P Doukhan

G Lang

RANDOMLY FRACTIONALLY INTEGRATED PROCESSES 1

Keywords: first keyword, second keyword, third keyword, very long keyword, very long keyword, very long keyword

may come

Limit theorems for sums of non linear function of ARFIMA processes with random Hurst exponents and Gaussian innovations

Paul Doukhan, Gabriel Lang, Donatas

INTRODUCTION

Philippe et al. [START_REF] Philippe | Invariance principle for a class of non stationary processes with long memory[END_REF] introduced time-varying fractional filters A(d), B(d) defined by

A(d)x t = ∞ j =0 a j (t)x t-j ,B ( d)x t = ∞ j =0 b j (t)x t-j , (1.1)
where d = (d t ,t ∈ Z) is a given function of t ∈ Z,anda 0 (t) = b 0 (t) := 1,

a j (t) := d t-1 1 d t-2 + 1 2 d t-3 + 2 3 ••• d t-j + j -1 j , (1.2)
b j (t) := d t-1

1 d t-j + 1 2 d t-j +1 + 2 3 ••• d t-2 + j -1 j ,j 1. (1.3) If d t ≡ d is a constant, then A(d) = B(d) = (I -L) -d
is the usual fractional integration operator (Lx t := x t-1 is the backward shift). Let a - j (t), b - j (t) be defined as in (1.2), (1.3), with d = (d t ) t∈Z replaced by -d = (-d t ) t∈Z . For arbitrary sequence d = (d t ) t∈Z such that d t ∈ Z (∀t ∈ Z), the coefficients in (1.2), (1.3) satisfy the orthogonality relation

n j =0 b - j (t)a n-j (t -j)= n j =0
a - j (t)b n-j (tj)= δ n .

In other words, the filters A(d), B(-d) are mutually inverse: A(d) -1 = B(-d), B(d) -1 = A(-d) (see Philippe et al. [START_REF] Philippe | Invariance principle for a class of non stationary processes with long memory[END_REF], [START_REF] Philippe | Almost periodically correlated processes with longmemory[END_REF]). The above mentioned papers studied long memory behavior and partial sums limits of nonstationary processes X A t = A(d)ε t , X B t = B(d)ε t ,where(ε t ) is a white noise, for certain classes of deterministic sequences d admitting (possibly different) Cesaro limits d + ,d -at +∞, -∞, respectively, and showed that the limit behavior of partial sums of X A t and X B t essentially depends on the limits d + ,d -alone.

The present paper studies long-memory properties of the randomly fractionally integrated processes

X A t := ∞ j =0 a j (t)ε t-j ,X B t := ∞ j =0 b j (t)ε t-j , (1.4)
where (ε t ,t ∈ Z) is an iid white noise, a j (t), b j (t) are given in (1.2)-(1.3), and d = (d t , ∈ Z) is an iid sequence with mean d = Ed 0 ∈ (0, 1/2) and finite variance (the sequences (ε t ,t ∈ Z) and d = (d t , ∈ Z) are assumed mutually independent). Then (X A t ), (X B t ) in (1.4) are well-defined, strictly stationary, and ergodic processes. We show in Section 1 that long-memory properties of X A and X B are very similar to FARIMA(0, d,0); in particular, partial sums of X A and X B converge to a fractional Brownian motion with Hurst parameter H = d + (1/2) ∈ (1/2, 1).

The main objective of the present paper is to study long-memory properties of nonlinear functionals (h(X A t )) t∈Z ,(h(X B t ) t∈Z of stationary processes in (1.4). There is a large literature concerning limit theorems for instanteneous nonlinear functionals of Gaussian and linear processes with long memory, see, e.g., Dobrushin and Major [START_REF] Dobrushin | Non-central limit theorems for non-linear functions of Gaussian fields[END_REF], Taqqu [START_REF] Taqqu | Convergence of integrated processes of arbitrary Hermite rank[END_REF], Ho and Hsing [START_REF] Ho | On the asymptotic expansion of the empirical process of long memory moving averages[END_REF], Surgailis [START_REF] Surgailis | Stable limits of sums of bounded functions of long-memory moving averages with infinite variance[END_REF], and the references therein. Consider a linear process Y t = ∞ j =0 a j ε t-j with slowly decaying coefficients a j ∼ c 0 j d-1

(c 0 = 0, d ∈ (0, 1/2) and iid innovations (ε t ) ∼ iid(0, 1). It was shown in these papers that the limit distribution of suitably normalized partial sums processes [Nτ] t=1 h(Y t ) is determined by the Appell rank of the nonlinear function h, or the integer

k * := min k 1: d k Eh(Y 0 + x) dx k x=0 = 0 , (1.5)
under some additional moment and regularity conditions on h and ε 0 . Note that, for Gaussian processes (Y t ) (with Y 0 ∼ N(0, 1)), the Appell rank in (1.5) coincides with the Hermite rank, or the smallest k 1 with c k = 0i nt h eH e r m i t e expansion h(x) = ∞ k=0 c k H k (x)/k!. Under the long memory condition k * (2d -1)< 1, the limit of partial sums of h(Y t ) is a so-called Hermite process of order k * (see Section 2 for definition).

To study the asymptotic behavior of nonlinear functionals of "random FARIMA" processes in (1.4), we assume that the iid sequence (d t ) is bounded and the iid sequence (ε t ) is Gaussian; moreover, our discussion is limited to the process (X A t ). Extensions to more general (d t ), (ε t ), and the filter (X B t ) are possible but not easy. The assumption of conditional gaussianity allows us to use conditional Hermite expansions and simplifies estimation of remainder terms. Our main result, Theorem 3.1, states that the limit distribution of partial sums [Nτ] t=1 h(X A t ) is determined by the integer k A * := min{k 1: β k = 0},whereβ k 's are defined via conditional Hermite expansion of h; it turns out that, under certain regularity conditions, the integer k A * can also be defined as

k A * = min k 1: d k Eh(X A 0 + xQ(0)) dx k x=0 = 0 , (1.6)
where Q(0) := lim j →∞ a j (0)/ψ j ( d),a n d

∞ j =0 ψ j ( d)L j = (1 -L) -d is FARIMA(0, d,0) filter. While k * in (1.

5) and k A

* in (1.6) look similarly, the former quantity is expressed via the marginal distribution of (Y t ) at t = 0 alone, and the latter involves the joint distribution (X A 0 ,Q(0)); moreover, the derivative in (1.6) does not seem to be related to any polynomial expansion of h.

LINEAR FUNCTIONALS

Everywhere below, (ε t ,t ∈ Z) ∼ iid(0, 1) is a standard iid sequence of rv's with zero mean and unit variance, and d = (d t , ∈ Z) is another sequence of iid rv's, with mean d = Ed 0 and variance σ 2 := E(d 0 -Ed 0 ) 2 < ∞; the sequences (ε t ) t∈Z and d = (d t ) t∈Z are assumed mutually independent. Let δ t := d t -Ed t denote the centered iid rv's. We also assume that d t ∈ Z -:= {0, -1, -2,...} a.s.

Let ψ j (d), j 0, be the FARIMA(0,d,0) coefficients defined by

(1 -z) -d = ∞ j =0 ψ j (d)z j .
Recall that, for 0 <d<1/2, the autocovariance of the process Y t := (1 -L) -d ε t decays as t 2d-1 , more precisely, (see Hosking [START_REF] Hosking | Fractional differencing[END_REF] and Kokoszka and Taqqu [START_REF] Kokoszka | Fractional ARIMA with stable innovations[END_REF]). Introduce the Hermite process of order k = 1, 2,...:

EY 0 Y t = ∞ j =0 ψ j (d)ψ t+j (d) = Ŵ(1 -2d) Ŵ(d)Ŵ(1 -d) t 2d-1 1 + O(t -1 ) (2.
J k (τ ) := 1 Ŵ(d) k R k τ 0 k i=1 (t -u i ) d-1 + dt W(du 1 )...W(du k ), (2.2)
givenbyak-tuple Itô-Wiener integral with respect to a standard Gaussian white noise W(ds) with zero mean and variance ds;

u d-1 + := u d-1 for u>0, := 0otherwise.The process J k in (2.2) is well defined for 1 k<1/(1 -2d) and is self-similar with index H = 1 -(1 -2 d)k/2.
EX A 0 X A t = ∞ j =0 Ea j a t+j , EX B 0 X B t = ∞ j =0
Eb j b t+j .

(ii) Let 0 < d<1/2.Then

EX A 0 X A t = EY 0 Y t 1 + O(t -1 log t) , EX B 0 X B t = c 2 B EY 0 Y t 1 + O(t -d ) (2.3)
as t →∞,whereEY 0 Y t is the autocovariance of FARIMA(0, d, 0)(see(2.1)), and the constant c 2 B is given in (2.8) below. Moreover,

N -d-(1/2) [Nτ] t=1 X A t → D[0,1] c( d)J 1 (τ ), (2.4
)

N -d-(1/2) [Nτ] t=1 X B t → D[0,1] c( d)c B J 1 (τ ), (2.5)
where

(J 1 (τ ), τ 0) is a fractional Brownian motion with (Hurst) parameter H = (1/2) + d (see (2.2)). Proof. (i) Assume first that d ∈ Z -.Define Q A (s, t) := a t-s (t) ψ t-s ( d) = s u<t 1 + δ u d + t -u -1 (s < t), (2.6)
Q B (s, t) := db t-s (t) d t-1 ψ t-s ( d) = s u<t-1 1 + δ u d + u -s + 1 (s < t -1), (2.7) Q B (s, s + 1) := 1. The expectations EQ 2 A (s, t) = s u<t 1 + σ 2 ( d + t -u -1) 2 c 2 A , EQ 2 B (s, t) = s u<t-1 1 + σ 2 ( d + u -s + 1) 2 c 2 B
are uniformly bounded in s<tby finite constants

c 2 A := i 0 1 + σ 2 ( d + i) 2 ,c 2 B := i 1 1 + σ 2 ( d + i) 2 , (2.8)
respectively. Therefore,

Ea 2 t-s (t) = ψ 2 t-s ( d)EQ 2 A (s, t) c 2 A ψ 2 ( d), Eb 2 t-s (t) = ψ 2 t-s ( d) Ed 2 t-1 d2 EQ 2 B (s, t) c 2 B d2 + σ 2 d2 ψ 2 t-s ( d),
implying ∞ j =0 Ea 2 j (t) < ∞ and ∞ j =0 Eb 2 j (t) < ∞ by the well-known property of FARIMA coefficients and, thus, the convergences of the series in (1.4). The stationarity and ergodicity properties of these series are easy. This proves part (i) for d ∈ Z -.I n the case d ∈ Z -, the above argument requires minor modifications.

(ii) Consider the covariance of X A .UsingEQ A (s, t) = 1, similarly as in the proof of (i), one has

EX A 0 X A t = s 0 ψ -s ( d)ψ t-s ( d)EQ A (s, 0)Q A (s, t) = EY 0 Y t + R A t ,
where

R A t := ∞ j =t+1 ψ j ( d)ψ t+j ( d) A j (t), A j (t):= j i=1 1+ σ 2 ( d +i -1)( d +t +i -1) -1.
Note that sup j>t | A j (t)|=O(t -1 log t); indeed,

1 1 + A j (t) exp j i=1 log 1 + σ 2 ( d + i -1)( d + t + i -1) P. Doukhan, G. Lang, D. Surgailis exp j i=1 σ 2 ( d + i -1)( d + t + i -1) exp C ∞ i=1 σ 2 i(t + i) = exp O(log t/t) = 1 + O log t t .
Then R A t = O(t 2 d-2 log t) (see (2.1)), proving the first asymptotic in (2.3). Next,

EX B 0 X B t = s -1 ψ -s ( d)ψ t-s ( d)EQ B (s, 0)Q B (s, t) d -1 d t-1 d2 + ψ 0 ( d)ψ t ( d) = c 2 B EY 0 Y t + R B t ,
where

R B t := ∞ j =0 ψ j ( d)ψ t+j ( d) B j and B j := j -1 k=1 1 + σ 2 ( d + k) 2 1 + σ 2 d( d + j) ∞ p=j 1 + σ 2 ( d + p) 2 -1 (j 1), B 0 := 1 -c 2 B , satisfy B j = O(j -1 ).FromthisR B t = O(t d -1
) easily follows, proving the second asymptotic in (2.3).

