ( 5
)
The control of ∆ n is guaranteed by Markov's inequality, i.e.

P ν

[ τ A ∑ i=1 f (X i ) ≥ x 3 ] ≤ E ν [ exp λ 0 τ A ∑ i=1 f (X i ) ] exp [ - λ 0 x 3 ] .
We deal similarly with the last term of ∆ n . We complement the data 1 + τ A (l n ) + 1 by observations up to the next regeneration time 1 + τ A (l n + 1) and obtain

P ν   n ∑ i=1+τ A (ln)+1 f (X i ) ≥ x 3   ≤ P ν   n ∑ i=1+τ A (ln)+1 f (X i ) ≥ x 3   ≤ P ν   1+τ A (ln+1) ∑ i=1+τ A (ln)+1 f (X i ) ≥ x 3   ≤ E A [ exp[λ 1 f (B 1 )] ] exp [ - λ 1 x 3 ] .
We note that although the Montgomery-Smith inequality allows to obtain easily Bernstein's bound for Markov chains, the constants are rather large. Interestingly, under an additional assumption on E A [τ A ] p we can obtain the Bernstein type inequality for regenerative Markov chains with much smaller constants for the dominating counterparts of the bound.

A4. (Block length moment assumption)

There exists a positive constant M 2 such that for any p ≥ 2

E A [τ A ] p ≤ p!M p-2 2 E A [τ 2 A ] and E ν [τ A ] p ≤ p!M p-2 2 E ν [τ 2 A ].
Before we formulate Bernstein's inequality for regenerative Markov chains we introduce a lemma which provides a bound for tail probability of

√ n ( ln n -1 α )
which will be cruciall for the proof of Bernstein's bound but also may be of independent interest. Lemma 3.1 Suppose that condition A4 holds. Then

P ν ( n 1/2 ( l n n - 1 α ) ≥ x ) is bounded by exp ( - 1 2
(αx √ n -2α) 2 ( E ν τ 2 A + ( n α + x √ n)E A τ 2 A ) + (αx √ n + E ν τ A )M 2 ( E ν τ 2 A + ( n α + x √ n)E A τ 2 A ) 1/2
) .

Proof of Lemma 3.1 is postponed to the Appendix.

Remark 3.3 Note that when n → ∞, the dominating part in the exponential term is of order

1 2 α 2 x 2 E A τ 2 A /α + α 1/2 xM 2 (E A τ 2 A ) 1/2 + O(n -1/2 ) = 1 2 α 2 x 2 E A τ 2 A /α(1 + αxM 2 (E A τ 2 A /α) -1/2 ) + O(n -1/2 ) = 1 2 (αx) 2 / (E A τ 2 A /α) (1 + αxM 2 (E A τ 2 A /α) -1/2 ) + O(n -1/2 ),
thus we have a Gaussian tail with the right variance for moderate x and an exponential tail for large x and, in consequence, the constants are asymptotically optimal.

Now we are ready to state an alternative Bernstein type inequality for regenerative Markov chains, where under additional condition on the length of the blocks we can obtain much better inequality in terms of constants. Theorem 3.3 Assume that X = (X n ) n∈N is a regenerative positive recurrent Markov chain. Then, under assumptions A1-A4 we have for any a > 0, for x > 0 and N > 0

P ν [ n ∑ i=1 f (X i ) -µ(f ) ≥ x ] ≤ 2 exp [ -x 2 2 × 3 2 (1 + a) 2 (⌊ n α ⌋ σ 2 (f ) + M 1 3 x 1+a ) ] + 18 exp [ -x 2 2 × 90 2 (1 + a) 2 ( N √ nσ 2 (f ) + M 1 90 x 1+a ) ] + P ν ( n 1/2 [ l n n - 1 α ] > N ) + C 1 exp [ - λ 0 x 3 ] + C 2 exp [ - λ 1 x 3 ] , ( 6
)
where

C 1 = E ν [ exp λ 0 τ A ∑ i=1 f (X i ) ] , C 2 = E A [ exp[λ 1 f (B 1 )] ] .
Remark 3.4 If we choose N = log(n), then by Lemma 3.1 we can see that

P ν ( n 1/2 [ ln n -1 α ] ≥ log(n) ) = o ( 1 n
) and in that case the second term in [START_REF] Chazottes | Concentration inequalities for Markov processes via coupling[END_REF] remains small uniformly in x.

Proof. We start by the obvious observation that

P ν [ n ∑ i=1 f (X i ) -µ(f ) ≥ x ] ≤ P A [ ln ∑ i=1 f (B i ) ≥ x/3 ] + P ν [ τ A ∑ i=1 f (X i ) ≥ x/3 ] + P A   n ∑ i=τ A (ln-1) f (X i ) ≥ x/3   . ( 7
)
Remark 3.5 Instead of dividing x by 3 in [START_REF] Clémençon | Moment and probability inequalities for sums of bounded additive functionals of regular Markov chains via the Nummelin splitting technique[END_REF], one can use a different splitting to improve a little bit the final constants.

The bounds for the first and last non-regenerative blocks can be handled the same way as in Theorem 3.2. Next, we observe that, for any a > 0, we have

P A [ ln ∑ i=1 f (B i ) ≥ x/3 ] ≤ P A    ⌊ n α ⌋ ∑ i=1 f (B i ) ≥ x 3(1 + a)    + P A   ln 2 ∑ ln 1 f (B i ) ≥ x 3(1 + a)   , ( 8
)
where

l n 1 = min( ⌊ n α ⌋ , l n ) and l n 2 = max( ⌊ n α ⌋ , l n ). We observe that ∑ ⌊ n α ⌋ i=1 f (B i )
is a sum of independent, identically distributed and sub-exponential random variables. Thus, we can directly apply Bernstein's bound and obtain

P A    ⌊ n α ⌋ ∑ i=1 f (B i ) ≥ x 3(1 + a)    ≤ 2 exp [ -x 2 2 × 3 2 (1 + a) 2 (⌊ n α ⌋ σ 2 (f ) + M 1 x/3(1 + a) ) ] . ( 9
)
The control of

∑ ln 2 ln 1 f (B i
) is slightly more challenging due to the fact that l n is random and correlated with the blocks itself. In the following, we will make use of the Montgomery-Smith inequality. Notice however, that since we expect the number of terms in this sum to be at most of the order √ n, this term will be much more smaller than the leading term (9) and will be asymptotically negligible. We have

P A   ln 2 ∑ ln 1 f (B i ) ≥ x 3(1 + a)   ≤ P A   ln 2 ∑ ln 1 f (B i ) ≥ x 3(1 + a) , √ n [ l n n - 1 α ] ≤ N   + P ν ( √ n [ l n n - 1 α ] > N ) = A + B. ( 10
)
Firstly, we will bound term A in (10) using Montgomery-Smith inequality and the fact that if

√ n [ ln n -1 α ] ≤ N, then l n 1 -l n 2 ≤ √ nN. P A   ln 2 ∑ ln 1 f (B i ) ≥ x 3(1 + a) , √ n [ l n n - 1 α ] ≤ N   ≤ P A ( max 1≤k≤N √ n k ∑ i=1 f (B i ) ≥ x 3(1 + a) ) ≤ 9P A   N √ n ∑ i=1 f (B i ) ≥ x 90(1 + a)   ≤ 18 exp [ -x 2 2 × 90 2 (1 + a) 2 ( N √ nσ 2 (f ) + M 1 90 x 1+a ) ] .
Lemma 3.1 allows to control term B.

Maximal inequalities under uniform entropy

In empirical processes theory for processes indexed by class of functions, it is important to assess the complexity of considered classes. The information about entropy of F helps us to inspect how large our class is. Generally, control of entropy of certain classes may be crucial step when investigating asymptotic behaviour of empirical processes indexed by a class of functions. In our setting, we will measure the size of class of functions F via covering numbers and uniform entropy number. The following definition is due to [START_REF] Van Der Vaart | Weak Convergence and Empirical Processes With Applications to Statistics[END_REF].

Definition 3.4 (Covering and uniform entropy number)

The covering number N p (ϵ, Q, F) is the minimal number of balls {g : ∥g -f ∥ L p (Q) < ϵ} of radius ϵ needed to cover the set F. The entropy (without bracketing) is the logarithm of the covering number. We define uniform entropy number as

N p (ϵ, F) = sup Q N p (ϵ, Q, F)
, where the supremum is taken over all discrete probability measures Q.

