Guégan Dominique

A prospective study of the k -factor Gegenbauer Processes with heteroscedastic errors and an application to inflation rates

In this paper we investigate some statistical properties of the k -factor Gegenbauer process with particular noises. We make an extensive study concerning the behavior of these models in the stationary and non stationary cases using Monte Carlo simulations. We apply some of the results on a data set of inflation rates.

Introduction

Since the sixties', a fairly large number of power spectra have been estimated using economic data and, as observed by [START_REF] Granger | The typical spectral shape of an economic variable[END_REF] for instance, these spectra exhibit a spike at the very low frequencies. At this time, the spectral methods are based upon the idea of decomposition of a stochastic process into a (possible non finite) number of orthogonal components, each of which being associated with a "frequency". It is well known that such a decomposition is always possible for processes that are covariance stationary, i.e., have variances and covariances independent of real time, and that the power spectrum records the contribution of the components belonging to a given frequency band to the total variance of the process. Thus the existence of peaks at low frequencies suggests that there exists, for these economic data, long term fluctuations associated with the frequency components. Generally the spectrum does not consist of a series of peaks of decreasing size corresponding to different cycles, but is rather a smooth, decreasing curve with no (significant) peaks corresponding to periods longer than the one of the lowest frequency component ( see for examples [START_REF] Granger | Spectral analysis of times series[END_REF]. It has, of course, long been realized that these cycles are not strictly periodic fluctuations and do not correspond to a specific seasonal component.

The idea of persistence and long memory behavior inside these kinds of data become more precise in the eighties' and at this time [START_REF] Granger | Long memory relationship and the aggregation of dynamic models[END_REF], [START_REF] Granger | An introduction to long memory time series models and fractional differencing[END_REF] and [START_REF] Hosking | Fractional Differencing[END_REF] propose a parsimonous representation of the long memory effect with the FARMA process. A possible trend in means within the data can be also the subject of certain confusion: the definition of such a notion is not obvious. It is not sufficient to think that a trend in mean can be approximated by a simple polynomial or exponential expression, and a trend in a short series would not be so considered if the series were longer. Indeed, in order to understand better the problem of existence of trend in mean in data, which is close to detect the existence of non stationarity, a lot of unit-root tests has been developed by [START_REF] Dickey | Likelihood ratio statistics for autoregressive time series with a unit root[END_REF], [START_REF] Tiao | Consistency properties of least squares estimates of autoregressive parameters in ARMA models[END_REF], [START_REF] Phillips | testing for a unit root in time series regression[END_REF] and Kwiakowski, [START_REF] Kwiatowski | Testing the null hypothesis stationarity against the alternative of a unit root[END_REF]. Thus, when some non stationarity or long memory behaviors are observed within real data, it is classical for practicians to use both approaches to try to detect effectively the existence of persistence within the data. It is important to think that a non stationary behavior is often associated with a linear approach, whereas the research of a long memory behavior is associated with a nonlinear point of view.

In that way, in economic policy, the development of the inflation rate plays a prominent role, and from the Maastricht treaty the convergence of inflation rates of prospective members of the European Monetary Union is a very important criterion. The inflation rates have been extensively studied in the literature and most of the authors detect non stationarity using unit root tests, for particular periods, see for instance [START_REF] Barsky | The Fisher hypothesis and the forecastability and persistence of inflation[END_REF], [START_REF] Rose | Is the real interest rate stable?[END_REF], [START_REF] Marinucci | testing for the long run relationship between nominal interest rates and inflation using cointegration techniques[END_REF], [START_REF] Kirchgasser | Are real interest rates stables? An international comparison[END_REF]. These results are very sensitive with respect to the different sample periods. Because of the contracdictory results, using these different unit root tests, the choice of modelling long memory behavior upon these kinds of data seem a more appropriate approach. Concerning inflation rates, this has been done in particular by [START_REF] Chung | Small sample biais in conditional sum of squares estimators of fractionally integrated ARMA models[END_REF], Baillie, Chung and Tieslau (1996a), [START_REF] Hassler | Long memory in inflation rates : international evidence[END_REF] and [START_REF] Franses | A periodic long memory model for quartely UK inflation[END_REF]. Other financial data sets have been also studied with this approach, see for instance, for stock markets, [START_REF] Smith | Forecasting cost incurred from unit differencing fractionally integrated processes[END_REF], [START_REF] Bollerslev | Modeling and pricing long memory in stock market volatility[END_REF], [START_REF] Breidt | Extremes of stochastic volatility models[END_REF] and Willinger, Taqqu and Teverovski (1999), for stock market trading volume, [START_REF] Bollerslev | Equity trading volumeand volatlity: latent information arrival and common long run dependencies[END_REF] and [START_REF] Lobato | Long memory in stock-market trading volume[END_REF] and for exchanges rates, Beran and Ocker (1999) and [START_REF] Ferrara | Forecasting financial time series with generalized long memory processes[END_REF], 2001b, 2002).

Among all the studies, some suggest that the long memory coefficient, called here d , can take some values greater than or equal to 1 2 / . This magnitude of long memory has been reported in some of the previous papers including [START_REF] Bollerslev | Modeling and pricing long memory in stock market volatility[END_REF], [START_REF] Hassler | Long memory in inflation rates : international evidence[END_REF], [START_REF] Ferrara | Forecasting financial time series with generalized long memory processes[END_REF] and [START_REF] Lobato | Long memory in stock-market trading volume[END_REF].

On the other side, the study of most of these real data prevails on the existence of more complex behavior including both persistence and heteroscedasticity. Thus, it appears pertinent to associate the possibility of modeling long term persistence with quasi periodic behavior and heteroscedasticity. These different phenomena have been already detected in a lot of financial data, see for instance, Avouyi-Dovi, Guégan and Ladoucette (2002), [START_REF] Campbell | Valuation ratios and the long run Stock market outlook: an update[END_REF], Ferrara and Guégan (2001a), [START_REF] Bollerslev | Intraday periodicity, long memory volatility and macroeconomic announcement effect in the U.S. Treasury bond market[END_REF], [START_REF] Bollerslev | Semiparametric estimation of long memory dependencies : the role of high-frequency data[END_REF], [START_REF] Lobato | Long memory in stock-market trading volume[END_REF], [START_REF] Chambers | Long memory and aggregation in macroeconomic time series[END_REF], [START_REF] Hassler | Long memory in inflation rates : international evidence[END_REF] and [START_REF] Ding | A long memory property of stock markets return and a new model[END_REF]. [START_REF] Guégan | A new model : the k-factor GIGARCH process[END_REF] in a recent article proposed a new model which permits to model long memory behavior with quasi periodic behavior in the conditional variance of the observed data. This model is an extension of the FIGARCH, FIEGARCH and the FARMA-GARCH models introduced respectively by Baillie, [START_REF] Baillie | Fractionally integrated generalized autoregressive conditional heteroscedasticity[END_REF], [START_REF] Bollerslev | Modeling and pricing long memory in stock market volatility[END_REF] and [START_REF] Ling | On fractionally integrated autoregressive moving average time series models with conditional heteroscedasticity[END_REF]. A general representation of the model proposed by [START_REF] Guégan | A new model : the k-factor GIGARCH process[END_REF] is given by the following expression:

2 1 ( ) ( 2 ) (1 ( )) i k d i t t i B I u B B X m B φ θ η = - + = + - C (
p p z z z φ φ φ = --⋅⋅⋅ - and 1 ( ) 1 q q z z z θ θ θ
= --⋅⋅⋅are p-th and q-th degree polynomials,

2

( 2 ) ,

i d I uB B - + i=1, .
.., k is a difference operator defined by means of Gegenbauer polynomials, that we present with more details below. This operator has a finite number of zeros or singularities of order 1 , , ,

k d d ⋅⋅⋅ (| i d |<1/2, i =1, .
..,k) on the unit circle which allows the modelling of long-short memory data containing seasonal periodicities, m is some constant. As for the random shocks, or innovations, ( )

t t
η , we will make the usual hypothesis that they are independent and have a conditional Gaussian law, with mean 0 and variance 1. The set t I is the σ -field generated by the past values ( 1 2 , ,

t t η η - -⋅⋅⋅
). This model can provide a useful way of analysing the relationship between the conditional mean and variance of a process exhibiting long memory and show decay in its level, yet with time varying volatility.