To show (2.4), decompose

X A t = Y t + Z A t ,
where

Z A t := ∞ j =0 ψ j ( d)(Q A (t -j,t) -EQ A (t -j, t))ε t-j is a short-memory process satisfying N t=1 Z A t = O P (N 1/2
), which follows by evaluating the covariance cov(Q A (s, 0), Q A (s, t)) (s 0 t) similarly as above. Then the convergence of finite-dimensional distributions in (2.4) is immediate from the corresponding convergence of FARIMA process (Y t ), and the tightness follows by the Kolmogorov criterion using the fact about the covariance of (X A t ) in (2.3). The proof of the tightness part in (2.5) is completely analogous; however, the convergence of finite-dimensional distributions is a little more complicated. Namely, one represents

X B t as X B t = Y B t + Z B t ,whereY B t = s t ψ t-s ( d)ε B s is FARIMA(0, d,0) process in strictly stationary backward martingale difference innovations (ε B s ) defined by ε B s := ε s Q B (s, ∞) = ε s j 1 1 + δ s+j -1 d + j , with variance E(ε B s ) 2 = c 2 B given in (2.8
). The fact that finite-dimensional ditributions of partial sums of (Y B t ) tend to those of the limit process in (2.5) can be easily proved using the scheme of discrete stochastic integrals as in Philippe et al. [START_REF] Philippe | Invariance principle for a class of non stationary processes with long memory[END_REF], [START_REF] Philippe | Almost periodically correlated processes with longmemory[END_REF]. The "remainder term" Z B t in the above decomposition of X B t is given by

Z B t := (ε B t -ε t ) + δ t d s<t ψ t-s ( d)Q B (s, t)ε s + s<t ψ t-s ( d )(Q B (s, t) -Q B (s, ∞))ε s .
The proof of the relation N t=1 Z B t = O P (N 1/2 ) follows using similar argument as in the proof of (2.3), and we omit the details for the sake of brevity. This completes the proof of Theorem 2.1.

NONLINEAR FUNCTIONALS

In this section, we study long-memory properties of nonlinear processes (h(X A t )) t∈Z , where h is a real function such that Eh 2 (X A 0 )<∞. As noted in Introduction, the discussion is limited to the process (X A t ) in (1.4), so we omit the superscript "A"i n the following notation, i.e., we write X t ≡ X A t ,Q(s,t)≡ Q A (s, t), etc. Because of the difficulty of dealing with nonlinear functionals, the assumptions on (d t ) and (ε t ) now are strenghtened as follows.

Assumption 1. The sequence (ε t ) is iid N(0, 1)-distributed. Assumption 2. The sequence (d t ) is iid, independent of (ε t ), with mean d ∈ (0, 1/2) and finite variance σ 2 = var(d 0 )<∞; moreover, there is a constant D<∞ such that

|δ t | D a.s. ( 3.1)
We use the following notation: ψ j := ψ j ( d),

Q(t) := Q(-∞,t)= ∞ i=1 1 + δ t-i d + i -1 , (3.2)
A 2 (t) := ∞ j =0 a 2 j (t) = ∞ j =0 Q 2 (t -j,t)ψ 2 j = ∞ j =0 ψ 2 j j i=1 1 + δ t-i d + i -1 2 . (3.3)
Also, let D t := σ {d s : s t} and D := t D t denote the sigma-algebras generated by the iid sequence (d t ). From Assumptions 1 and 2 it follows that the process (X t ) in (1.4) is a conditionally Gaussian process given the sigma-algebra D, with zero conditional mean and the conditional variance

A 2 (t), i.e., E[X t |D]=0, E[X 2 t |D]=A 2 (t).
For any A>0 and any real function

h(x), x ∈ R, with Eh 2 (X) < ∞ (X ∼ N(0,A 2 )),
we can write the Hermite expansion

h(x) = ∞ k=0 g k (A) k! H k (x; A), (3.4)
where

g k (A) := A -2k E h(X)H k (X; A) = 1 √ 2πA 1+2k R h(x)H k (x; A)e -x 2 /2A 2 dx, (3.5)
and H k (x; A) := A k H k (x/A), k 0, are Hermite polynomials with standard deviation A>0; H k (x), k 0, are standard Hermite polynomials with generating function

∞ k=0 z k H k (x)/k!=e zx-z 2 /2
. Finally,

β k := E g k (A(0))Q k (0) ,k = 0, 1,..., (3.6)
where g k (•), A(0), and Q(0) are defined in (3.5), (3.3), and (3.2), respectively. THEOREM 3.1. Let Assumptions 1 and 2 be fullfilled. Let h: R → R be a measurable function such that

E|h(BX 0 )| a < ∞ (3.7)
for some B>1 and a>2.Letk A * be the smallest integer k 1 such that β k = 0:

k A * := min k 1: β k = 0 , (3.8)
and let (1

-2 d)k A * < 1.
Then

N (1-2 d)(k A * /2)-1 [Nτ] t=1 (h(X t ) -Eh(X t )) → D[0,1] β k A * k A * ! J k A * (τ ), (3.9)
where J k (τ ) is a kth order Hermite process given in (2.2) with d = d.

Remark 3.1. We show in the proof of Theorem 3.1 below that the coefficients β k (3.6) are well defined for any k 1. Moreover, as noted in Introduction, under additional conditions on the function h(•), these coefficients can be identified with the derivatives in (1.6), i.e.,

β k = d k Eh(X 0 + wQ(0)) dw k w=0 . (3.10)
Indeed, X 0 = law A(0)Z,w h e r eZ ∼ N(0, 1) does not depend on (A(0), Q(0)) ≡ (A, Q). Then, assuming that the differentiations and integrations by parts below are legitim, we can rewrite the right-hand side of (3.10) as

R + ×R 1 √ 2π R h(Az + wQ)e -z 2 /2 dzP( dA, dQ) (k) = R + ×R Q k √ 2π R h (k) (Az)e -z 2 /2 dzP( dA, dQ) = R + ×R Q k √ 2πA k R h(Az)H k (z)e -z 2 /2 dzP( dA, dQ) = β k ; see definitions (3.6), (3.5). Remark 3.2. Condition E|h(BX 0 )| a < ∞ (B>1) entails E|h(X 0 )| a < ∞.I n - deed, E|h(BX 0 )| a = E 1 √ 2πA(0) h(BA(0)x) a e -x 2 /2A 2 (0) dx = E 1 √ 2πBA(0) h(A(0)x) a e -x 2 /2B 2 A 2 (0) dx B -1 E 1 √ 2πA(0) h(A(0)x) a e -x 2 /2A 2 (0) dx = B -1 E|h(X 0 )| a or E|h(X 0 )| a BE|h(BX 0 )| a < ∞.
The proof of Theorem 3.1 follows technical lemmas discussed in the following section. In these lemmas, Assumptions 1 and 2 are imposed without explicit reference to them. On the other hand, some of these statements hold under weaker conditions without the assumption of gaussianity of (ε t ) or boundedness of (d t ).

SOME TECHNICAL LEMMAS

Recall from (2.6) the definition Q(tj,t) ≡ Q j (t), i.e.,

Q j (t) = a j (t)/ψ j = j i=1 1 + δ t-i d + i -1 (j 1), Q 0 (t) := 1.
LEMMA 4.1. For any t ∈ Z, Q j (t) → Q(t) (j →∞) a.s. Moreover, for any p 2, there exists a constant C = C p such that, for any t ∈ Z and j 0,

E|Q j (t)| p C, EA 2p (t) C (4.1)
and such that

E Q j (t) -Q(t) p Cj -p/2 . (4.2)
Proof. Denote

ξ j := Q j +1 (t) -Q j (t) = Q j (t) δ t-j -1 d + j . P. Doukhan, G. Lang, D. Surgailis
Note that the random variables (ξ j ) j 1 are orthogonal (in fact, they are martingale differences) and, therefore,

Q j +1 (t) = 1 + j i=0 ξ i (4.3)
is the sum of orthogonal rv's. According to a result of Stout [START_REF] Stout | Almost Sure Convergence[END_REF], the series (4.3) converges a.s., provided that

∞ j =1 log 2 (j )Eξ 2 j < ∞. (4.4)
In our case,

Eξ 2 j = σ 2 ( d + j) 2 EQ 2 j (t), (4.5)
where

EQ 2 j (t) = j i=1 E 1 + δ t-i d + i -1 2 = j i=1 1 + σ 2 ( d + i -1) 2 <C (4.6)
is bounded. Therefore, (4.4) holds implying the first part of the lemma.

Let us prove the first bound in (4.1). We shall need the following general inequality: for any p 2, 0 <δ<1, and any rv δ with Eδ = 0, E|δ| p < ∞, there exists a constant C = C p such that

E|1 + aδ| p 1 + Ca 2 .
(4.7)

Indeed, write E|1 + aδ| p = 4 k=1 J k ,w h e r eJ 1 := E|1 + aδ| p I(aδ -1), J 2 := E(1 + aδ) p I(-1 <aδ -1/2), J 3 := E(1 + aδ) p I(|aδ| < 1/2), and J 4 := E(1 + aδ) p I(aδ > 1/2).T h e nJ 1 E|aδ| p Ca 2 , J 2 P(|aδ| > 1/2) 4a 2 Eδ 2 Ca 2 ,J 4 CE|aδ| p Ca 2 ,and

J 3 = 1+paEδI(|aδ| < 1/2)+O(a 2 ) = 1-paEδI(|aδ| 1/2)+O(a 2 ) = 1+O(a 2 ),
since E|δ|I(|aδ| 1/2) CaEδ 2 Ca 2 . This proves (4.7).

Applying (4.7) with a = 1 d+i-1 ,δ= δ t-i , we obtain

E|Q j (t)| p = j i=1 E 1 + δ t-i d + i -1 p j i=1 1 + C i 2 C
and, thus, prove the first bound in (4.1).

Consider the second bound in (4.1). Since A 2 (t) = ∞ j =0 Q 2 j (t)ψ 2 j , the Minkowski inequality and the previous bound together yield

E[A 2p (t)] 1/p ∞ j =0 ψ 2 j E[Q 2p j (t)] 1/p C ∞ j =0 ψ 2 j <C.
It remains to prove (4.2). In the case p = 2, we immediately get

E Q(t) -Q j (t) 2 = E ∞ i=j ξ i 2 = ∞ i=j Eξ 2 i C ∞ i=j j -2 Cj -1
due to (3.5) and (3.6). Let p>2. Since (ξ j ) is a martingale difference sequence, by the Bürkholder inequality we have

E|Q(t) -Q j (t)| p = E ∞ i=j ξ i p C p E ∞ i=j ξ 2 i p/2 .
By the Hölder inequality, we have

∞ i=j ξ 2 i = ∞ i=j (i + d) -1 (i + d) -1 δ 2 t-i-1 Q 2 i (t) ∞ i=j (i + d) -p/(p-2) (p-2)/p ∞ i=j (i + d) -p/2 |δ t-i-1 | p |Q i (t)| p 2/p Cj -2/p ∞ i=j (i + d) -p/2 |δ t-i-1 | p |Q i (t)| p 2/p .
Therefore, using (4.1), we get

E Q(t) -Q j (t) p Cj -1 ∞ i=j (i + d) -p/2 E|δ t-i-1 | p |Q i (t)| p Cj -1 ∞ i=j (i + d) -p/2 Cj -p/2 ,
proving the lemma. P. Doukhan, G. Lang, D. Surgailis LEMMA 4.2. Let p i ,k i 0, q i ∈ Z, i = 1, 2, be given integers, and let

φ i (t) := A q i (t)Q p i j i (t)M i (A(t)), (4.8)
where

M i (A) := R h(x)x k i e -x 2 /2A 2 dx. (4.9)
Then there exists a constant C = C(k i ,p i ,q i ,i = 1, 2)<∞ such that, for any t ∈ Z,j 1 ,j 2 0, cov(φ 1 (t), φ 2 (0)) Ct -1 log t.

Proof. Write the telescoping expansion

φ i (t) -Eφ i (t) = ∞ j =1 U i (t, j ), (4.10)
where

U i (t, j ) := E φ i (t)|D t-j -E φ i (t) | D t-j -1 = R h(x)x k i dx E A q i (t)Q p i (t)e -x 2 /2A 2 (t) D t-j -E A q i (t)Q p i (t)e -x 2 /2A 2 (t) D t-j -1 .
By orthogonality,

cov(φ 1 (t), φ 2 (0)) = ∞ j =1 E U 1 (t, t + j)U 2 (0,j) ∞ j =1 EU 2 1 (t, t + j) 1/2 EU 2 2 (0,j) 1/2 .
Thus, the lemma follows from the bound

EU 2 i (t, j ) Cj -2 . (4.11)
Fix t,j,i; then with k = k i ,p = p i ,q = q i , U(t,j) = U i (t, j ), by definition we have

U(t,j) = E V(t,j)|D t-j , (4.12)
where

V(t,j):= R h(x)x k dx A q (t)Q p (t)e -x 2 /2A 2 (t) -E A q (t)Q p (t)e -x 2 /2A 2 (t) F t-j (4.13)
and F s = σ {δ u : u = s}. With δ := δ t-j , write also

A 2 (t) = j -1 k=0 ψ 2 k k i=1 1 + δ t-i d + i -1 2 + 1 + δ d + j -1 2 ∞ k=j ψ 2 k k i=1,i =j 1 + δ t-i d + i -1 2 =: α 2 1 + 1 + δ d + j -1 2 α 2 2 , (4.14)
Ã2 (t) := A 2 (t) δ=0 = α 2 1 + α 2 2 , (4.15)
Q(t ) := Q(t) δ=0 = ∞ i=1,i =j 1 + δ t-i d + i -1 , (4.16)
where α 1 is σ {d u : u>t-j }-measurable, and α 2 is σ {d u : u<t-j }-measurable. Then θ(x,j) := A q (t)Q p (t)e -x 2 /2A 2 (t) -E A q (t)Q p (t)e -x 2 /2A 2 (t) F t-j = θ ′ (x, j )θ ′′ (x, j ), where

θ ′ (x, j ) := A q (t)Q p (t)e -x 2 /2A 2 (t) -Ãq (t) Qp (t)e -x 2 /2 Ã2 (t) , θ ′′ (x, j ) := E A q (t)Q p (t)e -x 2 /2A 2 (t) -Ãq (t) Qp (t)e -x 2 /2 Ã2 (t) F t-j .
We shall prove the bound

|θ(x,j)| Cj -1 (1 + x 2 ) Ãq (t)| Q(t)| p e -(x/B) 2 /2 Ã2 (t) . ( 4

.17)

It suffices to prove (4.17) for θ ′ (x, j ) only, as then the corresponding bound for θ ′′ (x, j ) is immediate. Clearly, (4.17) follows from e -x 2 /2A 2 (t)e -x 2 /2 Ã2 (t) Cj -1 x 2 e -(x/B) 2 /2 Ã2 (t) , (4.18)

A q (t) -Ãq (t) Cj -1 Ãq (t), (4.19)

Q p (t) -Qp (t) Cj -1 | Qp (t)|, (4.20)
which will be shown below.