In the following we state assumptions on the size of considered class of functions F. Rather than considering the assumptions A2 and A3, we impose the assumptions on the first and the last non-regenerative blocks for the envelope F of F. A2 ′ . (Non-regenerative block exponential moment assumption) There exists

λ 0 > 0 such that E ν [ exp [ 2λ 0 ∑ τ A i=1 F (X i ) ]] < ∞.
A3 ′ . (Exponential block moment assumption) There exists

λ 1 > 0 such that E A [ exp [ 2λ 1 F (B 1 ) ]] < ∞. A5. (Uniform entropy number condition) N 2 (ϵ, F) < ∞.
Before we formulate Bernstein deviation type inequality for unbounded classes of functions, we introduce one more piece of notation, let

σ 2 m = max f ∈F σ 2 (f ) > η > 0.
Theorem 3.5 Assume that X = (X n ) n∈N is a regenerative positive recurrent Markov chain. Then, under assumptions A1, A2 ′ , A3 ′ and A5 and for any 0 < ϵ < x and for n large enough we have

P ν [ sup f ∈F n ∑ i=1 f (X i ) -µ(f ) ≥ x ] ≤ N 2 (ϵ, F) { 18 exp [ - (x -2ϵ) 2 2 × 90 2 (nσ 2 m + M 1 (x -2ϵ)/90) ] +C 1 exp [ λ 0 (x -2ϵ) 3 ] + C 2 exp [ - λ 1 (x -2ϵ) 3 ]} , ( 11
)
where

C 1 = E ν [ exp 2λ 0 τ A ∑ i=1 F (X i ) ] , C 2 = E A [exp[2λ 1 |F | (B 1 )]]
and F is an envelope function for F.

Before we proceed with the proof of Theorem 3.

||f || C P (E ′ ) = sup x∈E ′ |f (x)| + sup x 1 ∈E ′ ,x 2 ∈E ′ ( f (x 1 ) -f (x 2 ) d(x 1 , x 2 ) p ) then we have M = sup x∈X F (x) < ∞ as well as L = sup f,g∈F ,f ̸ =g sup z |f (z)-g(z)| ||f -g|| C P (E ′ )
< ∞ so that we can directly control the empirical sum by the obvious inequality

sup f,g∈F 1 n n ∑ i=1 f (X i ) -g(X i ) ≤ L||f -g|| C P (E ′ ) .

It follows that if we replace the notion of uniform covering number N 2 (ε, F)

with respect to the norm ∥.∥ L 2 (Q) by the covering numbers N C p (ε, F) with respect to ||.|| C P (E ′ ) , then the results hold true for any n, provided that N 2 (ε, F) is replaced by N C p ( ε L , F) in the inequality. Proof of Theorem 3.5.

We choose functions

g 1 , g 2 , • • • , g M , where M = N 2 (ϵ, F) such that min j Q|f -µ(f ) -g j + µ(g 1 )| ≤ ϵ for each f ∈ F ,
where Q is any discrete probability measure. We also assume that g 1 , g 2 , • • • , g M belong to F and satisfy conditions A1, A2 ′ , A3 ′ . We write f * for the g j , where the minimum is achieved. Our further reasoning is based on the following remarks.

Remark 3.7 Let f, g be functions with the expectations µ(f ), µ(g) respectively. Then,

∥f -µ(f ) -g + µ(g)∥ L 2 ≤ ∥f -g∥ L 2 + ∥µ(f ) -µ(g)∥ L 2 ≤ 2∥f -g∥ L 2 .
In our reasoning, we will also make use of the following remark.

Remark 3.8 Assume that the functions f, g ∈ F and ∥f -g∥ 2,Pn < ϵ. Then, for n large enough (depending only on ϵ),

P(f -g) 2 ≤ P n (f -g) 2 + |(P n -P)(f -g) 2 | ≤ 2ϵ 2
since uniformly |(P n -P)(f -g) 2 | ≤ ϵ 2 by the uniform strong law of large numbers for regenerative Markov chains (see Theorem 3.6 from [START_REF] Levental | Uniform limit theorems for Harris recurrent Markov chains[END_REF]).

As a consequence, any ϵ-net in L 2 (P n ) is also √ 2ϵ-net in L 2 (P) (see also [START_REF] Kosorok | Introduction to empirical processes and semiparametric inference[END_REF], page 151 for some refinements in the i.i.d. case). Moreover, note that ∃ N such that ∀ n ≥ N ∥g i -g j ∥ 2,P ≤ ϵ and we have

∥g i -g j ∥ 2,Pn -∥g i -g j ∥ 2,P + ∥g i -g j ∥ 2,P ≤ 2ϵ.
Next, by the definition of uniform numbers and the Remark 3.8, we obtain

P ν [ sup f ∈F 1 n n ∑ i=1 (f (X i ) -µ(f )) ≥ x ] ≤ P ν { sup f ∈F [ 1 n n ∑ i=1 |f (X i ) -µ(f ) -f * (X i ) + µ(f * ) + 1 n n ∑ i=1 |f * (X i ) -µ(f * )| ] ≥ x } ≤ P ν [ max j∈{1,••• ,N 2 (ϵ,F)} 1 n n ∑ i=1 g j (X i ) -µ(g 1 ) ≥ x -2ϵ ] ≤ N 2 (ϵ, F) max j∈{1,••• ,N 2 (ϵ,F)} P ν { 1 n n ∑ i=1 g j (X i ) -µ(g 1 ) ≥ x -2ϵ } .
We set the notation that g j = g j -µ(g 1 ).

In what follows, our reasoning is analogous as in the proof of Theorem 3.2. Instead of taking any f ∈ F , we work with the functions g j ∈ F. Thus, we consider now the processes

Z n (g j ) = ln ∑ i=1 g j (B i ) (12)
and S n (g j ) = Z n (g j ) + ∆ n (g j ).

Under the assumptions A1, A2 ′ and A3 ′ for g j , we get the analogous to that from Theorem 3.2 Bernstein's bound for Z n (g j ), namely

P A [ ln ∑ i=1 g j (B i ) ≥ x -ϵ ] ≤ 18 exp [ - (x -2ϵ) 2 2 × 90 2 (nσ 2 (g 1 ) + M 1 (x -2ϵ)/90) ] . (13)
We find the upper bound for the remainder term ∆ n (g j ) applying the same reasoning as in Theorem 3.2. Thus,

P ν [ τ A ∑ i=1 g j (X i ) ≥ x -2ϵ 3 ] ≤ C 1 exp [ - λ 0 (x -2ϵ) 3 ] (14)
and

P A   n ∑ i=τ A (ln-1) ḡj (X i ) ≥ x -2ϵ 3   ≤ C 2 exp [ - λ 1 (x -2ϵ) 3 ] , ( 15
)
where

C 1 = E ν [ exp λ 0 τ A ∑ i=1 g j (X i ) ] , C 2 = E A [ exp[λ 1 g j (B 1 )] ] .
Finally, notice that

E ν [ exp λ 0 τ A ∑ i=1 g j (X i ) ] ≤ E ν [ exp 2λ 0 τ A ∑ i=1 F (X i ) ] < ∞ and E A [ exp[λ 1 g j (B 1 )] ] ≤ E A [exp[2λ 1 |F | (B 1 )]] < ∞
and insert it into ( 14) and ( 15) which yields the proof. Below we will formulate a maximal version of Theorem 3.3. Theorem 3.6 Assume that X = (X n ) n∈N is a regenerative positive recurrent Markov chain. Then, under assumptions A1, A2 ′ , A3 ′ , A4 -A5 and for any 0 < ϵ < x and for n large enough and N > 0 we have

P ν [ sup f ∈F n ∑ i=1 f (X i ) -µ(f ) ≥ x ] ≤ N 2 (ϵ, F) { 2 exp [ -(x -2ϵ) 2 2 × 3 2 (1 + a) 2 (⌊ n α ⌋ σ 2 (f ) + M 1 3 x-2ϵ 1+a ) ] + 18 exp   -(x -2ϵ) 2 2 × 90 2 (1 + a) 2 ( N √ nσ 2 (f ) + M 1 90 (x-2ϵ) 1+a )   +P ν ( n 1/2 [ l n n - 1 α ] > N ) + C 1 exp [ - λ 0 (x -2ϵ) 3 ] + C 2 exp [ - λ 1 (x -2ϵ) 3 ]} ,
where

C 1 = E ν [ exp 2λ 0 τ A ∑ i=1 F (X i ) ] , C 2 = E A [exp[2λ 1 |F | (B 1 )]] .
Proof. The proof is a combination of the proofs of Theorem 3.3 and Theorem 3.5. We deal with the supremum over F the same way as in Theorem 3.5. Then we apply Theorem 3.3.

We can obtain even sharper upper bound when class F is uniformly bounded. In the following, we will show that it is possible to get a Hoeffding type inequality and have a stronger control of moments of the sum S n (f ) which is a natural consequence of uniform boundedness assumption imposed on F.

A6. The class of functions F is uniformly bounded, i.e. there exists a constant D such that ∀f ∈ F |f | < D.