In this paper we are interested by the statistical properties of the model (1), in order to use it on real data. This long memory process whose covariance function is not absolutely summable is characterized by an explosive behavior of its spectral density in frequency j λ , 0 j λ π < < , 1 j k ≤ ≤ . We will present these properties for a specific class of function ϕ in (1), which corresponds to the GARCH models, introduced by [START_REF] Bollerslev | Generalized autoregressive conditional heteroscedasticity[END_REF]. All the results can be extended in a lot of cases that we precise latter. Apart from a theoretical discussion we present the models fitted on consumer price index and inflation rates. We apply fractionally integrated models to describe the behaviors of these series. We find that the difference parameters are significantly different from 1 as well as from 0, and we do not reject a priori the hypothesis of non stationarity. The same kind of assumption has been done for instance by [START_REF] Hassler | Long memory in inflation rates : international evidence[END_REF] and Baillie, Bollerslev and Mikkelsen (1996b).

As we have specified in the title, this work is a preliminary study on the k -factor Gegenbauer processes. We prove certain results, but a lot are still opened problems.

In a first step (Section 1) we present the model on which we work and we give some properties (existence of a stationary solution, asymptotic behavior of the autocovariance function and spectral density, ergodicity, mixing, asymptotic distribution for the k-factor GIGARCH process). Section 2 reports the results of several simulations experiments for different classes of GIGARCH processes defined in (1), showing trajectories, autocorrelation function, spectral densities and empirical distributions. Until now it does not exist an identification theory which permits to decide between the different classes of non linear models, thus the knowledge of the second order properties empirically and graphically of such a model can be fundamental to go through the choice of a model when we observe real data. We discuss, in Section 3, empirical results which explain how non stationary fractional integration may arise through specific series. Indeed this work is important because a lot of data sets can produce long memory behavior and non stationarity. Section 4 applies the GIGARCH model to different data sets on consumer price index and inflation rates. The Section 5 gives a summary of the approach developped in this paper and some possible extensions. We postponed the proofs of main results of section 1 in an appendix based in Section 6.

I -Properties of the stationary k -factor GIGARCH model.

In this section we precise the function ϕ in (1) and using the same notations as above, we consider the k -factor GIGARCH model introduced first in [START_REF] Guégan | A new model : the k-factor GIGARCH process[END_REF] and defined by the following expression (here

0 p q = = in (1)): 2 1 ( 2 ) i k d i t t i I u B B X η = - + = C (2)
where

2 ( | ) (0, ) t t t L I N
η σ and where:

2 2 2 0 1 1 r s t i t i j t j i j a a b σ η σ - - = = = + + ∑ ∑ . We assume that 0 0 a > , , i a R ∈ i = 1, ...r, , j b R ∈ j = 1, ..., s, 2 ( , ) i i d u R ∈ , i = 1, ..., k.
In this paper, the conditional law of the process ( )

t t
η is Gaussian, nevertheless this assumption can be relaxed (the theoretical results given above can be extended, but the computations are not straightforward and we do not give them in this paper). In the next section we examine some simulations in this extended context. We give now some statistical properties of the process (2) under smooth assumptions.

The k -factor Gegenbauer process defined by:

2 1 (1 2 ) i k d i t t i u B B X η = - + = , ∏ (3)
where ( )

t t
η is a Gaussian noise has been extensively studied in the literature, see for instance [START_REF] Giraitis | A generalized fractioanlly differencing approach in long memory modelling[END_REF], Wayne, Woodward, Cheng and Gray (1998) [START_REF] Ferrara | Processus longue mémoire généralisé: estimation, prévision et applications[END_REF], [START_REF] Collet | Forecasting with non Gaussian long memory processes[END_REF] and [START_REF] Collet | How shall we determine the number and the location of the Gegenbauer frequencies[END_REF]. This model exhibits both long term persistence and quasi periodic behaviour, [START_REF] Hosking | Fractional Differencing[END_REF] was the first to mention this phenomena. Here, we consider an extension of this model when the noise ( )

t t
η is dependent and follows a GARCH(r,s) model. In order to give the properties of the process (2), we introduce the Gegenbauer polynomials ( ) ( )

d u j j Z ψ , ∈ defined by: 2 ( ) 0 (1 2 ) d d u j j j uz z z ψ - , ≥ - + = , ∑ (4)
where 1 z | |≤ and 1 u | |≤ . They can be easily computed by the following recursion formula:

( , ) 0 ( , ) 1 ( , ) ( , ) ( , ) 1 2 1 2 (5) 1 1 2 ( 1) (2 1) d u d u d u d u d u j j j du d d u j j ψ ψ ψ ψ ψ - -   =   =   - -  = + - +   . An explicit expression of ( ) d u j ψ , is the following: 2 [ 2] ( ) 0 ( 1) (2 ) ( ) ( 2 ) ( ) k j k j d u j k u d k j k j k k ψ - / , = - Γ -+ = , ! -!Γ ∑ (6)
with ( ) Γ . the Gamma function. See for instance [START_REF] Magnus | Fomulas and Theorems for the Special Functions of Mathematical Physics[END_REF] or [START_REF] Rainville | Special functions, Mac Millan[END_REF] for further details on Gegenbauer polynomials. Now, we recall some well known properties of the process ( )

t t
η which appears in (1).

Because ( )

t t
η is conditionally Gaussian, we can write:

2 2 2 2 2 2 0 1 1 r s t t i t t i j t t j i j a Z a Z b Z η η σ - - = = = + + , ∑ ∑ (7)
where t Z is a standard normal random variable independent of 1 t I -. In a vector form, the expression (7) becomes:

2 2 2 2 1 1 ( 1)( 1) ( 1) 1 ( 1) 1 1 1 ( 1) 1 ( 1)( 1) ( 1) 1 0 0 0 0 t r t t s t r r r x r x r s r x r s s x a Z a Z b Z b Z I a a b b I - - - - - - - -   ⋅⋅⋅ ⋅⋅⋅   ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅     ⋅⋅⋅ ⋅⋅⋅   ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅    
where ( ) r r I × denotes the r r × identity matrix.

In (8), we got a relationship which is a stochastic differential equation. The existence of a solution for such an equation has been studied by Kesten (1973) for instance. [START_REF] Mikosch | Limit Theory for the Sample Autocorrelations and Extremes of a GARCH(1,1) Process[END_REF] apply his result to get a solution for a GARCH(1,1) model and [START_REF] Starica | Estimation of the Extreme Value Distribution for Constant Conditional Correlation Models[END_REF] for the GARCH(r,s) model. Then, under the previous assumptions, a solution for (7) can easily be derived and it has the following expression:

1 2 0 1 1 1 [ ( ) ] j t t i i t j j Z a A η δ ζ ∞ / = - - = ′ = + Π , ∑ (9)
where

1 1 ( ) r s a a b b δ ′ = ,⋅⋅⋅, , ,⋅⋅⋅,
. A solution for the process ( )

t t
X defined by (2) is given in the following proposition:

Proposition 1: Assume that 0 0 a > , 0 i a > , i = 1, ..., r and 0

j b > , j = 1, ..., s, 1 1 1 r s i j i j a b = = + < ∑ ∑ , 1 2 i d < / , i = 1, ...,
d u t n t n n j t t i t j j i X Z a A ψ η η δ ζ ∞ - = ∞ - - = =  =     = +   ∑ ∑ ∏ (10) where 1 1 1 1 1 ( ) ( ) ( ) 0 ... k k k k k d u d u d u j l l l …l j l l j … ψ ψ ψ , , , ≤ , , ≤ , + + = = , ∑ (11) with ( )
( )

i i d u l l Z ψ , ∈
the Gegenbauer polynomials defined in (6).

Proof: See the Appendix.

Now we precise the asymptotic behavior of the autocorrelation function and of the spectral density of the model (2). Their autocorrelation function resembles a superposition of hyperbolically damped sin waves, as also mentioned and illustrated in [START_REF] Guégan | A new model : the k-factor GIGARCH process[END_REF].

Proposition 2: Assume that the hypotheses of proposition 1 are verified, then the autocorrelation function

0 ( ) [ ] X h h E X X γ = of the process defined by (2), in the case 1 max 0 j k j d = ,,
> has the asymptotics:

2 1 1 0 ( ) (cosh (1)) j j d X j j j k d h h o γ ψ λ * - = ,, , > = + , ∑ (12)
as h → ∞ , with :

-

j j ψ ψ ′ = if 0 j λ = or π ; - 2 j j ψ ψ ′ = , if 0 j λ π < < , and 2 ( ) (1 2 ) sin( ) ( ) j j j f d d D j η λ π ψ * * ′ = Γ - ,
where ( )

f η λ is the spectral density of the process ( ) t t η . Moreover, -j j d d * = if 0 j λ π < < ; - 2 j j d d * = if 0 j λ = or π ,
and finally:

( ) | 2sin | | 2(cos cos ) | , j j d d j j l j l D j λ λ λ - - ≠ = - ∏ if 0 j λ π p p (13) | 2(cos cos ) | j d j l j l λ λ - ≠ = - ∏ , if 0 j λ = or π.
The spectral density ( )

X f λ of the process ( ) t t X is equal to: 2 1 ( ) ( ) | 2(cos cos ) | j k d X j j f f η λ λ λ λ - = = - ∏ , (14)
and has the asymptotics

2 2 ( ) ( ) ( ) | | j d X j f f D j η λ λ λ λ - - , as j λ λ → , j = 1, ..., k. (15)
Proof: See the Appendix.