According to (3.1), for any B>1 (arbitrary close to 1), we can find j 0 0s u c h that, for any j>j 0 ,

B -2 inf |u| D/j (1 + u) 2 sup |u| D/j (1 + u) 2 B 2 . (4.21)
Clearly, this implies

sup |u| D/j e -x 2 /2(α 2 1 +(1+u) 2 α 2 2 )
e -(x/B) 2 /2 Ã2 (t) (4.22) and

B -2 Ã2 (t) A 2 (t) B 2 Ã2 (t), B -1 | Q(t)| |Q(t)| B| Q(t)|. (4.23)
Let us prove (4.18). We have

e -x 2 /2A 2 (t) -e -x 2 /2 Ã2 (t) = exp - x 2 2(α 2 1 + (1 + z) 2 α 2 2 ) -exp - x 2 2(α 2 1 + α 2 2 ) = z 0 exp - x 2 2(α 2 1 + (1 + u) 2 α 2 2 ) ′ u du ,
where z := δ/( d + j -1). Using the expression of the derivative exp{-

x 2 2(α 2 1 + (1 + u) 2 α 2 2 ) } ′ u = exp - x 2 2(α 2 1 + (1 + u) 2 α 2 2 ) α 2 2 x 2 (1 + u) (α 2 1 + (1 + u) 2 α 2 2 ) 2
and estimating the right-hand side of the last equation by means of (4.22) and (4.23), relation (4.18) easily follows.

Next, with z = δ/( d + j -1),

A q (t) -Ãq (t) = α 2 1 + (1 + z) 2 α 2 2 q/2 -(α 2 1 + α 2 2 ) q/2 = (q -2)α 2 2 z 0 α 2 2 + (1 + u) 2 α 2 2 (q-2)/2 (1 + u) du,
and so (4.19) easily follows from (4.23). Finally, since Q(t) = Q(t)(1 + z), (4.20) is immediate from (4.23). This proves (4.17).

Next, from (4.17) and (4.13) we obtain

|V(t,j)| Cj -1 Ãq (t)| Q(t)| p R |h(x)| 1 +|x| k+2 e -(x/B) 2 /2 Ã2 (t) dx,
and so

EU 2 (t, j ) EV 2 (t, j ) Cj -2 E A 2q (t)| Q(t)| 2p R |h(x)| 1 +|x| k+2 e -(x/B) 2 /2A 2 (t) dx 2 ,
where we used the fact that, for all sufficiently large j j 0 , one has Ã2 (t) A

I := |h(x)|(1 +|x| k+2 )e -(x/B) 2 /2 Ã2 (t) dx I 1/a 1 I 1/a ′ 2
,where

I 1 := 1 √ 2π A(t)B |h(x)| a e -x 2 /2A 2 (t)B 2 dx = E |h(BX)| a | D , I 2 := √ 2πA(t)B a ′ /a 1 +|x| k+2 a ′ e -x 2 /2A 2 (t)B 2 dx,
and I 2 CA r (t) for suitable r>0(r can be explicitly found). Therefore (with q = 2q + 2r/a ′ ),

E A 2q (t) Q(t) 2p I 2 E A q (t) Q(t) 2p I 2/a 1 (EI 1 ) 2/a E A q (t) Q(t) 2p (a/(a-2) (a-2)/a ,
where the last expectation is finite by Lemma 4.1, and EI 1 < ∞ by condition (3.7). Lemma 4.2 is proved.

LEMMA 4.3. For any ǫ>0 and r>0,thereexistN 0 > 0 and r 0 > 0 such that, for all N>N 0 , the inequalities

a j (t) j ( d+ǫ)-1 , ∀1 t N, ∀j>( log N) r 0 , (4.24) ∞ j =0
a j (t)a j +s-t (s) |s -t| 2( d +ǫ)-1 , ∀1 t s N, s-t>(log N) r 0 (4.25) hold with probability not less than 1 -Ne -(log N) r .

Proof. Let ǫ ′ := ǫ/2, ǫ ′′ := ǫ ′ /2, ψj := ψ j ( d + ǫ ′ ), and Qj (t) := a j (t)/ ψj = j i=1 (1+(δ t-i -ǫ ′ )/( d +ǫ ′ +i -1)). In view of the assumption that |d t | D, one can P. Doukhan, G. Lang, D. Surgailis choose a (nonrandom) j 0 1 large enough so that |(δ t-iǫ ′ )/( d + ǫ ′ + i -1)| < 3/2 a.s. for all i>j 0 and t ∈ Z. Then, using the trivial bound |a j (t)| D j a.s., one obtains

| Qj (t)|=| Qj 0 (t)| j i=j 0 +1 1 + δ t-i -ǫ ′ d + ǫ ′ + i -1 C j 0 exp j i=j 0 +1 δ t-i -ǫ ′ d + ǫ ′ + i -1 , (4.26)
where C j 0 := D j 0 /| ψj 0 |. Introduce the following notation:

S j 0 ,j := j i=j 0 +1 δ t-i -ǫ ′ d + ǫ ′ + i -1 , T j 0 ,j := j i=j 0 +1 1 d + ǫ ′ + i -1 , α t,j := j i=1 (δ t-i -ǫ ′′ ).
Thus, | Qj (t)| C j 0 e S j 0 ,j . We want to evalute the probability of the event 1 t N j>K 0 {S j 0 ,j 0} or the probability of 1 t N j>K 0 {| Qj (t)| C j 0 }. Let j>K 0 (for K 0 1 specified below). Then S j 0 ,j =-ǫ ′′ T j 0 ,j + S j 0 ,K 0 +

j i=K 0 +1 α t,i -α t,i-1 d + ǫ + i -1 ,
where the last sum equals

α t,j d+ǫ ′ +j -1 - α t,K 0 d+ǫ ′ +K 0 + j -1 i=K 0 +1 α t,i
( d+ǫ ′ +i-1)( d+ǫ ′ +i) .Therefore, S j 0 ,j = S ′ j 0 ,j + S ′′ j 0 ,j ,where

S ′ j 0 ,j := -ǫ ′′ T j 0 ,j + S j 0 ,K 0 - α t,K 0 d + ǫ ′ + K 0 , S ′′ j 0 ,j := α t,j d + ǫ ′ + j -1 + j -1 i=K 0 +1 α t,i ( d + ǫ ′ + i -1)( d + ǫ ′ + i) .
Let ǫ,K 0 ,N :=

1 t N j>K 0 j -1 j i=1 δ t-i ǫ ′′ . (4.27)
By Bernstein's inequality for sums of bounded iid rv's, for any ǫ>0, one can find c 0 > 0a n dj 0 such that P(|j -1 j i=1 δ t-i | >ǫ ′′ ) e -c 0 j holds for all j j 0 and, therefore,

P( ǫ,K 0 ,N )>1 -Ne -c 1 K 0 (4.28)
holds for all N>N 0 and K 0 large enough and some c 1 > 0 independent of N,K.

By the definition of α t,j ,o nt h es e t ǫ,K 0 ,N , one has α t,j 0 (for all j>K 0 and 1 t N ) and, therefore, on the same set ǫ,K 0 ,N , one has S ′′ j 0 ,j 0f o ra l l 1 t N . It remains to evaluate S ′ j 0 ,j (on the set ǫ,K 0 ,N ). Clearly,

|S j 0 ,K 0 | C K 0 i=j 0 +1 i -1 C log K 0 a.s. and | α t,K 0 d+ǫ+K 0 | C a.
s. for some (nonrandom) constant independent of K 0 ,N.A l s o ,T j 0 ,j j j 0 x -1 dx = log jlog j 0 (j>j 0 ), and we obtain

S ′′ j 0 ,j -ǫ ′′ log j + C log K 0 + C 0 (4.29)
whenever j (eK 0 ) C/ǫ ′′ .Letr 0 := 2rC/ǫ ′′ and K 0 (N ) := (log N) r /c 1 .

Then j (eK 0 (N )) C/ǫ ′′ holds for j>( log N) r 0 and, moreover, c 1 K 0 (N ) > (log N) r holds for all N>N 0 large enough. We just proved that the inequality

P |a j (t)| <C j 0 | ψj |∀1 t N,∀j>( log N) r 0 > 1 -Ne -(log N) r (4.30)
holds for all N>N 0 large enough. The statement of the lemma concerning the event (4.24) now follows from the fact that C j 0 ψj = C j 0 |ψ j ( d + (ǫ/2))| <j d+ǫ-1 for all j>j 0 ( d,ǫ,C j 0 ) large enough. Next, consider the probability of (4.25). By (4.30), for s -t>( log N) r 0 and N> N 0 , the probability that the following inequalities hold

∞ j =0 a j (t)a j +s-t (s) C j 0 0 j (log N) r 0 a j (t) ψj+s-t + C 2 j 0 j>(log N) r 0 | ψj ψj+s-t | (4.31)
is not less than 1 -Ne -(log N) r . Using the trivial bound |a j (t)| C 2 j C 3 a.s. for some (nonrandom) C 2 ,C 3 > 0, we see that the right-hand side of (4.31) does not exceed

C 4 (log N) r 0 (1+C 3 ) |s -t| d+ǫ-1 +|s -t| 2( d+ǫ)-1 2C 4 |s -t| 2( d +ǫ)-1
for some (nonrandom) C 4 < ∞ and all |t -s| >(log N) r ′ 0 , r ′ 0 := r 0 (1 + C 3 )/ d.Toget the final bound as in (4.25), we replace the previous r 0 by r ′ 0 . Lemma 4.3 is proved.

PROOF OF THEOREM 3.1

Without loss of generality, assume that Eh(X t ) = 0. Recall the Hermite expansion in (3.4). Accordingly, we write the conditional Hermite expansion

h(X t ) = ∞ k=0 h k (t), h k (t) := 1 k! g k (A(t))H k (X t ; A(t)) (k 0), (5.1)
which converges conditionally in L 2 (i.e., with respect to the conditional probability

P D [•] = P[•|D]) a.
s. and, therefore, also unconditionally in L 2 for all t ∈ Z.B yt h e orthogonality property of Hermite polynomials and using the fact that A 2 (t) 1 a.s., we have

var D (h(X t )) = ∞ k=1 1 k! g 2 k (A(t))A 2 (t) ∞ k=1 1 k! g 2 k (A(t)), (5.2)
cov D h(X t ), h(X t ′ ) = ∞ k=1 1 k! g k (A(t))g k (A(t ′ )) ∞ j =0 a j (t)a j +t ′ -t (t ′ ) k . (5.3) Split h(X t ) = h ′ t + h ′′ t ,
where

h ′ t := 0 k k 0 h k (t), h ′′ t := k>k 0 h k (t). (5.4)
Let us show that there exist (nonrandom) k 0 1andr>0 such that

E N t=1 h ′′ t 2 = O N(log N) r . (5.5)
In other words, we want to show that partial sums of (h ′′ t ) are negligible with respect to partial sums of (h ′ t ) which will be shown below to give the limit law of partial sums of (h(X t )) as in Theorem 3.1.

To prove (5.5), split E( N t=1 h ′′ t ) 2 = 1 (N ) + 2 2 (N ),where

1 (N ) := 1 t,s N,|s-t| (log N) r 0 Eh ′′ t h ′′ s , 2 (N ) := 1 t<s N,s-t>(logN) r 0 Eh ′′ t h ′′ s ,
and where r 0 > 0 will be determined below. Clearly,

| 1 (N )| N(log N) r 0 E(h ′′ (0)) 2 N(log N) r 0 Eh 2 (X 0 ). Let us prove 2 (N ) = O(N (log N) r ).Sincek A * 1, 0 < d< 1/2, and (1 -2 d)k A
* < 1, one can take ǫ>0 small enough so that 0

< d + ǫ<1/2. Choose k 0 >k A * such that 1 -2( d + ǫ) k 0 > 1. (5.6) Let ρ N (τ ) := sup 1 t<s N s -t = τ ∞ j =0
a j (t)a j +s-t (s) .

By Lemma 4.3, for all ǫ>0a n dr>0, there exist r 0 > 0a n dN 0 > 0s uc ht ha t ,for any N N 0 ,

P( ǫ,N,r ) := P ρ N (τ ) τ 2( d+ǫ)-1 , ∀τ>( log N) r 0 1 -Ne -(log N) r . (5.7)
By the orthogonality property,

2 (N ) = s-t>(log N) r k>k 0 1 k! EG k (s, t), G k (s, t) := g k (A(s))g k (A(t)) ∞ j =0
a j (t)a j +s-t (s) k .