Theorem 3.7 Assume that X = (X n ) n∈N is a regenerative positive recurrent Markov chain. Then, under assumptions A1, A2 ′ , A3 ′ , A5 -A6 and for any 0 < ϵ < x, we have for n large enough

P ν [ sup f ∈F n ∑ i=1 f (X i ) -µ(f ) σ(f ) ≥ x ] ≤ N 2 (ϵ, F) { 18 exp [ - (x -2ϵ) 2 2n × 90 2 D 2 ] +C 1 exp [ - λ 0 (x -2ϵ) 3 ] + C 2 exp [ - λ 1 (x -2ϵ) 3 ]} , ( 16
)
where

C 1 = E ν exp |2λ 0 τ A D| , C 2 = E A exp |2λ 1 l(B 1 )D| .
Proof. The proof bears resemblance to the proof of Theorem 3.5, with few natural modifications which are a consequence of the uniform boundedness of F.

Under additional condition A4 we can obtain easily the bound with smaller constants, we follow the analogous way as in Theorem 3.6.

General Harris recurrent case

It is noteworthy that Theorems 3.2, 3.5, 3.7 are also valid in Harris recurrent case under slightly modified assumptions. It is well known that it is possible to retrieve all regeneration techniques also in Harris case via the Nummelin splitting technique which allows to extend the probabilistic structure of any chain in order to artificially construct a regeneration set. The Nummelin splitting technique relies heavily on the notion of small set. For the clarity of exposition we recall the definition.

Definition 3.8

We say that a set S ∈ E is small if there exists a parameter δ > 0, a positive probability measure Φ supported by S and an integer m ∈ N * such that

∀x ∈ S, B ∈ E Π m (x, B) ≥ δ Φ(B), (17)
where Π m denotes the m-th iterate of the transition probability Π.

We expand the sample space in order to define a sequence (Y n ) n∈N of independent r.v.'s with parameter δ. We define a joint distribution P ν,M of

X M = (X n , Y n ) n∈N . The construction relies on the mixture representation of Π on S, namely Π(x, B) = δΦ(B) + (1 -δ) Π(x,B)-δΦ(B) 1-δ
. It can be retrieved by the following randomization of the transition probability Π each time the chain X visits the set S. If X n ∈ S and • if Y n = 1 (which happens with probability δ ∈ ]0, 1[), then X n+1 is distributed according to the probability measure Φ,

• if Y n = 0 (that happens with probability 1-δ), then X n+1 is distributed according to the probability measure (1

-δ) -1 (Π(X n , •) -δΦ(•)).
This bivariate Markov chain X M is called the split chain. It takes its values in E × {0, 1} and possesses an atom, namely A = S × {1}. The split chain X M inherits all the stability and communication properties of the chain X. The regenerative blocks of the split chain are i.i.d. (in case m = 1 in ( 17)) (see [START_REF] Meyn | Markov chains and stochastic stability[END_REF] for further details). We will formulate a Bernstein type inequality for unbounded classes of functions in the Harris recurrent case (equivalent of Theorem 3.2). Theorems 3.5 and 3.7 can be reformulated for Harris chains in similar way. We impose the following conditions: AH1. (Bernstein's block moment condition) There exists a positive constant M 1 such that for any p ≥ 2 and for every f ∈ F

sup y∈S E y f (B 1 ) p ≤ 1 2 p!σ 2 (f )M p-2 1 . ( 18
)
AH2. (Non-regenerative block exponential moment assumption) There exists a constant λ 0 > 0 such that for every

f ∈ F we have E ν [ exp λ 0 ∑ τ S i=1 f (X i ) ] < ∞.

AH3. (Exponential block moment assumption) There exists a constant λ

1 > 0 such that for every f ∈ F we have sup y∈S E y [ exp[λ 1 f (B 1 )] ] < ∞. Let sup y∈S E y [τ S ] = α M < ∞.
We are ready to formulate a Bernstein type inequality for Harris recurrent Markov chains.

Theorem 3.9 Assume that X M is a Harris recurrent, strongly aperiodic Markov chain. Then, under assumptions AH1-AH3, we have

P ν [ n ∑ i=1 f (X i ) -µ(f ) ≥ x ] ≤ 18 exp [ - x 2 2 × 90 2 (nσ 2 (f ) + M 1 x/90) ] + C 1 exp [ - λ 0 x 3 ] + C 2 exp [ - λ 1 x 3 ] , (19)
where

C 1 = E ν [ exp λ 0 τ S ∑ i=1 f (X i ) ] , C 2 = sup y∈S E y [ exp[λ 1 f (B 1 )
] .

The proof of Theorem 3.9 is analogous to the proof of Theorem 3.

[ n ∑ i=1 f (X i ) -µ(f ) ≥ x ] ≤ 2 exp [ -x 2 2 × 3 2 (1 + a) 2 (⌊ n α ⌋ σ 2 (f ) + M 1 3 x 1+a ) ] + 18 exp [ -x 2 2 × 90 2 (1 + a) 2 ( N √ nσ 2 (f ) + M 1 90 x 1+a ) ] + P ν [ n 1/2 ( l n n - 1 α ) > N ] + C 1 exp [ - λ 0 x 3 ] + C 2 exp [ - λ 1 x 3 ] ,
where

C 1 = E ν [ exp λ 0 τ S ∑ i=1 f (X i ) ] , C 2 = sup y∈S E y [ exp[λ 1 f (B 1 )] ] .
. Remark 3.9 In the Theorem 3.9 we assumed that X M is strongly aperiodic.

It is easy, however, to relax this assumption and impose only the aperiodicity condition on Harris chain by using the same trick as in [START_REF] Levental | Uniform limit theorems for Harris recurrent Markov chains[END_REF]. Note that if X M satisfies M(m, S, δ, Φ) for m > 1, then the blocks of data are 1dependent. Denote by S = S ∪ { * }, where { * } is an ideal point which is not in S. Next, we define a pseudo-atom α M = S × {1}. In order to impose only and if 0 < x ≤ √ n(1 -α -1 ), then

P ν ( n 1/2 ( l n n - 1 α ) ≥ x ) = P ν ( l n ≥ n α + x √ n ) ≤ P ν ( l n ≥ [ n α + x √ n ]) ≤ P((∆τ 1 -E ν τ A ) + [ n α +x √ n] ∑ i=2 (∆τ i -α) ≤ n -([ n α + x √ n] -1)α -E ν τ A ),
where [.] is the integer part.

Since

n α + x √ n -1 ≤ [ n α + x √ n] ≤ n α + x √ n, we get n -([ n α + x √ n] -1)α -E ν τ A ) ≤ n -( n α + x √ n -2)α -E ν τ A = -αx √ n + 2α -E ν τ A .
It follows that

P ν ( n 1/2 ( ln n -1 α ) ≥ x ) ≤ P ( (∆τ 1 -E ν τ A ) + ∑ [ n α +x √ n] i=2 (∆τ i -α) ≤ -αx √ n + 2α -E ν τ A ) ,
where [.] is the integer part Now, we can apply any Bennett's or Bernstein's inequality on these centered i.i.d. random variables to get an exponential bound. This can be done since we assumed A4. Note that other bounds (polynomial for instance) can be obtained under appropriate modifications of A4. In our case we get

P((∆τ 1 -E ν τ A ) + [ n α +x √ n] ∑ i=2 (∆τ i -α) ≤ -αx √ n + 2α -E ν τ A ) ≤ exp ( - 1 2 (αx √ n -2α + E ν τ 2 A )/S 2 n 1 + (αx √ n -2α + E ν τ A )M 2 /S n ) ,
where

S 2 n = E ν τ 2 A + ([ n α + x √ n] -1)E A τ 2 A .
The above bound can be reduced to exp

( - 1 2 (αx √ n -2α) 2 ( E ν τ 2 A + ( n α + x √ n)E A τ 2 A ) + (αx √ n + E ν τ A )M 2 ( E ν τ 2 A + ( n α + x √ n)E A τ 2 A ) 1/2
)

Theorem 2 . 1 (

 21 Hoeffding's inequality) Let X 1 , X 2 , • • • , X n be independent identically distributed random variables with common expectation EX 1 and such that

  Notice that our bound is a deviation bound in that it holds only for n large enough. This is due to the control of the covering functions

	Remark 3.6 (under P n ) by a control under P (see Remark 3.8 in the proof ). However, by
	making additional assumptions on the regularity of the class of functions and
	by choosing the adequate norm, it is possible to obtain by the same arguments
	an exponential inequality valid for any n as in Zou, Zhang and Xu (2009)
	or Cucker and Smale (2002). See also examples of such classes of functions
	used in statistical learning in this latter. Indeed, if F belongs to a ball of
	a Hölder space C
	5, we indicate that under
	additional assumptions it is possible to obtain Bernstein type concentration
	inequality.