We remark that, as

i λ λ → , for 1 i … k = , , , the spectral density ( ) X f λ defined in (14)
becomes unbounded when 0 < d < 1/2. Thus, the spectral density of a k-factor Gegenbauer process with heteroscedastic noise clearly possesses k peaks on the interval [0,π].

We denote now n t A ⊗ the Kronecker product of n matrices t

A and we use it to study the moments of the process (2).

Proposotion 3: Assume that the hypotheses of proposition 1 are verified and that the process ( )

t t X is defined by (2), then, -If 2 ( ( )) t E A ρ ⊗
< 1, then the fourth-order moments of ( ) Proof: See the Appendix.

One of the goal of this paper is to give tools which permit to use this model to explain the behavior of certain data sets in finance or macroeconomics. If the notion of stationarity is important, in order to apply some results concerning the extreme value theory, it is important to know if this class of models is ergodic and mixing and on which conditions. The ergodicity and mixing properties of different classes of GARCH models is now established, for instance for the GARCH(1,1) model, we refer to [START_REF] Lu | A note on geometric ergodicity of autoregressive conditional heteroscedasticity model[END_REF], for the β-ARCH model to [START_REF] Guégan | Probabilistic properties of the β-ARCH model[END_REF] and for the general GARCH(p,q) model to [START_REF] Carrasco | Trial mode]-mixing and moment properties of various GARCH, Stochastic volatility and ACD models[END_REF] and references therein.

If | | t σ < ∞, 2 | | t E η < ∞ and 0 0 ( , )
η σ are initialized from the invariant measure, then the process ( )

t t X η
, is ergodic and mixing with exponential decay. These conditions are clearly verified for the process ( )

t t
η which appears in the model (2). On the other hand the mixing properties of the long memory process are not so documented. In a recent article [START_REF] Guégan | Non mixing properties of long memory processes[END_REF] establish the non mixing property of a Gegenbauer process defined by (3). We can establish the same property for the process (2) using the same way.

First of all, we recall some definitions concerning the notion of dependence for stationary processes. Let ( )

( ) sup | ( ) ( ) ( ) | n n k A F B F k P A B P A P B α ∞ -∞ + ∈ ∈ = ∩ - -the completely regular condition if lim ( ) 0 k r k →+∞ = , where : 1 2 , 1 2 ( ) sup | ( , ) | r k corr ξ ξ ξ ξ = ,
when the random variables 1 ξ and 2 ξ are respectively measurable with respect to the σ-algebras n F -∞ and n k F ∞ + . -the completely linearly regular condition if lim ( ) 0

k k ρ →+∞ =
, where :

1 _ 2 1 2 , ( ) sup | ( , ) | n n k L L k corr ξ ξ ρ ξ ξ ∞ ∞ + ∈ ∈ = .
We have the following relationships between these coefficients : α(k) ≤ r(k) and ρ(k) ≤ r(k); and for the Gaussian processes, ρ(k) = r(k) and α(k) ≤ r(k) ≤ 2πα(k), see for instance Ibrajimov and Rozanov (1974). Here we get the following result:

Proposition 4: Assume that the hypotheses of proposition 1 are verified, then the process defined by (2), is not completely linearly regular.

Proof: See the Appendix.

All the previous results are still true for the general process (2) with any finite p and q degrees in the polynomials φ(B) and θ(B). Using the same notations as before, we consider the following model:

2 1 2 2 2 2 0 1 1 ( ) ( 2 ) ( ( )) ( | ) (0, ) i t k d i t t i t t r s t i t i j t j i j B I u B B X m I B L I N a a b φ θ η η σ σ η σ = - - = =  - + = + -       = + +   ∑ ∑ C (16)
The next lemma summarizes some properties of this general model ( 16):

Lemma 1: Let the process ( )

t t
X defined by the equations ( 16). We assume that all roots of φ(B) and θ(B) lie outside the unit circle, 0 0 a > , 0 i a > , i = 1, ..., r and 0 j b > , j= 1, ..., s and 1 1

r s i j i j a b = = + ∑ ∑ < 1
. Now, if we assume that k=1 (for simplicity ), then, -If d < ½ and |u| < 1,the process defined by ( 16) has a unique second order stationary ergodic solution ( )

t t X η
, . The solution has a causal representation given by

1 ( , ) 0 ' 1/ 2 0 1 1 1 ( ) ( ) ( ( ) ) d u t n t n n j t t i t j j i X B B Z a A φ θ ψ η η δ ζ ∞ - - = ∞ - - = =  =     = +   ∑ ∑ ∏ (17)
where ( )

d u n
ψ , is given by ( 6).

-If d > -½ and |u| < 1, the process defined by ( 16) is invertible: that is ( ) t t η can be written as a function of ( ) t t X in the following way:

1 ( , ) 0 ( ) ( ) d u t n t n n X B B X φ θ π ∞ - - = = ∑ (18) where 1 1 1 1 1 ( ) ( ) ( ) 0 ... k k k k k d u d u d u j l l l …l j l l j … π ψ ψ -, -, , ≤ , , ≤ , + + = = , ∑ (19)
where ( ) d u n ψ , are the Gegenbauer polynomials defined in (11).

-The autocovarance function of ( 16) has the same asymptotics as in Proposition

( ) ( ) | | ( ) ( ) i X i e f f e λ λ θ λ λ φ - - = (20)
where ( )

X
f λ is given by ( 14).

-If ( ( )) -The process ( 16) is not completely linearly regular.

m t E A ρ ⊗ < 1 ,
All the properties given in this lemma are straightforward generalizations of the previous propositions.

Another kind of property which is important concerns the asymptotic distribution of the process ( ) t t X or of the maxima of the process defined by 1 max( ,..., )

n n M X X =
, n ≥ 2. This property is fundamental, in particular, in order to use these models in a context of risk management. Indeed, the computations of measures of risks are more and more important and these measures are based on the knowledge of the limiting distributions associated to a given model.

Here the process ( )

t t η which appears in (2) is a conditionally Gaussian GARCH model. If ( ) t t
η follows a GARCH(1,1) process, (this means, r = s = 1), then [START_REF] Mikosch | Limit Theory for the Sample Autocorrelations and Extremes of a GARCH(1,1) Process[END_REF] proves that :

( ) P x cx κ η - f where κ satifies 2 1 1 ( ) 1 E a Z b κ + = . ( 21
)
This indicates that the GARCH(1,1) model have extremely heavy tails. Now, we have seen, in proposition 1 that the process ( )

t t
X defined by (2) can be expressed like an infinite moving average of the process ( )

t t
η which is regularly varying at ∞ and -∞ with index κ/2. Then we can say that ( ) 21) is an extremal process generated by an extreme value distribution that we denote G and MDA(G) the corresponding domain of attraction. Then the extremal distribution function F of the process ( ) t t X belongs to MDA(G). This can derive for instance from Resnick (1987, chapter 4).

t t η in (
On the other side, the extreme value theory for Gaussian processes cannot be used here directly. Previously we have proven that the process (2) is not linearly completely regular, thus it does not verify the classical dependence conditions (see for instance [START_REF] Leadbetter | Extremes and Related Properties of Random Sequences and Processes[END_REF]). This condition is very important to establish the limit distribution of the maxima ( n M ) of stationary process ( )

t t
X . On the other side, it is easy to verify that for the process ( )

t t
X defined in (2) the [START_REF] Berman | Limit theorems for the maximum term in stationary sequences[END_REF] condition, ( ) ln 0 X n n γ → when n → ∞, holds. But the process (2) is not Gaussian and we cannot derive directly the limiting distribution of the maxima ( n M ) of the process using this result. Thus, the knowledge of the asymptotic distribution and the norming constants of the maxima ( n M ) of the process ( )

t t
X is an opened problem. Nevertheless, empirically it appears that this limiting distribution belongs to Gumbel domain, see the simulations below. It is also important to remark that, if the Berman's condition is theoretically verified for most of the classical time series models, nevertheless it can be difficult to observe its convergence towards zero for finite samples,. This has been illustrated for instance in [START_REF] Diop | Extreme distribution for generalzed stocastic volatility model[END_REF] for SVM models with GED distributions.