By the Cauchy-Schwarz inequality,

|G k (s, t)| g k (A(s))A k (t)g k (A(s))A k (t) 1 2 g 2 k (A(s))A 2k (s) + g 2 k (A(t))A 2k (t)
and, therefore, by (5.2)

k>k 0 E|G k (s, t)|I( c ǫ,N,r ) 1 2 E k>k 0 g 2 k (A(s))A 2k (s) k! + k>k 0 g 2 k (A(t))A 2k (t) k! I( c ǫ,N,r ) 1 2 E var D (h(X t )) + var D (h(X t )) I( c ǫ,N,r ).
Next, by using (5.7), E|h(X 0 )| a < ∞ (a>2), and the Hölder inequality, we have

E var D (h(X t ))I ( c ǫ,N,r ) E 2/a (E D h(X t )) a (P( c ǫ,N,r )) a/(a-2)
E 2/a |h(X 0 )| a Ne -(log N) r a/(a-2) , implying

k>k 0 1 t,s N 1 k! E|G k (s, t)|I( c ǫ,N,r ) CN 2+a/(a-2) e -c 2 (log N) r = O(N )
provided that r>1 + a/(a -2) was chosen large enough; c 2 := a/(a -2)>0. Finally, by the definition of ǫ,N,r ,fors -t>( log N) r 0 > 1, we obtain

k>k 0 1 k! E G k (s, t)I ( ǫ,N,r k>k 0 1 k! E g k (A(s))g k (A(t)) ρ k N (s -t)I( ǫ,N,r ) |s -t| 2( d+ǫ)-1 k 0 k>k 0 1 k! E g k (A(s))g k (A(t)) C |s -t| 2( d +ǫ)-1 k 0 k>k 0 1 k! E g 2 k (A(s)) + g 2 k (A(t)) C|s -t| -(1-2( d+ǫ))k 0 Eh 2 (X 0 ) C|s -t| -(1-2( d+ǫ))k 0 ,
where we recall that k 0 was chosen so that (1

-2( d + ǫ))k 0 > 1. Hence, 1 s,t N,s-t>(log N) r 0 k>k 0 1 k! E G k (s, t)I ( ǫ,N,r = O(N ),
thereby proving (5.5).

Let us prove that

N (1-2 d)(k A * /2)-1 [Nτ] t=1 h ′ t → fidi β k A * k A * ! J k A * (τ ), (5.8)
E N t=1 h ′ t 2 = O N 2-(1-2 d)k A * .
(5.9)

From (5.1), using the properties and notation of Wick products (see Surgailis [START_REF] Surgailis | Long-range dependence and Appell rank[END_REF]), we have

h ′ t = k 0 k=0 g k (A(t)) k! a j 1 (t) •••a j k (t) :ε t-j 1 •••ε t-j k :≡ k 0 k=0 1 k! Z k (t), (5.10)
where the middle sum is taken over all j 1 ,...,j k = 0, 1,.... Next, we decompose each of the "chaotic" terms k (t) as

Z k (t) = Z 0k (t) + Z 1k (t) + Z 2k (t), (5.11)
where

Z 0k (t) := E[g k (A(0))Q k (0)]H k (Y t , ), (5.12)
Z 1k (t) := (g k (A(t))Q k (t) -E[g k (A(t))Q k (t)])H k (Y t , ), (5.13)
Z 2k (t) := g k (A(t)) a j 1 (t) •••a j k (t)-Q k (t)ψ j 1 •••ψ j k ):ε t-j 1 •••ε t-j k : , (5.14)
where we used the fact that

ψ j 1 •••ψ j k :ε t-j 1 •••ε t-j k := H k (Y t , )
is a Hermite polynomial in the Gaussian FARIMA(0, d,0) process Y t := ∞ j =0 ψ j ε t-j with variance 2 := EY 2 0 = ∞ j =0 ψ 2 j . Note that Z 0k (t) = β k H k (Y t , ) ≡ 0f o r k<k A * , according to the definitions of k A * in (3.8) and k in (3.6). By (5.10) and (5.11), the proof of (5.8) and (5.9) reduces to in view of the fact that k A * < 1/(1 -2 d). Relations (5.15)-(5.17) are well known (see Taqqu [START_REF] Taqqu | Weak convergence to fractional Brownian motion and to the Rosenblatt process[END_REF]). Consider (5.18). Let φ t := g k (A(t))Q k (t); then, by independence of (d t ) and (ε t ),

N (1-2 d)(k/2)-1 [Nτ] t=1 H k (Y t ; ) → fdd J k (τ ), 1 k< 1 1 -2 d , (5.15)
E N t=1 H k (Y t ; ) 2 = O N 2-(1-2 d)k , 1 k< 1 1 -2 d , (5.16)
E N t=1 H k (Y t ; ) 2 = O(N log N), k 1 1 -2 d , (5.17)
E N t=1 Z 1k (t) 2 = O(N log N), k 0, ( 5
cov Z 1k (0), Z 1k (t) = cov(φ 0 ,φ t )cov H k (Y 0 , ),H k (Y t , ) .
Clearly, the above φ t is a particular case of (4.8) in Lemma 4.2 yielding cov(φ 0 ,φ t ) = O(t -1 log t), hence, also cov(Z 1k (0), Z 1k (t)) = O(t -1 log t) for all k 0, thus proving (5.18). Consider (5.19), which obviously follows from r k (t)

:= cov(Z 2k (0), Z 2k (t)) = O(t -1 log t).W ehave r k (t) = k! Eg k (A(0))g k (A(t)) j 1 ,...,j k 0 a j 1 (0) •••a j k (0) -Q k (0)ψ j 1 •••ψ j k × a t+j 1 (t) •••a t+j k (t) -Q k (t)ψ t+j 1 •••ψ t+j k .
For simplicity, we consider the case k = 2. Let Qj (t) := Q j (t) -Q(t) so that a j (t) = (Q(t) + Qj (t))ψ j .Thenr 2 (t) can be rewritten as

r 2 (t) = 2 j 1 ,j 2 ψ j 1 ψ j 2 ψ t+j 1 ψ t+j 2 t,j 1 ,j 2 , where | t,j 1 ,j 2 |:= E g 2 (A(0))g 2 (A(t)) Q(0) Qj 1 (0) + Q(0) Qj 2 (0) + Qj 1 (0) Qj 2 (0) × Q(t) Qt+j 1 (t) + Q(t) Qt+j 2 (t) + Qt+j 1 (t) Qt+j 2 (t) g 2 (A(0) b g 2 (A(t)) b Q(0) Qj 1 (0)+Q(0) Qj 2 (0)+ Qj 1 (0) Qj 2 (0)) b ′ × Q(t) Qt+j 1 (t) + Q(t) Qt+j 2 (t) + Qt+j 1 (t) Qt+j 2 (t) b ′
and where b>2, 1/b + 1/b ′ = 1/2. Let us check that there exists b>2s u c h that E|g k (A(0)| b < ∞(∀k 1).L e t2<b<a,w h e r ea>2i st h es a m ea si n the formulation of Theorem 3.1. By the Hölder inequality,

|E D h(X)H k (X, A)| b E b/a D |h(X)| a E b/a ′ D |H k (X, A)| a ′ (1/a + 1/a ′ = 1) and then E g k (A(0)) b = EA -2kb |E D h(X)H k (X, A)| b EA -2kb E b/a D |h(X)| a E b/a ′ D |H k (X, A)| a ′ E b/a |h(X)| a E (a-b)/a A -2kb E b/a ′ |H k (X; A)| a ′ a/(a-b) ,
where the first expectation on the last line is finite because of condition (3.7) of Theorem 3.1 (see also Remark 3.1), and the last expectation is dominated by E|A| q for suitable q<∞ and, therefore, is also finite by Lemma 4.1. From Eqs. (4.2) and (4.1), we have that Qj (t) p Cj -1/2 and Q(t) p C for all j,p > 1; hence, the Cauchy-Schwarz inequality yields

Q(0) Qj 1 (0) + Q(0) Qj 2 (0) + Qj 1 (0) Qj 2 (0)) b ′ C j -1/2 1 + j -1/2 2 , Q(t) Qt+j 1 (t)+Q(t) Qt+j 2 (t)+ Qt+j 1 (t) Qt+j 2 (t)) b ′ C (t +j 1 ) -1/2 +(t +j 2 ) -1/2 ,
and using arguments as in the proof of relation (4.25) and j>0 |ψ j ψ t+j | < ∞,w e finally obtain

|r 2 (t)| C j 1 ,j 2 0 ψ j 1 ψ j 2 ψ t+j 1 ψ t+j 2 j -1/2 1 + j -1/2 2 (t + j 1 ) -1/2 + (t + j 2 ) -1/2 = o(t -1 ), proving (5.19) for k = 2.
The case k 2 is considered similarly, but now one needs to use the Hölder inequality for products of k factors. Now we have r k (t) = k! j 1 ,...,j k ψ j 1 •••ψ j k ψ t+j 1 •••ψ t+j k t,j 1 ,...,j k , where t,j 1 ,...,j k is controlled as above: The two first factors were already estimated. In the last expression, S(t) is a sum of products of Q(t) and Qj (t) for j = t + j 1 ,...,t+ j k ,inwhichsome Qj (t) appears at least once by using the elementary identity x ky k = (xy)(x k-1 + ••• + y k-1 ). Exactly the same arguments thus yield this more general result, since S(t) b ′ C{(t + j 1 ) -1/2 +•••+(t + j k ) -1/2 }. Now we have k (t)| C j 1 ,...,j k 0

|ψ j 1 •••ψ j k ψ t+j 1 •••ψ t+j k | × j -1/2 1 +•••+j -1/2 k (t + j 1 ) -1/2 +•••+(t + j k ) -1/2
= o(t -1 ). P. Doukhan, G. Lang, D. Surgailis

The proof of the finite-dimensional convergence in Theorem 3.1 (3.9) follows from (5.8) and (5.4), (5.5), (5.9). The tightness in Theorem 3.1 (3.9) follows by the Kolmorogov's criterion, or E( [N(τ+h)] [N(τ)] X t ) 2 CN 1+γ ,γ := 1 -(1 -2 d)k A * > 0, which follows from (5.4), (5.5), (5.9), and the stationarity of (X t ). This completes the proof of Theorem 3.1.

We end the paper with few examples of nonlinear functions h in which the limit process in Theorem 3.1 (3.9) is identified.

Example 1. Let h(x) = x 2 .T h e nH 2 (x; A) = x 2 -A 2 ,g 2 (A) = 2,g 0 (A) = A 2 , β 0 = EA 2 (0), β 1 = 0, and β 2 = 2EQ 2 (0). The same coefficients can be obtain by differentiating the function Eh(X 0 + wQ(0)) = E(X 0 + wQ(0)) 2 = EA 2 (0) + w 2 EQ 2 (0) (see (3.10)). From Theorem 3.1 we obtain that, for 1/4 < d<1/2,

N -2 d [Nτ] t=1 (X 2 t -EX 2 t ) → D[0,1] E[Q 2 (0)]J 2 (τ ),
where J 2 is the Rosenblatt process.

Example 2. Let h(x) = x 3 .ThenH 3 (x; A) = x 3 -3xA 2 ,g 3 (A) = 6,g 2 (A) = 0, g 1 (A) = 3A 2 ,g 0 (A) = 0,β 0 = 0,β 1 = 3EA 2 (0)Q(0), β 2 = 0, and β 3 = 6EQ 3 (0). From Theorem 3.1 we obtain that, for 0 < d<1/2,

N -d-(1/2) [Nτ] t=1 X 3 t → D[0,1] 3 E A 2 (0)Q(0) J 1 (τ ),
where J 1 is a fractional Brownian motion. Moreover, if β 1 = 3EA 2 (0)Q(0) = 0a n d 1/3 < d<1/2, then

N -3 d+(1/2) [Nτ] t=1 X 3 t → D[0,1] E Q 3 (0) J 3 (τ ),
where J 3 is a Hermite process of order 3. where F(x)= P(Z x),Z ∼ N(0, 1),and

β k = 1 √ 2π
E e -y 2 /2A(0) Q k (0)A -k (0)H k (y/A(0)) (k 1).

In particular, β 1 ≡ β 1 (y) := (2π) -1/2 E[e -y 2 /2A 2 (0) Q(0)/A 2 (0)]. From Theorem 3.1 we obtain that, for any 0 < d<1/2,

N -d-(1/2) N t=1
I(X t y) -P(X t y) → fdd β 1 (y)J 1 (1), (5.20) where J 1 (1) is a normal random variable. It seems that the convergence in (5.20) can be extended to a functional convergence in the Skorohod space D( R) with the suptopology, using the argument of Dehling and Taqqu [START_REF] Dehling | The empirical process of some long-range dependent sequences with an application to U -statistics[END_REF]. Note that the limit process in the above equation is degenerate, similarly as in other papers on empirical processes under long memory (see Dehling and Taqqu [START_REF] Dehling | The empirical process of some long-range dependent sequences with an application to U -statistics[END_REF], Ho and Hsing [START_REF] Ho | On the asymptotic expansion of the empirical process of long memory moving averages[END_REF], Giraitis and Surgailis [START_REF] Giraitis | Central limit theorem for the empirical process of a linear sequence with long memory[END_REF], Doukhan et al. [START_REF] Doukhan | Asymptotics of weighted empirical processes of linear fields with long-range dependence[END_REF], [START_REF] Doukhan | Functional limit theorem for the empirical process of a class of Bernoulli shifts with long memory[END_REF], and the references therein).