P (E ′ ) on a compact set E ′ of an Euclidean space endowed with the norm

  2. We can obtain a bound with much smaller constants under an extra block moment condition. AH4. (Block length moment assumption) There exists a positive constant M 2 such that for any p ≥ 2 Assume that X M is a Harris recurrent, strongly aperiodic Markov chain. Then, under assumptions AH1-AH4, we have for any x > 0

	sup y∈S	E y [τ S ] p ≤ p!M p-2 2	E A τ 2 A ,
	E ν [τ S ] p ≤ p!M p-2 2	E ν τ 2 S .
	.
	Theorem 3.10 and for N ∈ R
	P ν

Keywords : uniform entropy, exponential inequalities, empirical processes indexed by classes of functions, regenerative Markov chain. Primary Class : 62G09, Secondary Class : 62G20, 60J10

Acknowledgment

This research was supported by a public grant as part of the Investissement d'avenir, project reference ANR-11-LABX-0056-LMH. The work was also supported by the Polish National Science Centre NCN ( grant No. UMO2016/23/N/ST1/01355 ) and (partly) by the Ministry of Science and Higher Education. This research has been conducted as part of the project Labex MME-DII (ANR11-LBX-0023-01).

aperiodicity in this case it is sufficient to consider two processes {E i } and

, for some k ≥ 0 and E i = * . Every function f : S → R will be considered as defined on S with identification f ( * ) = 0 (see also [START_REF] Levental | Uniform limit theorems for Harris recurrent Markov chains[END_REF] for more details concerning those two processes). Then, we prove Bernstein type of inequality similarly as we prove Theorems 3.2 and 3.9 applying all the reasoning to {E i } and {O i } separately, yielding to a similar inequality up to an additional multiplicative constant 2.

Appendix

Proof of Lemma 3.1. Let τ k be the time of the k-th visit to the atom A (S × {1} in the general case).

In the following we make use of the argument from [START_REF] Dedecker | Subgaussian concentration inequalities for geometrically ergodic Markov chains[END_REF] and observe that we have for any k ≤ n


MD Michael Benzaquen
email: michael.benzaquen@ap-hm.fr.

MD Luca Borradori

MD Philippe Berbis

MS Simone Cazzaniga

Rene Valero

MD Marie-Aleth Richard

MD, PhD Laurence Feldmeyer

MD Michaelbenzaquen

MD Ren Evalero

Dipeptidyl peptidase IV inhibitors, a risk factor for bullous pemphigoid: Retrospective multicenter

Keywords: bullous pemphigoid, case-control study, diabetes, dipeptidyl peptidase-4 inhibitor, gliptin, risk factor Boldface indicates statistical significance. CI, Confidence interval, DPP4i, dipeptidyl peptidase-4 inhibitor, OR, odds ratio. *Univariate conditional logistic regression analysis. y Multivariable

published or not. The documents may come

B ullous pemphigoid (BP) is the most frequent autoimmune subepidermal blistering disease that typically affects the elderly. Its cutaneous manifestations are polymorphic, ranging from pruritus with excoriated, eczematous, papular, and/ or urticaria-like lesions in the nonbullous phase to vesicles and bullae in the bullous phase. [START_REF] Joly | A comparison of two regimens of topical corticosteroids in the treatment of patients with bullous pemphigoid: a multicenter randomized study[END_REF] BP is associated with an immune response directed against 2 molecules, the BP antigen 180 (BP180 [also called BPAG2]) and the BP antigen 230 (BP230 [also called BPAG1]). [START_REF] Feliciani | Management of bullous pemphigoid: the European Dermatology Forum consensus in collaboration with the European Academy of Dermatology and Venereology[END_REF] Since the publication of the first case of BP associated with sulfasalazine in 1970, a wide range of drugs (spironolactone, furosemide, chloroquine, b-blockers, and several antibiotics) have been associated with the disease. [START_REF] Bastuji-Garin | Drugs associated with bullous pemphigoid. A case-control study[END_REF] Recently, several cases of BP have been reported in association with dipeptidyl peptidase-4 inhibitors (DPP4is), which are also known as gliptins. [START_REF] B En E | Bullous pemphigoid and dipeptidyl peptidase IV inhibitors: a case-noncase study in the French Pharmacovigilance Database[END_REF][START_REF] Garc Ia | Dipeptidyl peptidase-IV inhibitors induced bullous pemphigoid: a case report and analysis of cases reported in the European pharmacovigilance database[END_REF][START_REF] Keseroglu | A case of bullous pemphigoid induced by vildagliptin[END_REF][START_REF] Haber | Bullous pemphigoid associated with linagliptin treatment[END_REF][START_REF] Pasmatzi | Dipeptidyl peptidase-4 inhibitors cause bullous pemphigoid in diabetic patients: report of two cases[END_REF][START_REF] Skandalis | Drug-induced bullous pemphigoid in diabetes mellitus patients receiving dipeptidyl peptidase-IV inhibitors plus metformin[END_REF][START_REF] Aouidad | A case report of bullous pemphigoid induced by dipeptidyl peptidase-4 inhibitors[END_REF][START_REF] Attaway | Bullous pemphigoid associated with dipeptidyl peptidase IV inhibitors. A case report and review of literature[END_REF][START_REF] B En E | Bullous pemphigoid induced by vildagliptin: a report of three cases[END_REF][START_REF] Mendonc ¸a | Three cases of bullous pemphigoid associated with dipeptidyl peptidase-4 inhibitorsdone due to linagliptin[END_REF] DPP4is are oral antihyperglycemic drugs administered to patients with type 2 diabetes as monotherapy or in combination with other oral antihyperglycemic medications or insulin. DPP4 is an enzyme that inactivates incretins (glucagon-like peptide-1 and glucose-dependent insulinotropic polypeptide). DPP4is increase levels of incretins, thereby increasing insulin secretion, decreasing glucagon secretion, and improving glycemic control. Sitagliptin was first approved in 2006 by the US Food and Drug Administration, followed by saxagliptin (in 2009), linagliptin (in 2011), and alogliptin (in 2013). Three DPP4is are currently available on the French marketdsitagliptin and vildagliptin (since 2007) and saxagliptin (since 2009)dand 5 are available on the Swiss marketdthe 3 aforementioned DPP4is and linagliptin (since 2011) and alogliptin (since 2013)dboth of which are available only on the Swiss market. They are used alone or in association with metformin in the same tablet. [START_REF] B En E | Bullous pemphigoid and dipeptidyl peptidase IV inhibitors: a case-noncase study in the French Pharmacovigilance Database[END_REF] An increasing number of clinical reports and pharmacovigilance database analyses suggesting an association between DPP4i intake and BP have been published. Nevertheless, this has not been confirmed by a well-designed controlled study.

The main objective of our case-control study was therefore to retrospectively evaluate the association between DPP4i treatment and development of BP. The secondary end points were to determine a potential higher association for a specific DPP4i and to evaluate the disease course after DPP4i withdrawal.

MATERIALS AND METHODS

The investigations were conducted as a retrospective case-control study with a 1:2 design, comparing case patients with BP and diabetes with age-and sexmatched controls with type 2 diabetes from January

Data collection for cases and controls

The study was conducted in 3 university dermatologic departments (Bern, Marseille Nord, and Marseille La Timone). By using the database of the respective histopathology departments and clinical records, we identified all patients with BP diagnosed for the first time between January 1, 2014, and July 31, 2016. The diagnosis of BP was based on the following criteria developed by the French Bullous Study Group 14 : consistent clinical features, compatible histopathology findings, positive direct immunofluorescence studies, and in some cases, positive indirect immunofluorescence microscopy studies and/or positive enzyme-linked immunosorbent assayeBP180/ enzyme-linked immunosorbent assayeBP230 (MBL International, Japan). Among these patients with BP, we identified those having type 2 diabetes.

For these patients, we recorded age, sex, date of BP diagnosis, treatment of BP (with topical steroids, systemic corticosteroids, immune suppressors, or other treatments such as doxycycline or dapsone), evolution of BP (complete remission, partial remission, relapse, or death), comorbidities (including rheumatic, neurologic, cardiovascular, or digestive diseases and neoplasia), treatment with DPP4is, and other cotreatments (including diuretics, antibiotics, neuroleptics, nonsteroidal anti-inflammatory drugs, and antihypertensive drugs).

If a DPP4i was mentioned in the medical record, we examined the type of DPP4i, the chronology between BP diagnosis and onset of the DPP4i treatment, and the evolution after DPP4i withdrawal. Patients who had other autoimmune bullous diseases or did not otherwise fulfill the inclusion criteria were not included.

The control patients were obtained between January 1, 2014, and July 31, 2016, from the endocrinology departments of the same hospitals. For each case, 2 control patients with diabetes visiting the endocrinology department in the same 6-month period and matched to the case by sex and quinquennium of age were then randomly selected from all available patients satisfying the matching criteria. The patient files were reviewed for treatment of diabetes (specifically, the use of DPP4is), other cotreatments, and comorbidities. For the controls, we did not include case patients with any chronic skin diseases, including bullous dermatosis, at the time of the study.