II -Simulations

Using the package S-PLUS, see [START_REF] Ferrara | Analyser les séries chronologiques avec S-Plus : une approche paramétrique[END_REF], we have made some simulations to illustrate the properties of the k--factor GIGARCH model investigated in the previous section. To simplify, we have simulated, in this Section, the GIGARCH model defined in (1) with k=1, p = q= 0 and different noises ( ) t t η that we precise later. All along this section, the values of the long memory parameters are d = 0.4 and u = 0.45. On each figure, we present a realization of length n = 1000, the autocorrelation function, the spectral density and the histogram of the process ( )

t t X .
The simulated realizations shown in this section are based on m = 0. Thus, the expression of the Gegenbauer process which has been used in the following is: η is a Weibull distribution with parameters (1, 0.5)).

We observe on these three trajectories and on the corresponding autocorrelation functions, the existence of periodic behaviors. As we expected the spectral density explodes in the G-frequency η . In that last case the distribution law of the process ( )

t t X
is far from a Gaussian law. This will be very important for estimation theory as soon as actually most of the results concerning the properties of the estimated long memory parameter are based on Gaussian Gegenbauer processes. For the process whose illustration is given in figure (3), we can also observe that the trajectory displays a behavior which presents intermittencies or explosions. On the other side, the use of Student distribution for the noise (figure (2)), even with a small degree of freedom, here 4, does not provide behaviors far from the one observed with Gaussian noise, figure (1).

The presence of long memory behavior inside real data driven by non Gaussian law arises very often. This is well known in hydrology, see for instance [START_REF] Collet | Forecasting with non Gaussian long memory processes[END_REF] and references therein, and in finance, see Avouyi -Dovi, Guégan and Ladoucette (2002). The estimation of the long memory parameter in that case is more difficult. In Avouyi -Dovi, Guégan and Ladoucette (2002), we investigate empirically the importance of the law of the noise in the estimation procedures for the parameters. If we do not use Gaussian law, the variability of the estimates is great with respect to the method used. We present now some trajectories of the Gegenbauer processes ( 22) simulated from a sequence of conditionally Gaussian noises ( ) It is interesting to note -on these two figures ( 4) and ( 5) -the presence of a strong volatility on the trajectories. The spectral densities present an explosion always in the frequency 1 cos (0.90) and the autocorrelation functions decrease slowly but no so slowly than on the previous graphs. The histograms of the process ( )

t t X
are similar to those of a Gaussian law, in both cases, and in that cases, it seems possible to use the classical estimation theory to estimate the long memory parameters. In figure (6) we present the distribution of the maxima of 1000 points generated with the recursive scheme (2), respectively with r = 1, s= 0 and r = 1, s = 1. It appears that the distributions of these maxima belong to domain of Extreme value distributions, using QQ-PLOT method. Precisely, we get a Fréchet distribution for the maxima of 1000 points generated with the recursive scheme (23), with an ARCH(1). The estimation that we got for the parameter ξ is, ξ = 0.2234, (0.0385). The estimated values of the location parameter is µ = 6.3257, (0.0385) and of the scale parameter σ is, σ = 2.9021 (0.1197). We get a Gumbel distribution for the maxima of 1000 points generated with the recursive scheme (24), with an GARCH(1,1). Here the parameter ξ = 0.0651, (0.0237) is not significant. The values of the location parameter is µ = 14.4786, (0.2499) and the scale parameter σ = 6.9994 (0.1864). A complete study on the distributions of the maxima for more general classes of heteroscedatic models can be found in Avouyi -Dovi and Guégan (2000).

III -The non stationary Gegenbauer Processes

Very often, when we investigate real data and when we study the long memory behavior of these data, the long memory parameter d is greater than ½ . This means that we are in presence of a non stationary long memory process. This class of models

has not yet been really studied, nevertheless the importance of this kind of behavior requires that we look at it in more details. A natural approach when d > ½ is to make a transformation on the real data set to make them stationary. This approach has been a lot studied using unit roots tests. A competitive way consists to keep this "non stationarity " and to use it to adjust a model on the data including this "kind of non stationarity". This means that we need to develop new classes of non stationary processes. Here we are going to investigate, with an empirical approach, the second order properties of the non stationary long memory processes with independent noises and dependent noises. We are going to consider two cases, ½ < d < 1 and d > 1.

To make our study complete, we begin to present some simulations concerning the stationary FARMA processes (which are particular cases of non stationary Gegenbauer processes) defined by

( ) d t t I B X η - = (25)
where ( )

t t
η can be either a sequence of independant identically random variables or an heteroscedastic noise driven by (23) or by ( 24). Secondly we present some non stationary FARMA and Gegenbauer processes defined by (2) with the same kind of noises used before.

-1. Stationary FARMA processes defined by (25). η a Gaussian noise.

On figures ( 7), ( 8), ( 9), respectively we present a Gaussian FARMA(0, 0.4, 0) process, a Weibull FARMA(0, 0.4, 0) and a FARMA(0, 0.4, 0) process with an ARCH noise. The autocorrelation function decreases slowly and the spectral density explodes in zero in all cases. We do not observe a great difference in the behavior of the empirical distribution in figures ( 7) and (9). It is not so clear that we are close to the Normal distribution in figure (8). -2. Non stationary FARMA processes: ½ < < < < d < < < < 1. We use d = 0.7 in all simulations. In figures ( 10), ( 11) and ( 12), the slow decreasing rate of the autocorrelation function is amplified when we compare these figures with the three previous ones. In figure (11), the empirical distribution is far from the Gaussian one.

-3. Non stationary Gegenbauer processes: ½ < < < < d < < < < 1. We use d= 0.7 and u = 0.45 in all simulations. First of all, we use independent noises. We get the figures ( 13), (Gaussian noise) and the figure ( 14), (Weibull noise). The variability of the trajectories seem amplified. We have the same behavior as before for the autocorrelation function. For all the graphs the spectral density explodes in a frequency far from zero and the histogram is generally difficult to interpret.

-4. Non stationary FARMA and Gegenbauer processes: d > > > > 1. We use d = 1.2 in all simulations. In figures ( 17) and ( 18) we present non stationary FARMA processes with different noises and in figures ( 19) and ( 20) non stationary Gegenbauer processes ( u = 0.45) with the same noises as in the previous items. (We do not present here all the graphs, because they are very close to each other and not very informative). η an ARCH(1) noise with 1 0.8 a = .

If in the case of the non stationary FARMA processes, it seems possible to identify directly a model on the series, this appears impossible with the non stationary Gegenbauer processes. The series are too noisy. In that latter case, a preliminary transformation on the series will be necessary before investigate them. ( 0.90 )

t t I B B X η - + = , with ( ) t t
η an ARCH(1) noise with 1 0.8 a = .

These different graphs show that, when the long memory parameter is greater than 1, it will be necessary to transform the series first, before considering any identification procedure. It could be interesting, in the other cases, ½ < d < 1, to detect directly the values of the parameter d without transformation on the data. In that last case, even if the series are non stationary, keeping the real data set can be strongly informative in order to rebuild the series or to make predictions. Thus, a theory in that context seems really important. Let us discuss now this point.

While there are many methods in the time series literature that deal with certain type of trends and seasonnalities, the large variety of possible models is often confusing to the applied data analysis. Finding an appropriate model, and in particular, defining realistic forecasts, is therefore a challenging task in practice. A possibility to solve this problem is to set up a unified framework in which flexible modelling of deterministic and stochastic components is possible, and objective data driven inference can be made to decide which of the components (deterministic trends or seasonnalities, stochastic trends or seasonnalities, spurious effects, stationary and nonstationary components) may be present, see [START_REF] Guégan | Long memory behavior for simulated chaotic time series[END_REF][START_REF] Guégan | Le chaos en finance: Approche statistique[END_REF]. The model described here combine parametric modelling of stochastic components using stationary and non stationary fractional Gegenbauer factors with in addition non stationary conditional variance. Thus the nonstationary GIGARCH models can be a challenge for the problem discussed previously. The estimation theory needs to be done and it is in preparation.

Nevertheless because many time series appear to be non stationary in different degrees in the reality, the interest to extend the concept of long memory to non stationary processes has been already addressed in the literature from the Gaussian FARMA process (25). Most of the works concern results on the functional convergence of the partial sums of such model, see [START_REF] Robinson | Gaussian semiparametric estimation of non stationary time series[END_REF], [START_REF] Marinucci | testing for the long run relationship between nominal interest rates and inflation using cointegration techniques[END_REF], [START_REF] Lu | Asymptotics of non stationary fractional integrated series[END_REF] and [START_REF] Velasco | Gaussian semi parametric estimation of non stationary time series[END_REF]. These results are very important in estimation theory and for aplications, see for instance [START_REF] Ling | On fractionally integrated autoregressive moving average time series models with conditional heteroscedasticity[END_REF], Beran and Ocker (1999) and [START_REF] Beran | Estimation of the dominating frequency for stationary and non stationary fractional autoregressive models[END_REF]. In figure (21) we give the distribution of the 1000 maxima obtained from a non stationary Gegenbauer process with d = 0.7 and u = 0.45 and driven respectively with an ARCH(1) process and a GARCH(1,1) process. In these two cases it has not been possible to find precisely the distribution function of the maxima. The non stationarity of the process can affect the possibility of estimation of the limiting distribution of the maxima.