2 =

 2 .18) P. Doukhan, G. Lang, D. Surgailis E O(N log N), k 0, (5.19)

t,j 1

 1 ,...,j k g 2 (A(0)) b g 2 (A(t)) b S(0) b ′ S(t) b ′ .

Example 3 .

 3 Let h(x) = I(x y) be the indicator function. Then

  The process J 1 is a fractional Brownian motion (up to the constant E 1/2 J 2 1 (1)).

	Other properties of J k in (2.2) including the expressions for
	EJ 2 k (1) can be found in Taqqu [15].
	THEOREM 2.1. (i) Let d<1/2. Then the series X A t ,X B t in (1.4) converge a.s. and in L 2 for all t ∈ Z and define strictly stationary and ergodic processes with zero mean EX A t = EX B t = 0 and respective covariances

The research was partially supported by the bilateral France-Lithuania scientific project Gilibert and the Lithuanian State Science and Studies Foundation, grant no.T-10/06.

Randomly fractionally integrated processes

Randomly fractionally integrated processes


Sergio Ortiz

Mustapha-Kamel Dali-Yahia

Rapha Ël Grougnet

Yannick Estevez

Sylvie Michel

Sabrina Boutefnouchet

Gr Égory Genta-Jouve
email: gregory.genta-jouve@parisdescartes.fr

Combining both quantum mechanics and Hierarchical Organisation of Spherical Environments for natural products 2D NMR spectra simulation

Keywords:

Structure elucidation of unknown compounds has been the task of natural products chemists for a long time and their approach seems to be the most efficient when it comes to identification. Very recently, Kokkotou et al. proposed a work-flow based on an integrative approach for new metabolite identification. This approach, including prediction of 2D NMR HSQC spectra, demonstrated the necessity for implementation of new algorithms leading to the prediction of 2D NMR spectra. Herein we present a method based on both quantum mechanics (QM) and Hierarchical Organisation of Spherical Environments (HOSE) of the prediction 2D NMR spectra.

Introduction

Identification of small molecules, such as metabolites, remains a difficult task [START_REF] Bross-Walch | Strategies and tools for structure determination of natural products using modern methods of nmr spectroscopy[END_REF][START_REF] Hoffmann | Improving natural products identification through targeted lc-ms/ms in an untargeted secondary metabolomics workflow[END_REF]. It is one of the largest challenges in chemistry today and has even been considered as the main bottleneck in metabolomics for the last decade [START_REF] Neumann | Computational mass spectrometry for metabolomics: Identification of metabolites and small molecules[END_REF][START_REF] Salek | The role of reporting standards for metabolite annotation and identification in metabolomic studies[END_REF]. Structure elucidation of unknown compounds has been the task of natural products chemists for a long time and their approach seems to be the most efficient when it comes to identification [START_REF] Blunt | Marine natural products[END_REF][START_REF] Sumner | Modern plant metabolomics: advanced natural product gene discoveries, improved technologies, and future prospects[END_REF]. The main problem of the natural products approach is that it is time consuming. Indeed, the isolation and characterisation of a new compound is usually performed within months as it requires a sufficient amount of a pure compound to be successful. In the last decade, several hyphenated approaches have been proposed to overcome this issue, mainly using high performance liquid chromatography coupled to mass spectrometry (LC-MS) [START_REF] Cristiano | High resolution ultra high pressure liquid chromatography-time-of-flight mass spectrometry dereplication strategy for the metabolite profiling of brazilian lippia species[END_REF][START_REF] Scheubert | Computational mass spectrometry for small molecules[END_REF][START_REF] Wolfender | Current approaches and challenges for the metabolite profiling of complex natural extracts[END_REF]. Very recently, Kokkotou et al. proposed a work-flow based on an integrative approach for new metabolite identification [START_REF] Kokkotou | An integrated approach using uhplc-pda-hrms and 2d hsqc nmr for the metabolic profiling of the red alga Laurencia: Dereplication and tracing of natural products[END_REF]. This approach, including prediction of 2D NMR HSQC spectra, demonstrated the necessity for implementation of new algorithms leading to the prediction of 2D NMR spectra.

Herein we present a method based on both quantum mechanics (QM) and Hierarchical Organisation of Spherical Environments (HOSE) of the prediction 2D NMR spectra. We don't intend to describe here the theory of NMR calculations (DFT or HOSE), for more information, the reader are encouraged to look over [START_REF]Density functional calculations of nmr chemical shifts and esr g-tensors[END_REF] for an overview of the DFT calculations and [START_REF] Bremser | Hose -a novel substructure code[END_REF] for an HOSE description. The main difference between the two approaches is time. While computation of hundreds of spectra can be done in less than a second using HOSE, several hours are required for the calculation using DFT. The main steps involved in NMR calculation using DFT have been well described by Willoughby et al. [START_REF] Patrick | A guide to small-molecule structure assignment through computation of ( 1 h and 13 c) nmr chemical shifts[END_REF] and include a geometry optimization (of each conformer) followed by a frequency calculation. The NMR properties are then calculated (for each conformer) and the chemical shifts are computed using the Boltzmann equation.

In the current approach, the simulation of the 2D NMR spectra is based on the package Spinach [START_REF] Hogben | Spinach -a software library for simulation of spin dynamics in large spin systems[END_REF]. The focus of the present work will be on 2D NMR spectra as several approaches already exist for the prediction of accurate 1D NMR spectra [START_REF] Binev | Structure-based predictions of 1 h nmr chemical shifts using feed-forward neural networks[END_REF][START_REF] Binev | Prediction of 1 h nmr coupling constants with associative neural networks trained for chemical shifts[END_REF]. The magnetic parameters are computed with Gaussian 09 [START_REF] Frisch | Gaussian 09 Revision D.01[END_REF] (coordinates, g-tensor, hyperfine coupling tensors, chemical shielding tensors, scalar couplings) and nmr-shiftdb2 [START_REF] Kuhn | From chemical shift data through prediction to assignment and NMR LIMS -multiple functionalities of nmrshiftdb2[END_REF] (chemical shifts). To illustrate the methodology, the method has been tested on two compounds, caffeic acid (1) which is ubiquitous in plants and seco-tanapartholide A (2), a sesquiterpene lactone isolated from the aerial parts of Pentzia monodiana Maire (1929) (Asteracea) [START_REF] Ozenda | Flore et végétation du Sahara[END_REF]: High pressure extractions were performed on a Speed Extractor E-914 apparatus provided by Büchi (Fawil, Switzerland) equipped with four cells (120 mL) and a collector with four flat bottom vials (220 mL). Maximum pressure and temperature can be adjusted from 50 to 150 bar and from 30 to 200

6 4 5 3
• C, respectively. CPC was performed on a SCPC-250+1000B apparatus provided by Armen Instrument (Sain-Avé, France) fitted with two rotors, containing respectively 1953 or 2016 twin-cells. Rotation speed can be adjusted from 0 to 3000 rpm for the 250 mL rotor and from 0 to 1500 rpm for the 1 L rotor. A valve incorporated in the CPC apparatus allowed operation in ascending or descending mode. Fractions were collected by a Büchi 684 collector (Flawil, Switzerland). The experiments were conducted at room temperature. All the reagents and solvents were purchased from Carlo Erba-SDS. Cyclohexane and EtOAc were distilled before use. Water was distilled from deionized water. TLC were performed on Merck silica-gel 60 F 254 aluminum sheets, using vanillin/H 2 SO 4 as spray reagent.

NMR acquisition parameters

For compound 1: The pulse conditions were as follows: spectrometer frequency (SF) 399.83 MHz ( The upper phases were used as the mobile phases, whereas the lower phases were employed as the stationary phases. A sample of 400 mg of F17-22 was dissolved in a 1:1 (v/v) mixture of the biphasic system of S1.

The separation was performed in ascending mode on 250 mL rotor, flow rate of 5 mL min -1 at 1600 rpm. A total volume of 1.5 L of mobile phase was collected in fractions of 10 mL. Then the extrusion step was initiated in ascending mode using the lower phase as the mobile phase to ensure that any residual extract was recovered. Compound 1 was isolated from fractions 16 to 20 (23 mg).

Magnetic properties calculations

All density functional theory (DFT) have been performed using the Gaussian 09W package [START_REF] Frisch | Gaussian 09 Revision D.01[END_REF] and the B3LYP functional at the 6-311+g(d,p) levels of theory. Structure optimisation was followed with frequency calculation in order to check the absence of imaginary modes. Calculations of the magnetic properties (coordinates, g-tensor, hyperfine coupling tensors, chemical shielding tensors, scalar couplings) have been performed on the optimized conformer. The resulting set of tensor values were converted to chemical shifts by subtracting the shielding tensor value of tetramethylsilane (TMS) at the same level of theory. HOSE chemical shifts prediction was realised thanks to nmrshiftdb2 [START_REF] Kuhn | Facilitating quality control for spectra assignments of small organic molecules: nmrshiftdb2 -a free in-house nmr database with integrated lims for academic service laboratories[END_REF][START_REF] Steinbeck | Nmrshiftdbconstructing a free chemical information system with open-source components[END_REF] using the DFT optimised tridimentional structure as input (.mol). NMR spectra were plotted using the Spinach 1.6 toolbox [START_REF] Hogben | Spinach -a software library for simulation of spin dynamics in large spin systems[END_REF] running in the Matlab R environment.

Results and Discussion

In order to demonstrate the efficiency of the proposed methodology on a natural product, the whole work-flow was first applied to one of the most ubiquitous compound in plants: caffeic acid (1) and then to seco-tanapartholide A (2). While commercial caffeic acid (1) was used, the phytochemical investigation of the aerial parts of P. monodiana was undertaken to obtain compound 2. After extraction of the plant material with ethyl acetate under pressure (100 bar), a first fractionation on silica gel was performed, this step was then followed by a CPC separation that yielded seco-tanapartholide A (1).

After acquisition of 1 H-13 C HSQC spectrum of 1, the prediction of the different magnetic parameters was performed using both GIAO DFT (coordinates, g-tensor, hyperfine coupling tensors, chemical shielding tensors and scalar couplings) and the HOSE prediction (chemical shifts). In table 1 are summarized the experimental and predicted chemical shifts and couplings for compound 1 and 2.

Case study 1: caffeic acid (1) In order to evaluate how close the chemical shifts predictions were to the experimental data, the mean absolute error (MAE) and the corrected mean average error (CMAE) were calculated. As expressed in table 1, both 1 H and 13 C MAE values for the HOSE prediction were smaller from those obtained using DFT (MAE1 H = 0.08 and CMAE13 C = 1.94). Those results were also confirmed by the CMAE values (0.016 and 0.21 for 1 H and 13 C respectively).

A simulation of the HSQC spectrum was then performed to compare the both DFT and HOSE-DFT combined spectra. The current version of the Spinach toolbox [START_REF] Hogben | Spinach -a software library for simulation of spin dynamics in large spin systems[END_REF] is able to read Gaussian 09 outputs file (.out or .log) in order to get the parameters values needed for the NMR spectrum computation. At this step it was then possible to introduce the full chemical shift tensors (in ppm) predicted using the HOSE prediction and imported as a .txt file. The overall work-flow is described below (Figure 1). The HSQC simulations are presented in the figure 2. Crosspeaks between 1H and 13C for the HOSE values are not represented on figure 2 as no peak intensity and resolution were generated from this method. As expected from the values presented in table 1, a relatively good agreement was obtained between experimental HSQC spectrum and both simulated spectra. Only two protons were clearly separated from the experimental for the DFT prediction (H-2 and H-9 at δ H 5.8 and 6.4 ppm instead of 6.22 and 6.78 ppm). On the other hand, only two signals corresponding to one proton (H-9 at δ H 6.67 ppm) were clear separated from the experimental spectrum. This example clearly demonstrated the assets of the method. In order to test the validity of the method on another natural product, calculations were undertaken for a Case study 2: seco-tanapartholide A (2) In the present example, DFT predictions for 13 C shielding tensors were more accurate than the ones obtained using the HOSE algorithm, as MAE values were smaller, 2.91 and and 4.11 ppm for DFT and HOSE respectively. Nonetheless, it is well known that quantum mechanical methods (QM) give quite frequently larger MAE's [START_REF] Elyashberg | Empirical and dft giao quantum-mechanical methods of 13 c chemical shifts prediction: competitors or collaborators[END_REF]. This last assumption was confirmed by looking at the 1 H shielding tensors predictions. Indeed, MAE's values were greater for the DFT (0.36) than for HOSE (0.26). To 3, an overall good agreement was observed between the predicted and the experimental spectra with a higher degree of similarity for the QM predicted spectrum as expected after calculation of the accuracy (see Table 1). However, a disparity was observed between the predicted and the experimental spectra for the equivalent protons (methyls or linear methylenes). For instance, the three H-14 protons (methyl) are represented as mutliple signals at δ H 2.08, 1.95 and 1.71 ppm while on the experimental, only one cross peak was observed at δ H 2.10 ppm. To overcome this problem, the DFT GIAO full chemical shift tensors can be replace by empirical calculated ones (obtained using the HOSE code) and then submitted to the Spinach toolbox. While the full chemical shift tensors (in ppm) were read from the Gaussian output files (.out or .log), Zeeman interactions may also be specified as scalars [START_REF] Hogben | Spinach -a software library for simulation of spin dynamics in large spin systems[END_REF] and read from a .txt file using a short home made script. In the present case, hybridisation of both methods gave a slightly less accurate spectrum in term of chemical shifts, especially for the down-shielded signals at δ H 6.11 6.34 and 7.21 ppm. But one important improvement relied in the fact that the number of cross peaks observed on the HSQC spectrum fitted perfectly with the experimental spectrum (Figure 3). This last point was highlighted by Roussis' group in a recent paper in which they use HSQC spectra for dereplication purposes of the red alga Laurencia [START_REF] Kokkotou | An integrated approach using uhplc-pda-hrms and 2d hsqc nmr for the metabolic profiling of the red alga Laurencia: Dereplication and tracing of natural products[END_REF]. Using this method enabled the prediction of HSQC spectra but also any kind of 2D NMR spectra such NOESY, HMQC or HMBC as it relies on the Spinach toolbox formalism [START_REF] Edwards | Quantum mechanical nmr simulation algorithm for protein-size spin systems[END_REF]. Dereplication strategies using pattern recognition of NMR spectra have been thoroughly studied in the last decade [START_REF] Qiu | 2d nmr barcoding and differential analysis of complex mixtures for chemical identification: The ac-taea triterpenes[END_REF][START_REF] Hubert | Identification of natural metabolites in mixture: A pattern recognition strategy based on 13 c nmr[END_REF] and the proposed strategy is another tool available to the natural products chemists.