We then compared exposure to DPP4is between case patients and controls with adjustment for potential confounders.

Statistical analysis

Descriptive data were presented as number with percentages or means with standard deviations (SDs) for categoric and continuous variables, respectively. The Mann-Whitney U test was used to assess possible residual differences in the distribution of age between case patients and controls. Differences between case patients and matched controls across different levels of other factors were assessed by means of univariate conditional logistic regression analysis. Factors associated with DPP4i use were also investigated by means of the Pearson x 2 test or Fisher exact test, where required.

All factors with a P value less than .10 in the univariate case-control analysis and associated with DPP4i use, with a P value less than .10 at univariate level, were evaluated as possible confounding factors in multivariate conditional logistic regression models with backward stepwise selection algorithm. The factors retained for adjustment were neurologic and metabolic/endocrine comorbidities, as well as other dermatologic conditions unrelated to BP. The effect of DPP4i use on BP onset in diabetic patients was expressed in terms of an odds ratio (OR) along with its 95% confidence interval (CI) and P value. A stratified analysis by possible effect modifiers, including sex and age group, was also performed. All tests were considered statistically significant at a P value less than .05.

Before starting the study, we planned to recruit at least 183 patients (61 case patients and 122 controls) to detect an OR higher than 2.5 in a 1:2 matched case-control design, supposing to observe a 30% exposure to DPP4i use in the control group (a = 0.05, b = 0.20, multiple correlation coefficients \0.2). Analyses were carried out with SPSS software (version 20.0, IBM Corp, Armonk, NY).

RESULTS

From January 2014 to July 2016, BP was diagnosed in 165 patients (61 in Bern, 47 in Marseille Nord, and 57 in Marseille La Timone). Among these, 61 had diabetes (22 in Bern, 14 in Marseille Nord, and 25 in Marseille La Timone). We collected 2 matched controls for each case patient, resulting in a total of 122 controls.

Of the case patients, 50.8% were female, and the mean age was 79.1 plus or minus 7.0 years. The main comorbidities of cases were cardiovascular (86.9%), neurologic (52.5%), and metabolic and endocrine diseases other than diabetes (39.3%) and uronephrologic diseases (39.3%) (Table I).

In our 3 investigational centers, we collected 28 patients with diabetes and BP who were taking a DPP4i. DPP4is were used more frequently in case patients with BP (45.9%) than in controls (18%), and the difference was statistically significant (P \.001). Of the specific DDP4is, vildagliptin was more common in case patients (23%) than in controls (4.1%). For the other cotreatments, there was no statistical difference between case patients and controls, except for the use of antihistamines (P \ .001). There were no differences in other antidiabetic medications, including metformin, between case patients and controls (P = .08) (Table II).

All patients with BP were treated with high-potency topical steroids as first-line treatment. Systemic corticosteroids were used in half of them (50.8%), immunosuppressive agents in 32.8%, and other treatments such as doxycycline or dapsone in 34.4%. With treatment, 37.7% went into complete remission and 42.6% went into partial remission. Finally, there were no differences in treatment between the patients with diabetes and BP who had taken a DPP4i and the patients with diabetes and BP who had not taken a DPP4i (data not shown), an observation suggesting that presentation and initial severity of BP in these 2 groups were similar.

Abbreviations used:

BP: bullous pemphigoid CI: confidence interval DPP4i: dipeptidyl peptidase-4 inhibitor OR:

odds ratio SD:

standard deviation

DPP4is and BP

The univariate analysis of the association between DPP4i use and BP in diabetic patients yielded an OR of 3.45 (95% CI, 1.76-6.77; P\.001). After adjustment for possible confounding factors associated with BP onset and DPP4i use in multivariate analysis, the OR was 2.64 (95% CI, 1.19-5.85; P = .02) (Table III).

A more detailed analysis of DPP4i use revealed a higher association for vildagliptin, with a crude OR of 7.23 (95% CI, 2.44-21.40; P = .001) and an adjusted OR of 3.57 (95% CI, 1.07-11.84; P = .04). The study was underpowered to detect differences between other DPP4is, with linagliptin and alogliptin being only used in the Swiss cases.

Sex-stratified analysis indicated that the effect of a DPP4i on BP onset was higher in males (adjusted OR, 4.36; 95% CI, 1.38-13.83; P = .01) than in females (adjusted OR, 1.64; 95% CI, 0.53-5.11; P = .39). Age groupestratified analyses showed a stronger association for patients age 80 years or older, with an adjusted OR of 5.31 (95% CI, 1.60-17.62; P = .006).

Clinical course of patients with BP under treatment with a DPP4i

In our 3 centers, we collected a total of 28 patients with diabetes who developed BP under exposure to a DPP4i. The duration of DPP4i use before onset of BP ranged from 10 days to 3 years (median, 8.2 months).

Drug withdrawal was performed in 19 patients upon suspected DPP4i-associated BP. Complete (11 of 19 [58%]) or partial (7 of 19 [37%]) remission with some mild persistent disease was obtained for all patients but 1 (duration of follow up, 3-30 months; median, 16.4 months). First-line treatment was high-potency topical steroids and systemic corticosteroids in severe or refractory cases followed by a standard tapering schedule. [START_REF] Feliciani | Management of bullous pemphigoid: the European Dermatology Forum consensus in collaboration with the European Academy of Dermatology and Venereology[END_REF][START_REF] Joly | A comparison of oral and topical corticosteroids in patients with bullous pemphigoid[END_REF] No further therapy was necessary in these patients after DPP4i withdrawal to obtain BP remission. For 1 patient, sitagliptin was initially stopped, leading to a partial remission, but its reintroduction combined with metformin led to a relapse of the BP. Definitive discontinuation of sitagliptin and its replacement by repaglinide resulted in a partial remission of BP with 12-month follow-up. The clinical outcome in the 9 patients in whom DPP4is were not stopped was unfavorable. There were 3 deaths of unknown causes (33%), 1 relapse (11%), 4 partial remissions (45%), and 1 complete remission (11%).

DISCUSSION

Our study demonstrates that DPP4is are associated with an increased risk for development of BP, with an adjusted OR of 2.64. Association with vildagliptin was significantly higher than that with other DPP4is, with an adjusted OR of 3.57. Our findings further indicate that the rate of DPP4i intake in patients with BP is higher both in male patients and in patients older than 80 years. Finally, DPP4i withdrawal seems to have a favorable impact on the outcome of BP in patients with diabetes, as 95% of them went into remission after management DPP4 inhibition could enhance the activity of proinflammatory chemokines, such as eotaxin, promoting eosinophil activation in the skin, tissue damage, and blister formation. [START_REF] Forssmann | Inhibition of CD26/dipeptidyl peptidase IV enhances CCL11/ eotaxin-mediated recruitment of eosinophils in vivo[END_REF] Thielitz et al reported that DPP4is have an antifibrogenic activity by decreasing expression of transforming growth factor-b 1 and secretion of procollagen type I. [START_REF] Thielitz | Inhibitors of dipeptidyl peptidase IV-like activity mediate antifibrotic effects in normal and keloid-derived skin fibroblasts[END_REF] All these effects could be higher for vildagliptin than other for DPP4is because of molecular differences. Furthermore, vildagliptin administration in monkeys resulted in dose-dependent and reversible skin effects, such as blister formation, peeling, and erosions. [START_REF] Hoffmann | Vascular origin of vildagliptin-induced skin effects in cynomolgus monkeys: pathomechanistic role of peripheral sympathetic system and neuropeptide Y[END_REF] Finally and more importantly, DPP4 is a cell surface plasminogen receptor that is able to activate plasminogen, leading to plasmin formation. Plasmin is a major serine protease that is known to cleave BP180 within the juxtamembranous extracellular noncollagenous 16A domain. Hence, the inhibition of plasmin by a DPP4i may change the proper cleavage of BP180, thereby affecting its antigenicity and its function. [START_REF] Izumi | Autoantibody profile differentiates between inflammatory and noninflammatory bullous pemphigoid[END_REF] Our study has some limitations: we focused the analysis on DPP4i intake, whereas the potential isolated effect of metformin was not analyzed. Nevertheless, after DPP4i withdrawal, metformin was either continued (in those cases in which it was initially combined with a DPP4i) or newly started in 8 of our patients with BP. Among the latter, we observed 5 complete and 3 partial remissions on follow-up. In addition, metformin intake has not been implicated thus far in the development of BP. On the basis of these observations, it is unlikely that metformin plays a triggering role, but specific studies should be designed to examine the effect of metformin on its own. Finally, we included patients with BP identified by searching our histopathology databases. It is therefore possible that we missed a number of BP cases in which either the term pemphigoid was not used in the corresponding histopathologic report or BP was not clinically and/ or histopathologically considered.