IV -An illustrative example: are inflation rates non stationary and long memory?

In this section we are interested by the evolution of the Consumer price index (CPI) and inflation rate. Let denote the CPI by ( )

t t
V and the inflation rate by ( )

t t W , with 1 1 t t t t V V W V - - - =
. For all the models that we propose in the following, we first compute the empirical mean and we work with the centered process. -a. First we adjust a FARMA process:

1.002 ( ) ) confirms that the CPI is non stationary. Thus, we do not go on with this approach.

t t I B V ε - = (26)
-b. We decide to adjust a Gegenbauer process on the centered process ( ) t t V using the frequency λ = 0.05 (we choose a frequency which is close to zero). We get the following model: ) is significantly different of 0.5, thus this adjustment seems interesting. It confirms that if we do not consider that the explosion of the spectral density is not exactly in zero, then the Gegenbauer estimation can be considered. Here we get a non stationary Gegenbauer process. The variance of the residual seems important. Thus, we estimate the series of the residual using the recursive scheme ( 27) and we adjust on these residuals ( ) η , which appears in the expression of the ARCH(1) model ( 23) to be equal to one (this corresponds to the construction of the ARCH(1) process). Now in order to justify this adjustment, we need to study the distribution function of the residuals (by construction they have to be Gaussian). We give it in figure (23) using a Q-Qplot approach. V .

Clearly the distribution of the residual is far from the Gaussian distribution. Moreover, we observe that the estimation of the parameter 1 a is greater than 1. This value is not possible when we use ARCH(1) model (in that case the process in non mixing for instance, see [START_REF] Guégan | Probabilistic properties of the β-ARCH model[END_REF], thus we decide to differentiate the series ( )

t t V .
-c. Now we consider the differentiated process (( ) )

t t I B V -, its mean is m = 0.273. The representation of this process is given in figure (24). The trajectory does not present any feature of non stationarity. Nevertheless the autocorrelation function does not decrease quickly towards zero and the spectral density does not appear to be bounded. We first adjust a FARMA process. We get: ) . The estimation for d is significant. We observe that the variance of the residuals is smaller than the one obtained with the previous approach. We adjust on the residual ( ) Behavior of the ACF and of the QQ-plot of the distribution function of the residuals against Gaussian law, when we use a FARMA(0,0.415,0) with an ARCH(1) noise for the differentiated CPI (( ) )

0.415 ( ) ( ) t t I B I B V ε - - = (
t t I B V - . .
Here, the adjustment with a Gaussian law can be accepted. Now, to try to take into account some "cycle" observed on the autocorrelation function represented in figure (24), we adjust a Gegenbauer process in the frequency λ = 0.05 (close to zero). We get the following model: The values obtained for the different parameters are very close to those obtained with the model ( 26). Nevertheless comparing the figures ( 25) and ( 26), whether we have to decide between the models (28) and ( 29), we prefer the adjustment obtained for the model (29). 27), we observe that the trajectory does not present any trend, but the autocorrelation function does not decrease and the spectral density is not bounded into a frequency close to zero. We are going to try the same kinds of models as before to take this kind of non stationarity into account. -d. We first adjust a FARMA process. We get: 0.499 ( ) . The estimation obtained for the long memory parameter seems close to 0.5. Thus we prefer to stop the investigation of these data by this way and to try a Gegenbauer process.

t t I B W ε - = (
-e. With the Gegenbauer approach, we get the following model, using the frequency λ = 0.05. :

( 2 cos(0.05) ) . The value of 1 a ) gives interesting information on the volatility of the residuals. We compare the distribution of the residuals with the one of a Gaussian law using the QQ-plot method. This is given in figure (28). It seems that this modelization can be useful to make predictions for instance. These problems will be investigated in another paper.

t t I B B W ε - + = (
In the introduction, we have recalled different ways to investigate some kinds of non stationarity observed on the inflation rates. Here we apply Gegenbauer models to describe the fluctuations of the inflation rates. In this sense our approach is quite different of [START_REF] Hassler | Long memory in inflation rates : international evidence[END_REF] for instance. Indeed, we do not need to make any transformation on the process of inflation rates ( )

t t
W , and the model (31) permits to take into account both the long memory behavior (often assimilated to non stationarity) and the existence of cycle (also often assimilated to non stationarity). We think that this approach can be promising and can permit to propose new set of predictions taking into account the evolution of the different cycles displayed by the model (31).

V -Summary and Conclusion

In this paper, we have investigated some probabilistic and statistical properties of a kfactor Gegenbauer process driven with dependent noises. We have focused on an empirical investigation of these different properties in order to detect specific features which can permit to build an identification theory for these processes in the non stationary context. We have tried to adjust some of these models on inflation rates and our approach here remains prospective.

In this paper we have not studied the properties of the empirical moments of the process ( ) t t X and their asymptotic behavior, nor the estimation of the parameters. Some interesting problems concern the estimation of the parameters of the process (16) because to get nice properties of the estimates, we need to prove that the estimates obtained from the ARMA part on one side, from the long memory part on the other side and finally from the heteroscedastic part, are robust, altough the models are driven by dependent noises. This seems not do be done actually.

On the other side, we have not investigated the effect of aggregated time in the data, nor the aggregation with macro economic data, which is also an important point of view, to interpret any aggregated price levels. One important issue concerns how to aggregate prices when they respond to shocks. We know that in case of heteroscedastic models, this has been already studied, see for instance [START_REF] Drost | Temporal agregation of GARCH processes[END_REF], [START_REF] Guégan | Some recent developments in nonlinear time series[END_REF] and [START_REF] Duan | Augmented GARCH([Trial mode]) process and its diffusion limit[END_REF]. Recently this problem has been studied for high frequency data sets which present long memory behaviour, see for instance [START_REF] Willinger | Stock market prices and long range dependence[END_REF] and [START_REF] Bollerslev | Semiparametric estimation of long memory dependencies : the role of high-frequency data[END_REF] and for financial prices [START_REF] Chambers | Long memory and aggregation in macroeconomic time series[END_REF] and Avouyi-Dovi, Guégan and Ladoucette (2002 a). In the context of the k-factor GIGARCH model it is an open way of research.

All the properties that we have shown here for the k-factor GIGARCH process defined in (2) can be extended for more general expression for the volatility process t σ : for instance the EGARCH model, see [START_REF] Nelson | Conditional heteroscedastic in asset returns: a new approach[END_REF], the TARCH model, see [START_REF] Zakoian | Threshold heteroskedastic models[END_REF] or the Asymmetric power ARCH model, see [START_REF] Ding | A long memory property of stock markets return and a new model[END_REF].

VI -Appendix

A -Sketch of the proof of Proposition 1: For k = 1 in (2), the existence of the causal solution (10) is obtained as soon as , 0

n n t k t k k X ψ η - = = ∑
where the coefficients j ψ given by (11) satisfy 2 j ψ ∞ ∑ p . In the following the notation i ψ is used in place of ( , ) d u j ψ for simplicity. The uniqueness of the solution follows from the Theorem 12.4.1, in [START_REF] Brockwell | Time series: Theory and Methods[END_REF]. The stationarity of the solution is verified thanks to the condition imposed on the parameters, see [START_REF] Bollerslev | Generalized autoregressive conditional heteroscedasticity[END_REF] and [START_REF] Wayne | A k-factor GARMA long memory model[END_REF]. Moreover, as soon as there exist finite number of zeroes or peaks in the spectrum (k > 1) the proof requires a different approach in particular to get the asymptotics for (10) (we do not give them here), but this can be deduced directly from theorem 1 in [START_REF] Giraitis | A generalized fractioanlly differencing approach in long memory modelling[END_REF] using Darboux method, see [START_REF] Szegö | Orthogonal polynomials[END_REF].

B. Sketch of the proof of Proposition 2:

As soon as the spectral density of the process ( ) t t η has bounded derivative and is slowly varying in any point 0 ≤ λ ≤ π such that ( ) 0 f η λ ≠ , then the proof is similar to the lemma 3 in [START_REF] Giraitis | A generalized fractioanlly differencing approach in long memory modelling[END_REF]. The behavior of the autocovariance function follows from the Corollary 1 in the same article.