Figure 1 .

 1 Figure 1. Global work-flow used for the NMR spectra prediction.

F2Figure 2 .

 2 Figure 2. Overlapping of the experimental and predicted HSQC spectra of compound 1 (full scale available in SI).

Figure 3 .

 3 Figure 3. Overlapping of the experimental and predicted HSQC spectra of compound 2 (full scale available in SI).

Table 1 .

 1 1 H and13 C NMR data for compound 1 and 2

		Experimental		Predicted			Experimental		Predicted
				DFT	HOSE					DFT	HOSE
	position	1 H	13 C	1 H	13 C	1 H	13 C	position	1 H	13 C	1 H	13 C	1 H	13 C
	1	-	127.9	-	133.9	-	128.0	1	-	207.9	-	209.7	-	209.4
	2	7.02	114.7	7.1	116.2	6.27	115.2	2	6.07	133.0	6.33	138.6	6.11	132.5
	3	-	146.8	-	150.0	7.45	145.9	3	7.48	168.9	6.76	174.3	7.21	156.2
	4	-	147.2	-	151.4	-	146.5	4	-	79.5	-	83.70	-	66.9
	5	6.94	122.4	7.1	126.1	7.06	117.2	5	2.69	59.1	2.78	62.53	1.91	70.2
	6	6.78	116.1	6.4	119.9	-	123.2	6	4.46	82.5	4.72	83.42	4.38	74.1
	1'	7.53	146.5	7.8	155.2	-	144.9	7	3.46	41.9	2.57	44.31	3.32	45.7
	2'	6.22	115.4	5.8	114.6	6.82	116.5	8	1.89	28.9	1.55	29.43	1.98	25.2
	3'	-	171.2	-	166.6	6.67	171.5	9	2.60	42.0	2.43	37.29	2.25	42.7
			MAE	0.23	4.05	0.08	1.94	10	-	210.7	-	212.9	-	207.7
			CMAE	0.047	0.45	0.016	0.21	11	-	140.9	-	148.6	-	138.2
								12	-	168,9	-	172.9	-	170
								13a	6.36	123,0	5.81	123.6	6.35	119.9
								13b	5.78		6.83		5.31
								14	2.21	28.8	2.1	31.6	1.91	27
								15	1.58	27.2	1.28	27.0	1.19	28.9
										MAE	0.43	3.1	0.26	4.56
										CMAE	0.03	0.19	0.02	0.29
	illustrate the values presented in table 1, the prediction of
	the HSQC spectra of compound 2 was conducted and was
	presented below. As illustrated on figure

Acknowledgments

The authors want to thank l'Office du Parc National de l'Ahaggar (OPNA, Algérie for field collection. The authors are grateful to the University Paris Descartes for financial support.


Gilles Alcaraz

Lisenn Euzenat

Olivier Mongin

Claudine Katan

Isabelle Ledoux

Joseph Zyss

Mireille Blanchard-Desce
email: mireille.blanchard-desce@univ-rennes1.fr

Michel Vaultier

Improved transparency-nonlinearity trade-off with boroxine-based octupolar molecules

come   L'archive ouverte pluridisciplinaire

A series of octupolar molecules derived from the boroxine framework were designed and their optical nonlinearities were investigated by performing harmonic light scattering experiments in solution; the molecules were found to combine excellent transparency in the near UV-visible region (λ max ≤ 280 nm) and significant first-order hyperpolarisabilities (up to β(0) = 56 10 -30 esu).

The field of molecular nonlinear optics (NLO) has attracted increasing interest over the past twenty years, owing to its connection with fundamental issues such as charge transfer, higher order polarisabilities or conjugation, as well as to numerous applications in various fields such as telecommunications, 1 optical data storage, 2 microfabrication, biological imaging… Within this framework, organic chromophores have received major attention owing to their chemical flexibility. For instance, molecular engineering of "push-pull" compounds has proven particularly successful leading to chromophores displaying record first (β) and second (γ) hyperpolarisabilities. Yet, these chromophores are plagued by an intense absorption in the near-UV-visible region that significantly restricts their transparency range. Novel strategies were thus needed for optimising the efficiency-transparency trade-off.

Octupolar molecules, with 1,3,5-triamino-2,4,6-trinitrobenzene (TATB) as the prototypical structure, provide an interesting alternative for this purpose. 3 Following this approach, a variety of organic molecules with a two-dimensional (2D) character of β 4 as well as several 3D octupolar molecules 5 have been designed and investigated in recent years. [START_REF] Goovaerts | Cross in Handbook of Advanced Electronic and Photonic Materials and Devices[END_REF] Within this context, we have investigated a series of original and transparent octupolar molecules built from the boroxine ring. Their molecular design is based on the symmetrical grafting of three donating appendices on the electron-deficient boroxine core (Scheme 1). The boroxine ring usually leads to quasi planar structures, 7 a common feature with triazines whose octupolar derivatives have been widely studied. 4b,d-f In contrast, boroxine shows much lower aromatic character. 8 We emphasize that although boron inorganic derivatives have long been known to display interesting NLO properties, ‡ the potentialities of organic boroxine derivatives for NLO have not been considered yet.

Phenyl units were used as rigid links between the electronreleasing substituents and the boroxine core. Peripheral groups with increasing donating character were used with the aim of tuning the optical properties. Triarylboroxine derivatives are usually obtained from the corresponding boronic acids via a cyclocondensation reaction leading to the cyclic trimeric anhydride. We have implemented an innovative route that allows an efficient and versatile preparation of triarylboroxines via the borylation/hydrolysis sequence of p-substituted aryl halides 9 (Scheme 1). The presence of electron-donating substituents has been reported to increase the stability of triorganoboroxine derivatives towards hydrolysis. 10 All the derivatives investigated in the present study have been fully characterized by NMR, HRMS and elemental analysis prior to their optical study. X-Ray diffraction from single crystals of 1 shows that this molecule is quasi planar with the three phenyl rings lying approximately in the plane of the boroxine ring (Fig. 1). §

The absorption characteristics of the series of derivatives 1-3 are gathered in Table 1. The molecules show an intense absorption band in the UV region but maintain full transparency in the near UV-visible region. A slight positive solvatochromism is observed, indicative of a larger stabilisation of the excited state as compared to the ground state by a polar solvent. Increasing the donating strength of the substituents leads to a broadening and a bathochromic and hyperchromic shift of the absorption band and to a more pronounced solvatochromism.

D D D O B O B O B X X A 1: X = OMe 2: X = SMe 3: X = NMe 2 X B X H N Hal X Hal = Br, I i ii
Scheme 1 Reagents and conditions: i, i-Pr2NBH2 (2 equiv.), Et3N (5 equiv.), Pd (0) (5 mol%), dioxane, 15h, 78 °C; ii, H2O (1.5 equiv.), 16h, rt. Interestingly, HOMO-LUMO calculations ¶ reveal an intramolecular charge density shift from the donating appendices towards the boron atoms (Fig. 2).

The first hyperpolarisabilities β have been determined by performing harmonic light scattering (HLS) experiments in solution, which yield the HLS molecular averaged hyperpolarisability 〉 〈 2 β =β HLS . 11 HLS experiments were performed at 1.064 µm locating the second harmonic signal in the transparency region.# The corresponding static values β HLS (0) are calculated using a degenerated three-level dispersion factor. 3b The experimental values are compared to those reported for TIATB, 4a a soluble analogue of TATB, using the β X convention. 12 Examination of the series of molecules indicates that increasing the donating strength of the substituents results in a significant enhancement of β. Molecule 3 which bears the strongest electron-releasing subsituents of the series displays a molecular optical nonlinearity nearly twice larger than molecule 1, providing evidence that intramolecular charge transfer influences the nonlinear responses. We note that such boroxine derivative indeed leads to an improved efficiency-transparency trade-off as compared to triazine or TATB octupolar derivatives: molecule 3 displays a β(0) value about one order of magnitude larger than TIATB while remaining blue-shifted by nearly 100 nm and having a molecular weight less than twice larger.

Finally, our study demonstrates that the functionalisation of the boroxine core provides a promising and innovative route for the optimisation of the nonlinearity-transparency trade-off. It also opens the way for further engineering of the optical nonlinearity by boosting the multidimensional intramolecular charge transfer between the core and the peripheral groups. 4h,i This work was supported by the GDR MATFON. The calculations have been supported by CINES-France. We are grateful to T. Roisnel for X-rays diffraction experiments. show good agreement with the values derived from the absorption spectra for the lowest energy transition (Table 1). # This procedure avoids possible contamination of the HLS signal by two-photon excited fluorescence. In addition, the boroxines 1-3 did not show any detectable fluorescence when excited in their absorption band. ε max (M -1 .cm -1 ) FWHM a (cm -1 ) λcut-of (nm) Δν b (cm -1 ) βHLS (10 -30 esu) βHLS(0) (10 -30

Notes and references

Fig. 1

 1 Fig. 1 X-Ray structure of boroxine derivative 1.

  data and UV-Visible spectra for compounds 1-3; computational details. See http://**** ‡ Such as BBO, used in wavelength conversion devices. § Crystal data for 1: C21 H21 B3 O6, M=401.81, monoclinic, space group P21/n, a = 9.0568(2), b = 14.1334(4), c = 16.9564(7) Å, β = 104.5710(10)°, V = 2100.67(11) Å 3 , Z = 4, T = 293(2) K, Dc = 1.271 g cm -3 , R1 = 0.0502, wR2 = 0.1184 for I > 2σ(I). See http://**** for crystallographic data. ¶ The corresponding transition are symmetrically allowed and the calculated energy gaps (242 nm for 1, 268 nm for 2, and 269 nm for 3)

Fig. 2

 2 Fig. 2 Contour surfaces of HOMO (up) and degenerated LUMO, LUMO+1 (down) of organoboroxine 3.

Table 1

 1 Absorption and nonlinear properties of molecules 1-3 in chloroform

	Comp	λmax
	d	(nm)

  esu)

	1	241	4.0 10 4	1750	286	-	47	35
	2	270	4.6.10 4	3360	301	69	45	31
	3	281	5.8.10 4	4350	326	450	83	56
	TIATB 375					10	4.4

a Halfbandwith. b Solvatochromic shift = 1/λmax(CHCl3) -1/λ max(DMSO).


Laurent Porrès

Olivier Mongin

Claudine Katan

Marina Charlot

Thomas Pons

Jerome Mertz

Mireille Blanchard-Desce
email: mireille.blanchard-desce@univ-rennes1.fr

Enhanced Two-Photon Absorption with Novel Octupolar Propeller-Shaped Fluorophores Derived from Triphenylamine

Novel octupolar fluorophores derived from the symmetrical functionalization of a triphenylamine core with strong acceptor peripheral groups via phenylene-ethynylene linkers have been synthesized and shown to exhibit high fluorescence quantum yields, very large TPA cross-sections in the red-NIR region and suitable photostability.

Molecular two-photon absorption (TPA) has attracted growing interest over recent years owing to the many applications it offers both in material science and in biological imaging. These include microfabrication, 1 3D optical data storage, 2 optical power limitation, 3 localized photodynamic therapy, 4 and two-photon laser scanning fluorescence imaging. 5 These applications call for the design of specifically engineered compounds displaying enhanced TPA cross-sections. In the case of two-photon excited fluorescence (TPEF) microscopy, high performance fluorophores must exhibit both high fluorescence quantum yields (φ) and large TPA cross-sections (σ 2 ) in the red-NIR range (700-1200 nm), corresponding to the biological optimal window for reduced photodamage. 5 Photostability is also an important feature to consider for TPEF-based applications.