In conclusion, our findings in a case-control study confirm that DPP4is are associated with an increased risk for development of BP in patients with diabetes. Therefore, the prescription of a DPP4i, especially vildagliptin, should potentially be limited or avoided in high-risk patients, including males and those age 80 years or older. A larger prospective study might be useful to confirm our findings. with first-line therapeutic options (ie, topical and sometimes systemic corticosteroids).

An increasing number of reports have suggested that DPP4is trigger BP. Fourteen of the 19 BP cases described (74%) appeared to be related to vildagliptin intake. The median age of the affected patients was 72.5 years, with an almost identical number of males and females. [START_REF] Garc Ia | Dipeptidyl peptidase-IV inhibitors induced bullous pemphigoid: a case report and analysis of cases reported in the European pharmacovigilance database[END_REF][START_REF] Keseroglu | A case of bullous pemphigoid induced by vildagliptin[END_REF][START_REF] Haber | Bullous pemphigoid associated with linagliptin treatment[END_REF][START_REF] Pasmatzi | Dipeptidyl peptidase-4 inhibitors cause bullous pemphigoid in diabetic patients: report of two cases[END_REF][START_REF] Skandalis | Drug-induced bullous pemphigoid in diabetes mellitus patients receiving dipeptidyl peptidase-IV inhibitors plus metformin[END_REF][START_REF] Aouidad | A case report of bullous pemphigoid induced by dipeptidyl peptidase-4 inhibitors[END_REF][START_REF] Attaway | Bullous pemphigoid associated with dipeptidyl peptidase IV inhibitors. A case report and review of literature[END_REF][START_REF] B En E | Bullous pemphigoid induced by vildagliptin: a report of three cases[END_REF][START_REF] Mendonc ¸a | Three cases of bullous pemphigoid associated with dipeptidyl peptidase-4 inhibitorsdone due to linagliptin[END_REF] In our study, among the 28 diabetic patients developing BP under DPP4i exposure, males were more affected (56.7%) and the median age was 80 years.

Garcia et al 5 identified 170 cases of BP in patients taking a DPP4i in the EudraVigilance database, suggesting that the intake of DPP4is was more frequently associated with development of BP when compared with that of other drugs. In the latter, a disproportionally high number of cases of vildagliptin use were found. A French case-noncase study recording all spontaneous reports of DPP4i-related BP in the National Pharmacovigilance Database between April 2008 and August 2014 also provided evidence supporting an increased risk for development of BP associated with DPP4i exposure, especially vildagliptin. [START_REF] B En E | Bullous pemphigoid and dipeptidyl peptidase IV inhibitors: a case-noncase study in the French Pharmacovigilance Database[END_REF] Our present study confirms that the association with vildagliptin is stronger than that with the other DPP4is. This cannot be explained by an overprescription of vildagliptin compared with prescription of other DPP4is. In our control group, sitagliptin was the most prescribed DPP4i, with 14 diabetic patients (11.5%), whereas only 5 patients were treated by vildagliptin (4%). Increased prescribing of sitagliptin was confirmed by an analysis of drug sales in France published by the French National Agency for Medicines and Health Products Safety in 2014. In this survey, sitagliptin was the most prescribed DPP4i and the 24th highest-earning drug in 2013, whereas vildagliptin was not ranked. A recent retrospective study suggests that DPP4i-associated BP is frequently noninflammatory or pauci-inflammatory and characterized by small blisters, mild erythema, and a limited skin distribution. The latter is potentially related to a distinct reactivity profile of autoantibodies to BP180. [START_REF] Izumi | Autoantibody profile differentiates between inflammatory and noninflammatory bullous pemphigoid[END_REF] Although in our retrospective evaluation, there was no apparent difference in clinical presentation and initial management between patients with diabetes and BP who had been treated with DPP4i and patients with diabetes and BP who had not been treated with DPP4i (data not shown), prospective studies are required to address the question of whether BP associated with the intake of a DPP4i has unique clinical and immunologic features.

The pathophysiologic mechanisms linking DPP4i intake and BP development remain unclear.

  1, 2014, to July 31, 2016. All study procedures adhered to the principles of the Declaration of Helsinki. The French Committee for the Protection of Persons (RO-2016/37) and the Ethics Committee of the Canton of

	Bern (KEK-2016/01488) approved the study. The
	French Advisory Committee
	on Information Processing
	in Material Research in the
	Field of Health and the
	French Commission for
	Information Technology and
	Civil Liberties also authorized
	this study.

Table I .

 I Demographics and comorbidities of selected cases and controls

			Controls	Cases		Total
	Demographic characteristic/comorbidity	N	%	N	%	N	%	P*
	Sex
	Male	60	49.2%	30	49.2%	90	49.2%	d
	Female	62	50.8%	31	50.8%	93	50.8%
	Age, y (mean, SD = 7)	79.3	7.0	78.7	7.2	79.1	7.0	.63
	\75	30	24.6%	17	27.9%	47	25.7%
	75-84	62	50.8%	29	47.5%	91	49.7%
	$85	30	24.6%	15	24.6%	45	24.6%
	Comorbidities
	Neurologic	47	38.5%	32	52.5%	79	43.2%	.06
	Cardiovascular	108	88.5%	53	86.9%	161	88.0%	.75
	Rheumatic	36	29.5%	11	18.0%	47	25.7%	.10
	Digestive	34	27.9%	19	31.1%	53	29.0%	.65
	Metabolic and endocrine y	85	69.7%	24	39.3%	109	59.6%	<.001
	Pulmonary	27	22.1%	17	27.9%	44	24.0%	.41
	Uronephrologic	45	36.9%	24	39.3%	69	37.7%	.74
	Neoplasia	29	23.8%	12	19.7%	41	22.4%	.49
	Dermatologic z	5	4.1%	12	19.7%	17	9.3%	.03
	Other	35	28.7%	23	37.7%	58	31.7%	.18

SD, Standard deviation. *Mann-Whitney U test was used to assess possible residual differences in the distribution of age between cases and age-and sex-matched controls. Differences between cases and matched controls across different levels of other factors were assessed by means of univariate conditional logistic regression analysis. Boldface indicates statistical significance. y Except for diabetes. z Except for BP.

Table II .

 II DPP4i use and other cotreatments in selected cases and controls

			Controls		Cases		Total
	Treatment	N	%	N	%	N	%	P*
	DPP4i							<.01
	None	100	82.0%	33	54.1%	133	72.7%
	Vildagliptin	5	4.1%	14	23.0%	19	10.4%
	Sitagliptin	14	11.5%	10	16.4%	24	13.1%
	Linagliptin	3	2.5%	3	4.9%	6	3.3%
	Saxagliptin	0	0.0%	1	1.6%	1	0.5%
	Cotreatment
	Diuretics	69	56.6%	28	45.9%	97	53.0%	.17
	Antihypertensives/antiarrhythmic agents	101	82.8%	47	77.0%	148	80.9%	.36
	Neuroleptics	46	37.7%	26	42.6%	72	39.3%	.52
	Antiaggregants/anticoagulants	85	69.7%	45	73.8%	130	71.0%	.56
	NSAIDs	12	9.8%	0	0.0%	12	6.6%	.14
	Analgesics	22	18.0%	12	19.7%	34	18.6%	.79
	Statins	71	58.2%	31	50.8%	102	55.7%	.34
	Antihistamines	5	4.1%	19	31.1%	24	13.1%	<.001
	Antidiabetics y	122	100.0%	51	83.6%	173	94.5%	.08
	Endocrine or metabolic treatment z	45	36.9%	27	44.3%	72	39.3%	.32
	Proton pump inhibitors	59	48.4%	28	45.9%	87	47.5%	.75
	Others	50	41.0%	23	37.7%	73	39.9%	.67

DPP4i, Dipeptidyl peptidase-4 inhibitor; NSAID, nonsteroidal anti-inflammatory drug. *Boldface indicates statistical significance. y Except for DPP4i. z Except for diabetes.

Table III .