C. Sketch of the proof of Proposition 3:

For simplicity, we assume that k = 1 in (2). It is known that the 2mth-order moments of the process (8) are finite, as soon as ( ( )) 1 m t E A ρ ⊗ p , see for instance [START_REF] Bougerol | Stationarity of GARCH processes and of some non negative time series[END_REF]

k k t k t k k k E ψ ψ η η ∞ - - ≤ ∑ 2 2 4 ( ) ( ) k t k E ψ η ∞ = ∑ ,
where k ψ is given by ( 6). The proof for the eighth-order moment is the same as soon as 4 ( )

t E η ≤ ∞ .
D. Sketch of the proof of Proposition 4: It has been established that if a stationary process (with a spectral density f) is completely linearly regular, then the function r f is summable for every positive constant r, see for instance Ibrajimov and Rozanov (1974). Thus a stationary process, with spectral density f, is not completely linearly regular if there exists a positive constant p such that the function p f is not summable. Now, from proposition 3, we know the asymptotic behavior of the spectral density of the process ( )

Figure 1 :

 1 Figure 1: Realization, Autocorrelation function, spectral density, histogram of 2 0.4 ( 0.90 )

Figure 3 :

 3 Figure 3: Realization, Autocorrelation function, spectral density, histogram of 2 0.4 ( 0.90 )

  autocorrelation function decreases very slowly. The influence of the distribution law of the noise ( ) t t η on the one of the process ( ) t t X appears mainly in figure (3) as soon as we use a Weibull distribution for ( ) t t

  Figure 4: Realization, Autocorrelation function, spectral density, histogram of 2 0.4 ( 0.90 )

  Figure 5: Realization, Autocorrelation function, spectral density, histogram of 2 0.4 ( 0.90 )

  Figure 6: Distribution of 1000 maxima of the process

  Figure 7 : Realization, Autocorrelation function, spectral density, histogram of 0.4 ( )

Figure 8 :

 8 Figure 8 : Realization, Autocorrelation function, spectral density, histogram of 0.4 ( )

  Figure 10 : Realization, Autocorrelation function, spectral density, histogram of 0.7 ( )

Figure 11 :

 11 Figure 11: Realization, Autocorrelation function, spectral density, histogram of 0.7 ( )

Figure 12 :η

 12 Figure 12 : : Realization, Autocorrelation function, spectral density, histogram of 0.7 ( )

  Figure 13 : : Realization, Autocorrelation function, spectral density, histogram of 2 0.7 ( 0.90 )

  Figure 16: : Realization, Autocorrelation function, spectral density, histogram of 2 0.7 ( 0.90 )

Figure 17 :

 17 Figure 17: Realization, Autocorrelation function, spectral density, histogram of 1.2 ( )

Figure 18 :

 18 Figure 18 : Realization, Autocorrelation function, spectral density, histogram of 1.2 ( )

Figure 19 :

 19 Figure 19 : Realization, Autocorrelation function, spectral density, histogram of 2 1.2 ( 0.90 )

Figure 20 :

 20 Figure 20 : Realization, Autocorrelation function, spectral density, histogram of 2 1.2

  Figure 21 : Distribution of 1000 maxima of the process

Figure 22 :

 22 Figure 22: : Realization, Autocorrelation function, spectral density, histogram of the CPI ( )

  of the estimate d

Figure 23 :

 23 Figure 23: Behavior of the ACF and of the QQ-plot of the distribution function of the residuals against Gaussian law, when we use a Gegenbauer (0.45; 1.02) with an ARCH(1) noise for the CPI ( ) t t

Figure 24 :

 24 Figure 24: Realization, Autocorrelation function, spectral density, histogram of the differentiated CPI (( ) ) t t I B V from 01/01/70 to 30/11/98.

  Figure 25:Behavior of the ACF and of the QQ-plot of the distribution function of the residuals against Gaussian law, when we use a FARMA(0,0.415,0) with an ARCH(1) noise for the differentiated CPI (( ) )

)

  is significant. We can keep it. The variance of the residual is greater than the one obtained for the previous model. Thus, we adjust on the residual ( ) -PLOT of the distribution function of the residuals with respect to a Gaussian law is given in figure (26).

Figure 26 :

 26 Figure 26 : Behavior of the ACF and of the QQ-plot of the distribution function of the residuals against Gaussian law, when we use a GARMA(0.05,0.415) with an ARCH(1) noise for the differentiated CPI (( ) ) t t I B V -.

Figure 27 :

 27 Figure 27 : Realization, Autocorrelation function, spectral density, histogram of the inflation rates ( ) t t W from 01/01/70 to 30/11/98.

  obtained for the long memory parameter d is significant. We decide to try to adjust on the residuals ( )

Figure 28 ::

 28 Figure 28 : : Behavior of the ACF and of the QQ-plot of the distribution function of the residuals against Gaussian law, when we use a GARMA(0.05,0.275) with an ARCH(1) noise for the inflation rates ( ) t t W .

  i = 1, ..., k is fixed. Hence, there exists an i for which we can find

  then the 2mth-order moments of the processes ( )

	t t X and
	( ) t t η are finite.

Professor, E.N.S. Cachan, Department of Economy and Managment, 61 Avenue du President Wilson, 94235, Cachan Cedex, France, e-mail : guegan@ecogest.ens-cachan.fr

0.275


D Blanke

D Bosq

D Guégan
email: guegan@ensae.fr

Modelization and Nonparametric Estimation for a Dynamical System with Noise

Keywords: Invariant Measure, Nonparametric Estimation, Dynamical System, Small Noise, Deconvolution

We examine the effect of two specific noises (either known or small ones) on a dynamical system. We obtain consistent estimates with their rates of convergence for the invariant density in that context.

Introduction

In this paper, we focus on estimation of the invariant measure of a dynamical system disturbed by noise. Let us recall that a discrete time dynamical system is usually defined via a measurable mapping ϕ : E → E where E is a closed subset of IR d such that the state of the system at time t is given by:

x t = ϕ (t) (x 0 ) (1.1)
where ϕ (t) = ϕ o ϕ • • • o ϕ (t times) and x 0 ∈ E is the state of the system at time t = 0. Note that (1.1) implies x t = ϕ(x t-1 ), t ≥ 1.

(1.2)

Nevertheless a purely theoretical system like (1.2) is quite unrealistic since observations x t are in general corrupted by some noise.

We assume therefore that we observe a "noisy" trajectory y 1 , y 2 , • • • , y n which leads to the more natural model:

y t = ψ(y t-1 , δ t ) , t ≥ 1 (1.3)
where ψ is a measurable function: E × F → E, (F ∈ B IR d ) and where (δ t , t ≥ 1) is the noise which pollutes the system. In the following, typical kinds of noise will be introduced and studied.

In this paper, we are interested in ergodic dynamical system, thus we consider a probability space (E, B(E), µ) where E is a Borel set of IR d and B(E) is its Borel σ-field. We thus re-write (1.2) as:

X t = ϕ(X t-1 ), t ≥ 1 (1.4)
where X 0 is a given E-valued random variable.

In the same way, (1.3) becomes

Y t = ψ(Y t-1 , ∆ t ), t ≥ 1 (1.5)
where (Y t ) and (∆ t ) are sequences of random variables. Now, we intend to estimate nonparametrically the density f of the (possible) invariant measure µ associated to models such as (1.4) when observations are given by (1.5).

Only two particular but important cases of (1.5) (see (2.1) and (2.3)) will be considered in the sequel.

Many works have been devoted to dynamical systems, we refer to Lasota-MacKey (1994) and the references therein for further reading. Note that in this paper, we have rather followed a statistical approach of such dynamical systems. The plan of our paper is the following: in Section 2 we specify the noises which will be studied. In Section 3 we give conditions for existence and uniqueness of an invariant measure. Section 4 and 5 deal with estimation of f for two specific systems: known and small noises. Proofs appear in Section 6.

Classification of Noises

One may encounter many kind of noises in experimental systems, we now specify those which are studied in our paper.

• First suppose that we observe a dynamical system such (1.4) but with errors-invariables. Thus, we get the following model:

Y t = X t + ε t , t ≥ 0 X t = ϕ(X t-1 ), t ≥ 1. (2.1)
Note that (2.1) gives one possible model which is a particular case of (1.5) since one may write

Y t = ψ(Y t-1 , ∆ t ) with ψ(y, δ) = ϕ(y) + δ , ∆ t = ϕ(X t-1 ) -ϕ(Y t-1 ) + ε t , t ≥ 1 , (2.2)
provided that Y t ∈ E i.e. ε t satisfies the condition X t + ε t ∈ E for t ≥ 1.

This latter condition is somewhat restricting, but it can be relaxed by supposing that E = IR d , even if the system lives in some subset of IR d . In the particular case where ε t is bounded ( ε t ≤ ε) it is only necessary to suppose that ϕ has a natural extension to E ε = {x : inf y∈E x -y ≤ ε} and then to work over this set.