The optimization of molecular TPA has largely focused on one-dimensional dipolar 6,7 or quadrupolar 3,7,8 structures, and so far only little attention has been devoted to the design of optimized octupolar structures, 9 although it has been realized that increased dimensionality and branched structures 10,11 could lead to highly effective multiphoton absorption. In this perspective, we have investigated a series of novel three-branched octupolar fluorophores with the aim of optimizing their TPEF cross-section (σ 2 φ) in the NIR region, as well as ensuring suitable photostability. Our trigonal derivatives are derived from the symmetrical functionalization of an electron-donating triphenylamine core with three conjugated branches (Scheme 1) bearing electron-withdrawing (A) end-groups (12-15). Although octupolar derivatives based on triphenylamine cores have been shown to lead to high quadratic polarizabilities, 12 only few examples of such derivatives have been shown to display large TPA crosssection, 9a-c calling for further engineering of the basic structures.

To examine the role of octupolar symmetry, we also have investigated a structurally-related analogue bearing only electron-releasing D peripheral groups (11). Following the same line of reasoning, we have selected strong acceptor peripheral groups using in particular the triflyl moiety (compounds 13, 15), a powerful acceptor, [START_REF] Hansch | Hammett constants: σ p = 0.96 for SO 2 CF 3 and σ p = 0.72 for SO 2 Me[END_REF] which has not been considered yet in molecular engineering of TPA. Rigid (11-13) or semi-rigid (14, 15) conjugated rods built from phenylene-ethynylene and phenylene-vinylene units were used to allow Intramolecular Charge Transfer (ICT) to take place between the core and the peripheral groups while maintaining fluorescence. In order to enhance the TPA cross-section, we have investigated molecules of increasing size from 2.9 to 4.3 nm in diameter. B3LYP ab-initio calculations indicate that the molecules adopt a propellershaped structure (Figure 1), the core nitrogen being trigonal and the conjugated branches twisted by about 40°. The synthesis of these fluorophores is based on the three-fold Pd(0)-catalyzed cross-coupling reaction of the nitrogen-cored building block 10 (Scheme 1) with aromatic iodide (1, 14 6, 7) or bromide (2 15 -3 16 ) derivatives. The synthesis of the triphenylamine reagent 10 was achieved by triiodination 17 of triphenylamine, followed by a triple Sonogashira coupling with ethynyltrimethylsilane. The extended building blocks 6 and 7 were obtained from the halo derivatives 4 and 5, respectively, in a three-step sequence (Michaelis-Arbuzov reaction, oxidation and Horner-Wadsworth-Emmons condensation with 4-iodo-benzaldehyde). All new fluorophores have been fully characterized by NMR, HRMS and elemental analysis data.

The photophysical properties of the new series of trigonal fluorophores are shown in Table 1 andFigure 2. We observe that all fluorophores display a broad absorption in the near UV-visible violet region and exhibit high fluorescence quantum yield (about 0.8) independently of their end-groups (D or A) or conjugated branch nature. In contrast their fluorescence lifetimes are significantly affected by the nature of the peripheral groups. We observe that the octupolar derivatives (12-15) display longer fluorescence lifetimes than the analogue bearing only donor peripheral groups (11). Interestingly, the powerful triflyl acceptor leads to a noticeable increase of the fluorescence lifetime, while an increase in the length of the branches by the addition of a phenylene-vinylene unit results in a decrease of the fluorescence lifetime.

Increasing the polarity of the solvent leads to a slight bathochromic shift of the absorption band of octupolar fluorophores (12-15), and to a more pronounced red shift in their emission band (Table 1). Such positive solvatochromic behavior can be related to an ICT phenomenon taking place between the fluorophore center and periphery. 18,19 The TPA spectra of fluorophores 11-15 in the 740-950 nm range were determined by investigating their TPEF in solution according to the protocol described by Webb and coworkers 20 in the femtosecond regime. We note that We note that the character of the end-groups significantly affects the TPA response. The tris-acceptor octupolar fluorophores display much larger TPA crosssections than their tris-donor counterpart (11). For instance, molecule 13 shows a TPA cross-section more than oneorder of magnitude larger than molecule 11 of similar size. Moreover we observe from the comparison of fluorophores 12 and 13 that increasing the acceptor character of the endgroup does lead to an important increase in the TPA crosssection (Figure 3), in correlation with a concomitant increase in the Stokes shift (Table 1) and a red shift of both the absorption and emission spectra (Figure 2).

λ (nm) a log ε b OPA em λ (nm) c stokes shift (cm -1 ) b,d φ b,e τ (ns) b,f σ 2 (GM)
By increasing the distance between the core and peripheral electroactive groups, we aimed at extending the multidimensional charge transfer phenomenon and increasing the nonlinear response. 23 This strategy turned out to be effective, as observed from the comparison of fluorophores 12 and 14, which bear similar acceptor endgroups: increasing the molecular weight of the molecule by 30% (and increasing the size by about 50%) leads to a 4-(respectively 6-) fold enhancement in the TPA crosssection at 2 OPA abs λ (respectively 740 nm). This also leads to a red-shift and broadening of both absorption and emission bands (Figure 2), as well as to an increase in the Stokes shifts, consistent with an increase of the core-to-periphery ICT. A similar trend is noted when comparing fluorophores 13 and 15 (Table 1) -but to a lesser extent -while the main effect lies in the major broadening of the TPA spectrum towards the NIR region. As a result, the elongated octupolar fluorophore 15 bearing strong acceptor peripheral groups maintains sizeable TPA cross-section at 900 nm (Figure 3). Finally, we stress that the use of phenyleneethynylene linkers in the conjugated branches (in place of the more classical phenylene-vinylene linkers) leads to improved photostability, as indicated by the absence of photodegradation either upon pulsed-laser or prolonged lamp irradiation for derivatives 12 and 13.

In summary, we have designed and investigated novel multipolar fluorophores derived from a triphenylamine core. By enhancing the multidimensional ICT between the core and the peripheral groups, we produced fluorophores that combine very large TPA cross-sections in the red-NIR region, high fluorescence quantum yields and suitable photostability. This strategy led to octupolar fluorophores that exhibit some of the highest TPA cross-sections measured to date in the NIR region and in the femtosecond regime. 24 Furthermore, our study provides evidence that the TPA cross-sections of such derivatives can be significantly enhanced in the NIR region with elongated derivatives bearing strong electron-withdrawing peripheral groups, leading to new directions for further engineering of their TPA properties. In particular by modifying and lengthening the conjugated branches, even larger TPA cross-sections should be attainable. 25

Figure 1 .

 1 Figure 1. B3LYP optimized geometry of fluorophore 13.

  Scheme 1 a

a

  One-photon absorption maximum.b In toluene.c One-photon emission maximum. d Stokes shift = (1/λ abs -1/λ em ). e Fluorescence quantum yield relative to fluorescein in 0.1 N NaOH as a standard (λ ex = 470 nm). f Experimental fluorescence lifetime. g TPA cross-sections; 1 GM = 10 -50 cm 4 .s.photon -1 ; TPEF measurements were performed using a mode-locked Ti:sapphire laser delivering 80 fs pulses at 80 MHz, calibrating with fluorescein.the TPA spectra of the octupolar fluorophores 12-15 (Figure3) exhibit a relative maximum close to twice OPA abs λ , indicating that the lowest energy excited-state is both onephoton and two-photon allowed.[START_REF]This excited state is doubly degenerated[END_REF]

Figure 2 .

 2 Figure 2. Normalized absorption and emission spectra of fluorophores 11-15 in toluene.

Figure 3 .

 3 Figure 3. TPA spectra of fluorophores 11-15 in toluene.

Table 1

 1 Photophysical and Two-Photon Absorption Characteristics of Fluorophores 11-15.

	compd	OPA abs

Acknowledgment. We acknowledge financial support from CNRS (ATIP and NOI programs), and Rennes Métropole. L. P. and M. C. received fellowships from MENRT and DGA, respectively. We thank K. Rousseau and E. Ronceray for assistance in the synthesis. Calculations were supported by the "Centre Informatique National de l'Enseignement Supérieur" (CINES-France).

Supporting Information Available: Computational details. Photophysical methods (absorption, fluorescence and TPEF). Description of all new fluorophores. This material is available free of charge via the Internet at http://pubs.acs.org.


Laurent Porrès

Claudine Katan

Olivier Mongin

Thomas Pons

Jerome Mertz

Mireille Blanchard-Desce

Two-photon absorption and fluorescence in nanoscale multipolar chromophores: effect of dimensionality and charge-symmetry

Keywords: two-photon absorption (TPA), two-photon-excited fluorescence (TPEF), quadrupole, octupole, dendrimer

come

Introduction

# Present address: Boston University, BME Dept., 44 Cummington St., Boston, MA 02215 Journal of Molecular Structure, Volume 704, Issues 1-3, 18 October 2004, Pages 17-24, http://dx.doi.org/10.1016/j.molstruc.2003.12.036 Molecular two-photon absorption (TPA) has attracted increasing interest over recent years owing to the many applications it offers. These include microfabrication [1], 3D optical data storage [2], photodynamic therapy [3], and two-photon laser scanning fluorescence imaging [4]. These applications call for the design of specifically engineered compounds displaying enhanced TPA cross-sections. In the case of two-photon-excited fluorescence (TPEF) microscopy, novel fluorophores combining high fluorescence quantum yield (φ) and TPA cross-section (σ 2 ) orders of magnitude larger than endogenous chromophores in the red-NIR range (700-1200 nm) are desirable for non-damaging biological imaging. TPA also offers promising potentialities for optical power limitation [5]. Such a nonlinear process offers the possibility of maintaining high transparency in ambient light and achieving efficient and immediate protection against the high intensity delivered by pulsed lasers. Candidate molecules for optical power limitation should exhibit high solubility, high linear transmission at low intensities (i.e. low linear absorption) together with very large multiphoton absorptivities at the intensities delivered by high power pulsed lasers.

Until recently, optimization of molecular TPA has focused mainly on one-dimensional dipolar [6,7], or quadrupolar [7,8] structures. However, it has been realized lately that twodimensional [9], branched [10] and dendritic [11,12] structures could lead to highly effective multiphoton absorption. In this context, we have been interested in the elucidation of the role of dimensionality and charge-symmetry on molecular TPA. We have investigated a series of structurally-related chromophores of different symmetry (quadrupolar, C 2v , octupolar...) and shape (rod-like, propeller-shaped, Y-shaped, dendritic…). Their design is based on the functionalization of either 1-D linear or 2-D branched conjugated backbones with electronreleasing or electron-withdrawing peripheral groups (Fig. 1a). The structures were built from biphenyl and its trigonal analogue 1,3,5-triphenylbenzene units as core/node moieties. The triphenylbenzene moiety acts as a trigonal node that maintains a large distance between the conjugated branches, thus preventing strong through-space electronic interactions. B3LYP abinitio calculations indicate that the branched chromophores derived from the trigonal core adopt a propeller-shaped structure, the conjugated branches being planar and twisted by about 38° with respect to the core phenyl ring (Fig. 1b). Likewise, the quadrupolar chromophore 1 derived from the biphenyl core moiety displays two planar conjugated branches twisted by 34° in the middle of the biphenyl moiety. Arylene-vinylene units were used as connecting spacers between the core/nodes and the peripheral groups to ensure effective electronic conjugation between end-groups and core/node moieties and allow multidimensional intramolecular charge transfer to take place within the chromophores. The one-and two-photon absorption and fluorescence properties have been investigated and compared in order to evaluate the role of dimensionality and chargesymmetry on both their photophysical and TPA properties. In addition, HOMO-LUMO calculations have been carried out to provide additional information on the topology of the electronic charge redistribution occurring in those chromophores upon excitation.

Journal of Molecular

Experimental and computational details

Chromophores

Molecules 1-4 (Fig. 1a) have been synthesized using the synthetic approach reported in ref. [12]. Their purity was checked by NMR, HRMS and elemental analysis prior to photophysical studies.

Computational details

Calculations and especially geometry optimization were conducted using the Gaussian 98 package [13] with the B3LYP [14] density functional and the 6-31G* basis set. For the sake of simplicity, calculations have been conducted on chromophores where the alkyl solubilizing chains have been replaced by methyl substituents. Illustrations have been obtained with MOLEKEL [START_REF] Molekel ; Flükiger | Swiss Center for Scientific Computing[END_REF].

Photophysical methods

UV/VIS spectra were recorded on a Jasco V-570 spectrophotometer.

Steady-state and time resolved fluorescence measurements were performed at room temperature in diluted solutions (ca. 10 -6 M) using an Edinburgh Instruments (FLS 920) spectrometer in photon-counting mode. Emission spectra were obtained, for each compound, at λ ex = λ max (abs) with Aλ ex = 0.1 to minimize internal absorption. Fluorescence quantum yields were measured on degassed samples at room temperature; fluorescein in 0.1 N NaOH was used as a standard (φ = 0.90 at λ ex = 470 nm) and refractive index correction was performed [START_REF] Demas | [END_REF]. The lifetimes values were obtained from the reconvolution fit analysis of the decay profiles with the F900 analysis software and the fitting results were judged by the reduced chi-square value. TPEF measurements were conducted using a mode-locked Ti:sapphire laser operating between 740 and 1000 nm and delivering 80 fs pulses at 80 MHz, following the experimental protocol described in detail by Xu and Webb [17]. The quadratic dependence of the fluorescence intensity on the excitation intensity was verified for every data point, indicating that the measurements were carried out in intensity regimes in which saturation or photodegradation did not occur. TPEF measurements were calibrated relative to the absolute TPEF action cross-section determined by Xu and Webb for fluorescein in water (pH = 11) in the 690-1000 nm range [17,18]. For each data point, an additional control was performed by comparing with the TPEF action cross-section of rhodamine B in methanol [17,18]. The experimental uncertainty amounts to ± 10%.