 III Univariate and multivariate analysis of the association between DPP4i use and BP in patients with diabetes, overall and in strata of sex and age group

						DPP4is could induce BP de novo or accelerate the
						development of BP in susceptible individuals. Many
						cell types, including keratinocytes, T cells, and
						endothelial cells, constitutionally express DPP4.
			Controls		Cases	Univariate analysis*	Multivariate analysis y
	Strata	DPP4i use	N	%	N	%	OR (95% CI)	P	OR (95% CI)	P
	Overall	No	100 82.0% 33 54.1%	1		1
		Yes	22 18.0% 28 45.9% 3.45 (1.76-6.77)	<.001 2.64 (1.19-5.85)	.02
	Overall (detailed) No	100 82.0% 33 54.1%	1		1
		Vildagliptin	5	4.1% 14 23.0% 7.23 (2.44-21.40) <.001 3.57 (1.07-11.84) .04
		Sitagliptin	14 11.5% 10 16.4% 1.82 (0.73-4.54)	.20	2.13 (0.77-5.89)	.15
		Linagliptin/	3	2.5%	4	6.6% 5.10 (0.98-26.62)	.053	2.90 (0.47-17.74) .25
		saxagliptin
	Males	No	51 85.0% 13 43.3%	1		1
		Yes	9 15.0% 17 56.7% 5.85 (2.13-16.08)	.001 4.36 (1.38-13.83) .01
	Females	No	49 79.0% 20 64.5%	1		1
		Yes	13 21.0% 11 35.5% 2.00 (0.78-5.15)	.15	1.64 (0.53-5.11)	.39
	Age \80 y	No	49 79.0% 18 56.2%	1		1
		Yes	13 21.0% 14 43.8% 2.47 (1.00-6.13)	.05	1.53 (0.52-4.52)	.44
	Age $80 y	No	51 85.0% 15 51.7%	1		1
		Yes	9 15.0% 14 48.3% 4.50 (1.58-12.77)	.005 5.31 (1.60-17.62) .006


Intellectual disability in patients with MODY due to hepatocyte nuclear factor 1B (HNF1B) molecular defects

Introduction

Research design and methods

The study population consisted of 107 adult patients with diabetes in whom a molecular abnormality of HNF1B had been identified, as described elsewhere [START_REF] Bellanné-Chantelot | Large genomic rearrangements in the hepatocyte nuclear factor-1beta (TCF2) gene are the most frequent cause of maturity-onset diabetes of the young type 5[END_REF]. The phenotype of the HNF1B-related syndrome was assessed through a questionnaire, filled in by referring physicians, that comprised clinical, biological and morphological items. ID was defined as limitations in intellectual functioning and in adapting to environmental demands, beginning early in life, and was appraised by the need for educational support, protected work or assistance in daily activities, and by the social skills of the patients [START_REF]Diagnostic and statistical manual of mental disorders (DSM 5)[END_REF]. Learning disability (LD) was defined as persistent difficulties in reading, writing, or mathematical-reasoning skills [START_REF]Diagnostic and statistical manual of mental disorders (DSM 5)[END_REF].

Of these 107 patients, 14 had access to detailed evaluations by a geneticist and a neurologist who were blinded to the patient's HNF1B genotype. In case of clinical suspicion of cognitive defects, the evaluation was completed by neuropsychological testing, including the Wechsler Adult Intelligence Scale, Third Edition (WAIS-III), and by further testing of executive functions (Trail-Making Test), memory [84-item Battery of Memory Efficiency (BEM 84)] and visuospatial function [Rey Complex Figure Test and Recognition Trial (RCFT)]. In patients presenting with ID or LD, single-nucleotide polymorphism (SNP) array analyses were performed, using the HumanCytoSNP-12 v2.1 array scanner and assay (Illumina, Inc., San Diego, CA, USA), after excluding fragile X syndrome. Results were analyzed with GenomeStudio software, version 3.1.6 (Illumina). All patients gave their written informed consent.

The frequency of ID was assessed in two control groups of adult patients with diabetes followed in our department: 339 consecutive patients with type 1 diabetes (T1D); and 227 patients presenting with phenotypes suggestive of MODY, including 31 glucokinase (GCK)-MODY, 42 HNF1A-MODY, 13 HNF4A-MODY, five ATP-binding cassette subfamily C member 8 (ABCC8)-MODY, two insulin (INS)-MODY and 134 genetically screened patients with no identifiable molecular aetiology (referred to as MODY-X).

Results are reported as means ± SD or as frequencies (%). Comparisons between groups were made by non-parametric tests or by Fisher's exact test.

Intellectual disability (ID) is characterized by impairments of general mental abilities that have an impact on adaptive functioning in conceptual, social and practical areas, and which begin in the developmental period [START_REF]Diagnostic and statistical manual of mental disorders (DSM 5)[END_REF]. It affects 1-3% of the general population [START_REF] Maulik | Prevalence of intellectual disability: a meta-analysis of population-based studies[END_REF]. Chromosomal aberrations or mutations in almost 500 genes have been associated with ID. Among these genes, several are also involved in diseases with phenotypes that may overlap with ID, such as autism spectrum disorders (ASD) and schizophrenia.

Molecular defects of the hepatocyte nuclear factor 1B (HNF1B) have been associated with a syndrome that includes maturity-onset diabetes of the young 5 (MODY5 or HNF1B-MODY), kidney structural abnormalities, progressive renal failure, pancreatic hypoplasia and exocrine dysfunction, abnormal liver tests and genital tract abnormalities [START_REF] Bellanné-Chantelot | Large genomic rearrangements in the hepatocyte nuclear factor-1beta (TCF2) gene are the most frequent cause of maturity-onset diabetes of the young type 5[END_REF]. In half the cases, the HNF1B-related syndrome is due to HNF1B heterozygous mutations whereas, in the others, it is associated with HNF1B whole-gene deletion [START_REF] Bellanné-Chantelot | Large genomic rearrangements in the hepatocyte nuclear factor-1beta (TCF2) gene are the most frequent cause of maturity-onset diabetes of the young type 5[END_REF]. In all cases examined thus far, the latter results from a 17q12 deletion of 1.4-2.1 Mb, encompassing 20 genes including HNF1B [START_REF] Bellanné-Chantelot | Large genomic rearrangements in the hepatocyte nuclear factor-1beta (TCF2) gene are the most frequent cause of maturity-onset diabetes of the young type 5[END_REF][START_REF] Loirat | Autism in three patients with cystic or hyperechogenic kidneys and chromosome 17q12 deletion[END_REF][START_REF] Laffargue | Towards a new point of view on the phenotype of patients with a 17q12 microdeletion syndrome[END_REF][START_REF] Clissold | Chromosome 17q12 microdeletions but not intragenic HNF1B mutations link developmental kidney disease and psychiatric disorder[END_REF].

Autism and/or ID have been described in patients with various HNF1B -related phenotypes, such as HNF1B-MODY [START_REF] Raile | Expanded clinical spectrum in hepatocyte nuclear factor 1bmaturity-onset diabetes of the young[END_REF], cystic kidney disease [START_REF] Loirat | Autism in three patients with cystic or hyperechogenic kidneys and chromosome 17q12 deletion[END_REF][START_REF] Nagamani | Clinical spectrum associated with recurrent genomic rearrangements in chromosome 17q12[END_REF][START_REF] Dixit | 17q12 microdeletion syndrome: three patients illustrating the phenotypic spectrum[END_REF] and müllerian aplasia [START_REF] Cheroki | Genomic imbalances associated with müllerian aplasia[END_REF], always in the context of 17q12 deletion. On the other hand, in a large population study, the 17q12 deletion was recognized as a strong risk factor for ID, ASD and schizophrenia, being identified in 1/1000 of children referred for those conditions [START_REF] Moreno-De-Luca | Deletion 17q12 is a recurrent copy number variant that confers high risk of autism and schizophrenia[END_REF].

Whether the neurocognitive phenotypes associated with the 17q12 deletion result from deletion of HNF1B itself or another deleted gene, or from a contiguous gene syndrome, remains unknown [START_REF] Loirat | Autism in three patients with cystic or hyperechogenic kidneys and chromosome 17q12 deletion[END_REF][START_REF] Nagamani | Clinical spectrum associated with recurrent genomic rearrangements in chromosome 17q12[END_REF][START_REF] Moreno-De-Luca | Deletion 17q12 is a recurrent copy number variant that confers high risk of autism and schizophrenia[END_REF].

To investigate the role of HNF1B abnormalities in the occurrence of cognitive defects, the frequency of ID was assessed according to the presence of HNF1B mutations or deletion in a large cohort of adult patients with HNF1B-MODY.

Results

The main characteristics of the 107 patients are shown in Table 1. ID was reported in 15 (14 proband) patients (14%). LD was noticed in a further nine patients (Table S1; see supplementary material associated with the article online). Overall, cognitive defects were thus observed in 24/107 patients (22.4%). Common causes of ID were ruled out by the search for fragile X syndrome and SNP array analyses, which excluded other large genomic deletions in all tested patients.

The frequency of ID was significantly higher in HNF1B-MODY patients than in those with T1D [8/339 (2.4%), OR: 5.9, 95% CI: 2.6-13.6; P < 10 -4 ], or in those with other monogenic diabetes or MODY-X [6/227 (2.6%), OR: 6.0, 95% CI: 2.3-16.0; P = 0.0002].

HNF1B-MODY patients with or without ID were similar as regards gender, age at diabetes diagnosis, duration and treatment of diabetes, frequency and severity of renal disease, frequency of pancreatic morphological abnormalities and livertest abnormalities, and frequency of arterial hypertension and dyslipidaemia (Table 1). HbA 1c levels at the time of the study were higher in the patients with ID (9.4 ± 3.0% vs 7.3 ± 1.4%; P = 0.005).