System (2.1) corresponds to model with measurement errors. This case appears, for instance, when one wants to simulate systems, in particular aperiodic ones (like chaotic systems, see [START_REF] Guégan | Nonparametric forecasting techniques for mixing chaotic time series[END_REF]. Now, if in (2.1), ε t is independent of X t and ε t has a known invariant measure ν, t ≥ 1, existence of an invariant measure, say µ for (X t ) implies the same property for (Y t ) and the invariant measure associated with (Y t ) is µ * ν where * denotes convolution product. If the characteristic function of ν does not vanish, then µ * ν determines µ: in that case one deals with a deconvolution problem.

• Now, we consider a general model which corresponds to propagation of errors. We take model (1.5) with ψ(y, δ) = ϕ(y) + δ and E = IR:

Y t = ϕ(Y t-1 ) + ∆ t . (2.3)
Assume Y 0 is observed, we get:

Y 1 = ϕ(Y 0 ) + ∆ 1 , thus Y 2 = ϕ[ϕ(Y 0 ) + ∆ 1 ] + ∆ 2 .
For a general t, the relation between Y t and Y 0 is intricate. However, this representation may be simplified by using successive approximations (see appendix for details):

Y t = ϕ (t) (Y 0 ) + ξ t , t ≥ 1 (2.4)
where

ξ t = ∆ t + Γ t-1 ∆ t-1 + Γ t-1 Γ t-2 ∆ t-2 + • • • + Γ t-1 • • • Γ 1 ∆ 1 , with Γ t = (ϕ ′ oϕ (t) )(Y 0 ) ,
where ϕ ′ , the derivative of ϕ, is supposed to exist except in a countable set of points.

Notice that if ϕ is linear, the models given by (2.3) and (2.4) coincide. Models such as (2.4) can be easily found in experimental situations, for example we refer to the Couette-Taylor fluid flow experiment described in Brandstäter and Swinney (1987), other examples of deterministic noise amplifiers can be found in [START_REF] Deissler | Deterministic noise amplifiers[END_REF]. In these experimental systems, the smallness of the noise is fundamental. Thus, in order to modelize the smallness of ∆ we consider a sequence of observed r.v.'s (Y 1n , • • • , Y nn ) associated with the noise (∆ 1n , • • • , ∆ nn ) where the r.v.'s ∆ in are i.i.d., zero-mean, with variance σ 2 n and such that (∆ in ) is independent of Y 0 for any i. Now, in order to control the noise in (2.4) we can make a classical assumption upon ϕ ′ (which is satisfied in usual cases) given by 1 ≤ ϕ ′ ∞ < ∞ where • ∞ denotes the essential supremum. Then the conditional variance of ξ t = ξ tn , (introduced in (2.4)) with respect to Y 0 is:

Var (ξ tn |Y 0 ) = σ 2 n (1 + Γ 2 t-1 + • • • + Γ 2 t-1 • • • Γ 2 1 ) ≤ σ 2 n n • ϕ ′ 2(n-1) ∞ ,
and since Var (ξ tn ) = E (Var (ξ tn /Y 0 )) + Var (E (ξ tn /Y 0 )), we get

Var (ξ tn ) ≤ σ 2 n n • ϕ ′ 2(n-1)
∞ .

Hence we have, Var (ξ tn ) → 0 as n → +∞, provided lim

n→+∞ nσ 2 n ϕ ′ 2(n-1) ∞ = 0.
3 Existence and uniqueness of invariant measure

In this section we give a result about existence and uniqueness of invariant measure for the dynamical system (1.5). We use the so-called FOAIS operator associated to ψ (defined in (1.5)), see Lasota-Mackey (1994, p. 414). Here, we relax somewhat assumptions concerning this operator in order to extend Theorem 12.5.1 in Lasota-Mackey.

Let us suppose that (∆ t , t ≥ 1) is a sequence of i.i.d. random variables with common distribution ν, that for each t ≥ 1, (Y t ) admits a distribution measure µ t , and that (∆ t ) and (Y t ) are independent. Note that the model (2.2) does not satisfy such assumptions but as seen before, the invariant measure will exist as soon as X t has an invariant measure and (ε t ) are i.i.d. variables independent of (X t ).

Let Π be the FOAIS operator defined over P(E) (the space of probability measures over (E, B E )) such that:

(Πµ)(B) = E dµ(x) F 1 B [ψ(x, y)]dν(y) = (1 B oψ)d(µ ⊗ ν), B ∈ B E . Therefore Πµ t = µ t+1 , t ≥ 0 .
We now specify our main assumptions: Assumptions A3.1 (i) T here exists µ 0 belonging to P(E) such that f or all η > 0, there exists a bounded B in B E and such that (Π t µ 0 )(B) ≥ 1 -η, f or any t ≥ 0.

(ii) : Π is continuous over P(E) with respect to the weak topology.

These assumptions are slightly more general than those in Lasota-Mackey (1994, p. 417), since ψ is not assumed to be continuous with respect to the first argument. We have:

Proposition 3.1 If assumptions A3.1 hold, then Π has an unique invariant measure.

Model with errors-in-variables

Suppose that we have n observations Y 0 , . . . , Y n-1 from the model (2.1):

Y t = X t + ε t , t = 0, 1, . . . , n -1 (4.1)
with

X t = ϕ(X t-1 ) t = 1, . . . , n -1. (4.2)
We make the following general assumptions about (4.1). If we denote by µ the invariant measure of (4.2), our goal is then to estimate its density f (if it exists) with respect to Lebesgue measure over E, when only observations Y 0 , . . . , Y n-1 are available and the law of the noise is known. Note that this latter condition is somewhat strong but it permits to ensure the identifiability of our problem. Moreover this condition will be satisfied when for example, one may preliminarily calibrate the measuring instrument. We now make the following assumptions upon the dynamical system {X t , t ∈ Z Z}: Assumptions A4.2 (i) ϕ preserves µ and µ has a density f with respect to Lebesgue measure over E.

Assumptions

(ii) X 0 has density f 0 and f or any t ≥ 1, X t has density Π t f 0 where Π is the socalled F robenius operator (see Lasota-M acKey p. 202) such that

+∞ t=0 Π t f 0 -f ∞ < +∞.
(iii) T here exists a sequence c n → +∞ def ined by

c n = E×E 0≤i,j≤n-1 i =j dP (X i ,X j ) (u, v) -dµ(u)dµ(v)
and such that c n = o(n 2 ) as n → +∞.

Note that these assumptions are quite similar to those used in the no-noise case. We refer to Smili (1990), [START_REF] Bosq | Optimal asymptotic quadratic error of density estimators for strong mixing or chaotic data[END_REF][START_REF] Bosq | Nonparametric Statistics for Stochastic Processes[END_REF] and [START_REF] Bosq | Nonparametric estimation of the chaotic function and the invariant measure of a dynamical system[END_REF] for examples of systems satisfying assumptions A4.2. Furthermore if c n is of same order as n 2 , we get a degenerate case.

We finally remark that if f t , g t are the respective densities of X t and Y t , assumptions A4.1 imply that g t = f t * ξ ε for any t, so we deal here with the classical deconvolution problem but in a non-stationary context. (1993), as well as in a continuous time context, see [START_REF] Blanke | Estimation de la densité pour des processus bruités à temps continu[END_REF][START_REF] Blanke | Estimation de la densité pour des trajectoires non directement observables[END_REF]. These works are related both with density and regression estimation. The nonparametric kernel-type density estimator is defined by:

Definition of the estimator

f n (x) = 1 nh d n n-1 i=0 W h x -Y i h n , x ∈ IR d (4.3) with W h (x) = 1 2π d IR d e -i<t,x> φ K (t) φ ε (t/h n ) dt
where φ K is the Fourier transform of a kernel K, φ ε is the characteristic function of the noise ε and < ., . > denotes the scalar product over IR d .

In the following, we will choose K as a kernel product function K = K ⊗ . . . ⊗ K where K is a real symetric bounded density such that lim

u→0 |u| K(u) = 0 and IR u 2 K(u) du < +∞.
Then, we may write:

W h (x) = d j=1 W h (x j ) with W h (x j ) = 1 2π IR e -itx j φ K (t) φ ε (t/h n ) dt.
Furthermore we will suppose in the sequel that | φ ε (t)| = 0 and that φ K (.)

φε(./h) ∈ L 1 (IR d ) ∩ L ∞ (IR d
) for all real t and positive h. Note that the first condition is close to the "minimal" condition (i.e. | φ ε (t)| = 0 for almost all t) which ensures identifiability of our problem (see [START_REF] Devroye | Consistent deconvolution in density estimation[END_REF]. Note that this assumption excludes characteristic functions with compact support such for instance the Beta distribution

β(1/2, 1/2) defined on [-1, 1].