Journal of Molecular

Results and discussion

Photophysical properties

The photophysical characteristics of chromophores 1-4 are summarized in Table 1.

The chromophores show an intense absorption band in the near UV or visible region depending on their structure (Fig. 2a). Interestingly, we observe from Table 1 and Fig. 2a that increasing the branched character clearly results in improved transparency: a definite blueshift of the absorption band is observed in the series 1-4, paralleling the increasing number of branches in the chromophores. As a result branched chromophores 3-4 maintain very good transparency in the visible region. Such an effect can be of particular interest when applications such as optical power limiting are considered.

The fluorescence characteristics are dependent on dimensionality: 2-dimensional chromophores 2-4 show much lower fluorescence quantum yield than the 1-dimensional quadrupolar chromophore 1. In contrast, we observe that branched chromophores 2-4 exhibit similar fluorescence spectra (Fig. 2b) and quantum yields (Table 1). We note that dendritic All chromophores investigated in the present work show a common feature: they exhibit large Stokes shifts values as evidence of a significant nuclear reorganization taking place after excitation prior to emission. Such behavior can be related to the electronic redistribution occurring upon excitation. Indeed HOMO-LUMO calculations reveal that pronounced intramolecular charge transfer phenomena occur within the chromophores upon excitation: a pronounced charge density shift from the electron-releasing peripheral groups to the core is observed both in 1-D quadrupolar chromophore 1 and in 2-D octupolar chromophore 3 (Fig. 3a,b). In the case of Y-shaped chromophore 2, a directional intramolecular charge transfer from the donating end-groups towards the acceptor moiety is observed (Fig. 3c). Such a phenomenon is confirmed by the positive solvatochromism shown by chromophore 2: increasing the solvent polarity leads to a slight red-shift of its absorption band while a definite bathochromic shift of the emission band is observed (Fig. 4). This is consistent with the larger stabilization of the polar emitting excited-state with respect to the ground state by polar solvents. Interestingly, this dipolar type charge transfer phenomenon is observed though the donating and accepting moieties are not directly conjugated.

Two-photon absorption

The TPA spectra of chromophores 1-4 are shown in Fig. 5a. We note that for molecule 1, the TPA spectrum does not show a maximum in the 740-900 nm range, the TPA magnitude still increasing at 740 nm. This indicates that the first TPA maximum is located at lower wavelength whereas the lowest energy one-photon absorption maximum is observed at 401 nm. This shows that the lowest-energy two-photon allowed excited state is located at higher energy than the one-photon allowed excited state. Such behavior is in agreement with earlier theoretical and experimental observations reported for symmetrical quadrupolar chromophores [8a,c,d,g]. In contrast, we observe that the TPA spectra of the branched chromophores 2 and 3 exhibit a local maximum close to twice the one-photon maximum absorption wavelength (i.e. X and Y GM at 770 nm = 2×385 nm), indicating that for these molecules the lowest energy excited-state is both one-photon and two-photon allowed. For Journal of Molecular Structure, Volume 704, Issues 1-3, 18 October 2004, Pages 17-24, http://dx.doi.org/10.1016/j.molstruc.2003.12.036 dendritic molecule 4, a plateau is observed in the wavelength range corresponding to twice the one-photon maximum absorption (i.e. at 750 nm = 2×375 nm).

We note from Fig. 5a that the 1-D quadrupolar chromophore 1 shows higher TPA cross-section at 740 nm than the other branched structures. However, this observation is no longer valid when shifting to higher wavelengths: all 2-D branched structures become stronger two-photon absorbers than their 1-D quadrupolar analogue at λ > 780 nm. Indeed, although the maximum TPA of the branched structures are lower than those of the quadrupolar chromophore (Table 1), the branched-structures definitely show broader and redshifted TPA band in the NIR region. As a result, the dendritic chromophore 4, which exhibits lower TPA cross-section than chromophore 1 at 740 nm, also shows a three times larger TPA cross-section at 800 nm. Hence increasing the branched character seems to be a promising route for improving the TPA-transparency trade-off, in particular in the context of broadband optical limitation.

It is particularly interesting to note that Y-shaped chromophore 2 and three-branched octupolar chromophore 3 exhibit very similar TPA spectra, molecule 2 showing only slightly lower TPA cross-section while having a 20% lower molecular weight and showing similar transparency. This is particularly attractive in terms of optimizing the normalized TPA molecular efficiency σ 2 /MW as shown in Fig. 5b. Along the same line of reasoning, comparison of the normalized TPA efficiency of octupolar chromophore 3 and of its secondgeneration dendritic analogue 4 shows that increased branching character, although leading to enhanced TPA in the NIR region (Fig. 5a) does not yield a net gain in normalized TPA efficiency (Fig. 5b). Interestingly, in the NIR region both the three-branched octupolar chromophore 3 and the Y-shaped chromophore 2 appear as the best candidates in terms of normalized TPA-transparency trade-off.

Conclusion

The present study provides evidence that modulation of the nonlinear absorptivity/transparency range trade-off can be achieved by playing on the charge symmetry and dimensionality in a series of structurally-related chromophores. Indeed, increasing the branched character clearly results in improved transparency, but also in a significant broadening of the TPA spectra in the NIR region. Hence although the 1-D quadrupolar chromophore show the highest TPA cross-sections close to the visible red, the branched chromophores offer a promising alternative for enhanced TPA in the NIR region while

  Structure, Volume 704, Issues 1-3, 18 October 2004, Pages 17-24, http://dx.doi.org/10.1016/j.molstruc.2003.12.036 Two-photon excitation cross-sections of chromophores 1-4 were determined by investigating their two-photon-excited fluorescence (TPEF) in solution. These measurements provide the TPEF action cross-section σ 2 Φ. The corresponding σ 2 values can be derived by determining the fluorescence quantum yield Φ from standard fluorescence measurements.

chromophores 3 (

 3 octupole: first generation) and 4 (second generation) exhibit similar fluorescence lifetimes, although chromophore 4 has an extinction coefficient about three Journal of Molecular Structure, Volume 704, Issues 1-3, 18 October 2004, Pages 17-24, http://dx.doi.org/10.1016/j.molstruc.2003.12.036 times larger than chromophore 3. Since these chromophores display similar emission spectra and identical fluorescence quantum yields, one would expect a much lower radiative lifetime for chromophore 4 and hence lower fluorescence lifetime. This behavior suggests some excitation localization in dendritic chromophore 4.

cFig. 1a .

 1a Fig. 1a. Series of structurally-related quadrupolar (1), Y-shaped (2), octupolar (3) and dendritic (4) chromophores.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Solvatochromic behavior of chromophore 2.

Structure, Volume 704, Issues 1-3, 18 October 2004, Pages 17-24, http://dx.doi.org/10.1016/j.molstruc.2003.12.036

Table 1

 1 One and two-photon photophysical data of molecules 1-4 in toluene

	Compound	λ abs (nm)	ε (cm -1 .mol -1 .L)	λ cut-off (nm)	a	λ em (nm)	Stokes Shift b (cm -1 )	Φ c	τ d (ns)	e (GM) σ 2
	1	401	83000	455		455	3000	0.81	0.82	905
	2	385	91000	436		445	3500	0.37	0.79	405
	3	384	131000	433		438	3200	0.36	0.63	510
	4	375	360000	432		442	4000	0.37	0.60	810

a Wavelength at which the transmittance is 95%.

b

Stokes shift.

Acknowledgements

MBD acknowledges financial support from CNRS (ATIP and NOI grants), and Rennes Métropole. L.P. received a fellowship from MENRT. Calculations were supported by the "Centre Informatique National de l'Enseignement Supérieur" (CINES-France).


Cécile Trân

Pablo Cruz

Mickaël Hedde

Claire Jouany

Eric Lecloux

Johanne Nahmani

Eric Chauvet

Antoine Lecerf

Benjamin Pey
email: benjamin.pey@ensat.fr

Litter and fauna functional identity effects on litter degradation

Litter degradation depends on litter functional groups (a) and on particular litter and plants traits (b) …

Main issue : is functional biodiversity as a key driver of organic matter degradation ?

• Soil fauna represents a huge diversity of soil animals involved in a great number of soil ecosystem services (Lavelle et al. 2002, Kibblewhite et al., 2008) • In spite of their crucial issues, the relations between soil fauna biodiversity and soil behaviour are still poorly understood and very poorly predicted.

• The use of functional groups and trait-based approaches improved the understanding of soil invertebrate diversity effect on soil functions compared to historiccaly taxonomic approaches.

• Soil macrodetritiovres roles are pivotal in sofar as they are implied in organic matter degradation and participate to the regulation of biogeochemical cycles of elements such as carbon and nitrogen • Studies using a functional approach mainly concern litter decomposition in forest systems. Meadow systems are scarcely studied. • There are only scarce studies which dela with both functional diversity of litter and soil invertebrates.

Objectives : assess functional identity of litter and soil invertebrate effects on degradation in meadows

• A first step to contribute to the undestanding of the relations between soil invertebrates and litter functional diversity on decomposition in meadow systems • 1) Assess the effect of functional identity of litters (through functional group of litters and trait-based approaches) on litter decomposition by soil invertebrates • 2) Assess the effect of functional identity of macrodetritivores (trought their functional group) on litter decomposition by soil invertebrates

Materials & Methods : combining litter and soil macrodetritivore with different functional identities (FI)

Authors thank the IDEX "Nouveaux entrants" program for the funding support

Necessity to take all trophic levels into account

 Both the litter and macrodetritivore functional identities are necessary to explain litter degradation  The macrodetritivore functional identity effect is however predominant compared to the effect of litter functional group  Interactions between the litter and macrodetritivore functional identities explain about 10% of litter degradation (results non presented)

Litter selection

Four species of forage grasses were chosen for their litter. They belong to different plant functional groups (FG) based on a trait analysis related to palatability (Cruz et al., 2010). We made the hypothesis that they will contribute differently to the litter decomposition)

Holcus lanatus L. Brachypodium pinnatum P. Beauv.

Festuca rubra L.

Bromus erectus Huds.

Experimental design and analysis

Macrodetritivore selection

Three species of macrodetritivores were chosen for (i) their dissimilar ecological and morphological characteristics and (ii) because they are common in temperate climate. We made the hypothesis that they will contribute differently to the litter decomposition. They are thus considered as three different functional groups (FG).

Armadillidium vulgare We performed a two-ways ANOVA on litter consumption with the functional group of litter and the functional group of macrodetritivore as factores. The consumption values were corrected with respect to the the control in order to remove the effect of factors other than macrodetritivore effects on degradation. The consumption litter was expressed as dried litter consumed per gram of biomass per day. Results demonstrate that 12% of the litter consumption was explained by the litter functional group.

The Festuca rubra and the Holcus lanatus present a higher degradation than Bromus erectus and Brachypodium pinnatum. Results for faecal pellet production were similar.

The ANOVA results demonstrate that 40 % of the litter consumption was explained by the macrodetritivore functional group. The effect of macrodetritivore has a higher effect than the litter functional group.

+ Chemical traits + Physical traits

Linear correlations of litter consumption and faecal pellet production against litter and plant trait data were performed. Only significative correlations with a minimum R 2 of 20% were presented. First result is that there is no trait which is correlated neither to litter consumption nor faecal pellet production for the whole invertebrates. Second results are that consumption and production were mostly explained by chemical traits (mean) for A. vulgare and P. Muscorum whereas by physical traits (standard errors) for G. marginata.

Litter traits = C : carbon, N : Nitrogen, P : Phosphorus. Plant traits = TMS : Dried matter content of water saturated leaf, RES : leaf resistance to breaks, SSF : specific leaf area

Trait-based approaches to accurate results

 Some chemical traits (C, P, C/N) mainly explained the litter degradation for some functional groups of macrodetritivores  Some physical traits of plasticity (dTMS, dRES, dSSF) explained the litter degradation for some others functional groups of macrodetritivores  Necessity to used a trait-based approaches for the effect of macrodetritivore functional identity

Perspectives

 Scientific (this experiment) : mesure morphological traits (e.g. size of buccal pieces), faecal pellet traits (e.g. C and N) to better understand the effect of macrodetritivore functional identity effect  Scientific (general) : continue to acquire trait data on litter and macrodetritivore to better understand the relationship between soil fauna biodiversity and organic matter  Applied : to better manage meadows systems

  were put in monospecific combinations in plastic boxes (120 x 90 x 50 mm) = 4 litter FG * 3 macrodetritivore FG = 12 treatments. Four additionnal treatments were made of each of litter FG without fauna. Four replicates were made. on the litter : C, N, P and water holding capacity. Mean litter and plant measures (the study of Cruz et al, 2010) were used as litter traits. Standard errors (d) were used as traits to express litter plasticity.