Of the 15 patients presenting with ID, six had HNF1B coding mutations (three missense, two splicing defects, one deletion of exon 5) and nine had a whole-gene deletion (Table S1). Thus, the frequency of ID was not statistically different between patients with HNF1B mutation (11%) or deletion (17%; P = 0.42; Table 1).

Discussion

Our study showed that ID affects 14% of adult patients with HNF1B-MODY, which is higher than the 1-3% reported in the general population [START_REF] Maulik | Prevalence of intellectual disability: a meta-analysis of population-based studies[END_REF] and than the 2.4-2.6% observed in our two control groups of adult patients with other diabetes subtypes.

The main characteristics of the HNF1B-MODY patients with ID did not differ from those without ID, except for the poorer glycaemic control observed in the former.

In patients with HNF1B-related syndrome, the occurrence of cognitive defects has been noted almost exclusively in paediatric series. ID/ASD has been reported in two adolescents with renal cystic disease, livertest abnormalities and diabetes [START_REF] Raile | Expanded clinical spectrum in hepatocyte nuclear factor 1bmaturity-onset diabetes of the young[END_REF]; developmental delay and/or learning difficulties were quoted in three young patients presenting with multicystic renal disease [START_REF] Nagamani | Clinical spectrum associated with recurrent genomic rearrangements in chromosome 17q12[END_REF]; and speech delay in two children with renal cystic disease [START_REF] Dixit | 17q12 microdeletion syndrome: three patients illustrating the phenotypic spectrum[END_REF]. In a series of 86 children with HNF1B-related renal disease, three cases of ASD were noted [START_REF] Loirat | Autism in three patients with cystic or hyperechogenic kidneys and chromosome 17q12 deletion[END_REF]. The systematic evaluation of 28 children with HNF1B-associated kidney disease also suggested an increased risk of neuropsychological disorders in those harbouring the 17q12 deletion [START_REF] Laffargue | Towards a new point of view on the phenotype of patients with a 17q12 microdeletion syndrome[END_REF]. A recent study performed in a UK cohort reported the presence of neurodevelopmental disorders in eight out of 20 patients with renal abnormalities or diabetes due to HNF1B whole-gene deletion [START_REF] Clissold | Chromosome 17q12 microdeletions but not intragenic HNF1B mutations link developmental kidney disease and psychiatric disorder[END_REF]. In all these reports, cognitive defects were observed in the context of the 17q12 deletion.

Conversely, the 17q12 deletion has also been reported in children evaluated for ID, beyond the setting of HNF1B-related syndrome. Indeed, an association between the deletion and cognitive defects has been confirmed in paediatric cases with no renal abnormalities [START_REF] Roberts | Clinical report of a 17q12 microdeletion with additionally unreported clinical features[END_REF][START_REF] Palumbo | Variable phenotype in 17q12 microdeletions: clinical and molecular characterization of a new case[END_REF]. In one population study, the 17q12 deletion was detected in 18/15,749 children referred for ASD and/or ID, but in none of the controls [START_REF] Moreno-De-Luca | Deletion 17q12 is a recurrent copy number variant that confers high risk of autism and schizophrenia[END_REF]. However, detailed phenotypes, available for nine children, were suggestive of the HNF1B-related syndrome, as all but one showed multicystic renal disease and/or kidney morphological abnormalities, and one had diabetes. Altogether, these observations strongly suggest that cognitive defects are part of the phenotype associated with the 17q12 deletion.

Whether cognitive defects may result from molecular alterations of HNF1B itself remains unsolved. Learning difficulties have been reported in two patients with HNF1B frameshift mutations: one was a man with polycystic kidney disease [START_REF] Bingham | Mutations in the hepatocyte nuclear factor-1b gene are associated with familial hypoplastic glomerulocystic kidney disease[END_REF]; the other was a woman with renal disease, diabetes, and livertest and genitaltract abnormalities [START_REF] Shihara | Identification of a new case of hepatocyte nuclear factor-1beta mutation with highly varied phenotypes[END_REF]. ID has also been reported in two patients with HNF1B-kidney disease due to point mutations [START_REF] Faguer | Diagnosis, management, and prognosis of HNF1B nephropathy in adulthood[END_REF]. However, in these four patients, a search for other causes of cognitive defects was not performed. In the above-mentioned UK study, no neurodevelopmental disorders were reported in 18 patients with intragenic HNF1B mutations [START_REF] Clissold | Chromosome 17q12 microdeletions but not intragenic HNF1B mutations link developmental kidney disease and psychiatric disorder[END_REF]. Conversely, in our study, ID was observed in 6/54 patients (11%) with an HNF1B point mutation, a frequency three times greater than in the general population, and common causes of ID were ruled out in four of them. These discrepancies might be explained by the small number of patients (n = 18) with HNF1B mutations in the UK study, and by the fact that neurocognitive phenotypes might be milder in patients with mutations.

Thus, our observations may suggest the involvement of HNF1B defects in the occurrence of cognitive defects in patients with HNF1B-MODY. The links between HNF1B molecular abnormalities and intellectual developmental disorders remain elusive. Nevertheless, it should be noted that HNF1B is one of the evolutionarily conserved genes involved in the hindbrain development of zebrafish and mice [START_REF] Makki | Identification of novel Hoxa1 downstream targets regulating hindbrain, neural crest and inner ear development[END_REF]. However, the role of HNF1B in the human brain has yet to be established.

In our study, because of geographical remoteness, only a small number of patients had access to detailed neurological evaluation. However, the absence of selection bias is supported by the similar spectrum of HNF1B-related syndrome in patients evaluated by either examination or questionnaire (Table S2; see supplementary material associated with the article online). Moreover, the accuracy of the diagnosis made by referring physicians-ID vs no ID-was confirmed in all patients who underwent neurological evaluations.

Conclusion

ID is more frequent in adults with HNF1B-MODY than in the general population or in patients with other diabetes subtypes. Moreover, it may affect patients with HNF1B point mutations as well as those with 17q12 deletion. Further studies are needed to refine the cognitive phenotypes of HNF1B-related syndrome and to precisely define the role of HNF1B itself in their occurrence.

Table 1

 1 Main characteristics of 107 HNF1B-MODY patients according to the presence (+) or absence (-) of intellectual disability (ID).

	Total	ID+	ID-	P a

Values are expressed as n or as mean ± SD.

a ID+ vs ID-. b Estimated glomerular filtration rate <60 mL/min/1.73 m 2 [Chronic Kidney Disease Epidemiology Collaboration (CKD-EPI) formula], or dialysis or renal transplantation.

p CH de Valenciennes, 59322 France q CHU Pellegrin, 33076 Bordeaux, France r CHU Godinne, UCL Namur, 5530 Belgium s CHU Tenon, 75020 Paris, France t CH de Roanne, 42300 France u CH Belle-Isle, 57045 Metz, France v CH Saint-Joseph, 75014 Paris, France w CH de Lannion-Trestel, 22300 France x CH Louis Pasteur, 28630 Chartres, France y CH Emile Muller, 68100 Mulhouse, France z CH Saint-Philibert, 59160 Lomme, France aa CH Laënnec, 60100 Creil, France bb CHU de Caen, 14003 France cc CH de Compiègnes, 60200 France dd CHU de Strasbourg, 67000 France ee CHU de Poitiers, 86021 France ff CH Lucien Hussel, 38200 Vienne, France gg CH Bretagne Sud, Lorient, 56322 France hh CH de Blois, 41016 France ii CHIC Poissy-Saint-Germain-en-Laye, 78300 France jj CHU Cochin, 75014 Paris, France kk CHU Haut-Lévêque, 33600 Bordeaux, France ll CHU Louis Pradel, 69500 Lyon, France mm CHRU Clocheville, 37000 Tours, France nn CH d'Avignon, 84000 France oo CH Robert Bisson, 14100 Lisieux, France pp CHU Ambroise Paré, 92104 Boulogne, France qq CH de Saint-Nazaire, 44600 France rr CH Sud-Francilien, 91100 Corbeil, France ss CHU de Nantes, 44093 France tt CHRU Jean Minjoz, 25030 Besançon, France uu CHU d'Angers, 49100 France vv CHU de Brest, 29200 France ww CHU de la Conception, 13005 Marseille, France xx CHRU de Lille, 59000 France yy CH Manchester, 08011 Charleville-Mézières, France zz CHU Lapeyronie, 34090 Montpellier, France

Acknowledgements

We thank A. Faudet, C. Vaury and S. Clauin, of the Centre of Molecular Genetics, Hôpital Pitié-Salpétrière, Assistance Publique-Hôpitaux de Paris, for their technical assistance.

Funding source

None.

Disclosure of interest

The authors declare that they have no competing interest.

Appendix A. Supplementary data

Supplementary material (Tables S1 andS2) related to this article can be found, in the online version, at http://dx.doi.org/10.1016/j.diabet.2016.10.003.