Properties of kernel W h

We consider two general classes of noises specified below by (4.4) and (4.6). As usual, noises satisfying (4.4) and (4.6) will be respectively refered in the sequel as "ordinary smooth noise distributions" and "supersmooth noise distributions".

Ordinary smooth noise distributions

We make the following assumptions on the characteristic functions φ ε (t) and φ K (t) (see [START_REF] Masry | Multivariate probability density deconvolution for stationary processes[END_REF]Masry , 1993)).

Assumptions A4.3 φ K (.) and φ ε (.) are twice continuously differentiable with bounded derivatives such that :

(i) | φ ε (t)| > 0 ∀ t ∈ IR, (ii) ∃ β ≥ 1 such that t β φ ε (t) → A 1 as |t| → +∞ with |A 1 | > 0, (4.4)
(iii)

IR |t| β-2 | φ K (t)| dt < +∞ if β > 1, IR |t| 2β | φ K (t)| 2 dt < +∞, (iv)
IR |t| β-1 | φ ′ K (t)| dt < +∞, IR |t| β | φ ′′ K (t)| dt < +∞.
Condition (ii) specifies the asymptotic behaviour of the noise, it includes, in particular Laplacian densities (β = 2) and also the Gamma ones Γ (λ,t) (β = t). Assumptions (iii) and (iv) are technical and are satisfied by e.g. Gaussian kernels K.

Under such assumptions [START_REF] Masry | Multivariate probability density deconvolution for stationary processes[END_REF]Masry ( , 1993a) ) gives the following useful properties of W h : Lemma 4.1 U nder assumptions A4.3 and for any 1 ≤ q ≤ +∞ we get

W h q = O h -β n n → +∞. (4.5)

Supersmooth noise distributions

We now consider noises satisfying the following assumptions.

Assumptions A4.4 (i) For all real t, | φ ε (t)| > 0, furthermore there exist positive constants B 1 , a, β and a real constant β 0 such that

| φ ε (t)| ≥ B 1 |t| β 0 exp -a|t| β as t → +∞, (4.6)
(ii) φ K (.) has a compact support ] -τ, τ [, (iii) φ K is an even, real, decreasing and bounded function over

[0, +∞[ with φ K (0) = 1, φ K admits (p + 1) bounded derivatives such that φ K (τ ) = . . . = φ (p-1) K (τ ) = 0 and φ (p) K (τ ) = 0.
Assumption (i) specifies the asymptotic behaviour of the noise and will be satisfied for e.g. Gaussian noises (with β 0 = 0 and β = 2) or those following a Cauchy law (β 0 = 0 and β = 1). Conditions (ii) and (iii) are technical and will be fulfilled for K with Fourier transform such that φ

K (t) = (τ 2 -t 2 ) 2 1 I [-τ,τ ] (t).
Properties of kernel W h for such noises are given by [START_REF] Stefanski | Rates of convergence of some estimators in a class of deconvolution problems[END_REF] and extended to any dimension d in the following lemma.

Lemma 4.2

U nder assumptions A4.4, as h n → 0 we get

W h ∞ = O (nΛ n ) 1/2 h d n (4.7) W h q = O (nΛ n ) 1/2 h (1-β/q)d n for 2 ≤ q < ∞, (4.8)
where Λ n is defined by

Λ n = n -1 h 2d[(r+1)β+β 0 -1]
n e 2ad(τ /hn) β . (4.9)

Asymptotic results of convergence for f n

A surprising result concerns the asymptotic bias of the estimator f n defined by (4.3) which does not depend on the noise distribution. We denote by C 2,d (b) the space of twice continuously differentiable real valued functions f , defined on IR d , and such that f ∞ ≤ b and f (2) ∞ ≤ b where f (2) denotes any partial derivative of order 2 for f .

Theorem 4.1 If f ∈ C 2,d (b) and if h n → 0 such that nh 2 n → +∞, then assumptions A4.1 and A4.2(i)-(ii) imply that E f n (x) -f (x) = O(h 2 n ), n → +∞.
In order to study the asymptotic variance of our estimator, we now have to treat separately the two classes of noise distributions introduced above.

Ordinary smooth noise distributions

First we deal with ordinary smooth noise distributions of order β satisfying relation (4.4).

Theorem 4.2 U nder assumptions A4.1, A4.2, A4.3 and if ξ ε is bounded, then for any h n → 0 such that nh (2β+1)d n → +∞, we get

Var f n (x) = O 1 nh (2β+1)d n + O c n n 2 h 2βd n , n → +∞.
Note that the covariance term depends on c n : c n = O (n) is a sufficient condition for the variance of our estimator to tend to zero. Moreover, the smallest is c n , more rates will be close to the optimal ones of the i.i.d. case (see [START_REF] Fan | On the optimal rates of convergence for nonparametric deconvolution problems[END_REF]. Theorems 4.1 and 4.2 imply the following corollary which gives the asymptotic quadratic error of the estimator.

and h n = n -1 4+d(2β+1) , E f n (x) -f (x) 2 = O n -4 4+d(2β+1) ; (2) if n -4+2d(β+1) 4+d(2β+1) c n → +∞ then for h n = c 1 4+2βd n n -1 2+βd E f n (x) -f (x) 2 = O c n n 2 2 2+βd
.

Remark that if condition (1) is satisfied in corollary 4.1, then we find again the optimal rates (i.e. independent of the choice of the estimator) of the i.i.d. case with d = 1 (see [START_REF] Fan | On the optimal rates of convergence for nonparametric deconvolution problems[END_REF]. Such assumption will be satisfied for e.g. the r-adic function defined by ϕ r (x) = rx (mod 1) since it can be shown that c n does not exceed n (see Smili, 1990). Furthermore, the no-noise case may be deduced from corollary 4.1 by putting β = 0, thus, under asumptions of theorems 4.1 and 4.2 we get that

(i) c n = O n 4+2d 4+d
implies a mean-square error of order O n -4 4+d , (ii) n -4+2d 4+d c n → +∞ implies a mean-square error of order O (n -2 c n ) .

Super smooth noise distributions

We now establish the asymptotic mean square error of our estimator when the noise has a super smooth distribution of order β given by (4.6).

Theorem 4.3 U nder assumptions A4.1 A4.2 and A4.4, as n → +∞ we have

Var f n (x) = O (Λ n ) + O Λ n h (1-β)d n n + O 1 nh d n + O Λ n c n n ,
where Λ n is given by (4.9).

In order to get convergence of the variance to zero, we now have to choose h n decreasing to zero logarithmically, thus the term of bias becomes dominant and then the mean-square error is given by with θ, θ ′ such that 0 < θ < 2 and 0 < θ ′ < 2 -θ, we have

E f n (x) -f (x) 2 = O 1 ln n 4/β .
We obtain the same rates as Fan (1991) who has shown in the i.i.d. case with d = 1 that these rates are optimal in presence of such kind of noise. Remark that condition upon c n is still satisfied for the r-adic function.

Models with small noise

We have seen in the previous section that rates of convergence for our estimator are very sensitive with respect to the law of the noise. In particular for a Gaussian noise, rates seem too poor (corollary 4.2) for ensuring good estimation. However frequently, one may suppose that the noise has low level and then it is negligible. In this section, we intend to fix the maximum level of noise for which usual nonparametric estimation remains consistent with classical rates.

In this section, we consider models (2.1) and (2.3) with respectively "small" ε t and ∆ t . For sake of simplicity, only real-valued processes will be considered but our results can be extended to any dimension d.

As we no more follow a deconvolution approach, we will use for both models the classical kernel density estimator (see [START_REF] Rosenblatt | Remarks on some nonparametric estimates of a density function[END_REF])

f n (x) = 1 nh n n-1 i=0 K x -Y i h n (5.10)
where the Y i 's (i = 0, . . . , n) are real-valued observations arising respectively from models (2.1) and (2.3) and where K is a real positive lipschitzian kernel satisfying the same conditions as kernel K introduced in section 4.

Model with errors-in-variables

Fan (1992) shows that for i.i.d. variables and small Gaussian noise, the deconvolution kernel estimator reaches the optimal i.i.d. rates. We now generalize this result in several ways: we consider a large class of densities for the noise (including the Gaussian one) and then, we study the asymptotic behaviour of the usual kernel estimator (whose use is easier than the deconvolution one) for systems given by (4.1) and (4.2). Note that for stationary, mixing or irregularly observed continuous-time processes, similar results may be found in Blanke (1997).

In concret terms, we suppose that we observe Y 0 , . . . , Y n-1 given by We now show that if σ is small enough (σ = β n → 0 as n → +∞), we may consistently estimate the invariant density f when only observations Y 0 , . . . , Y n-1 are given. Note that such models may be associated with instruments which are more and more accurate.

Y i = X i + σε i . ( 5
It is important to note that in assumptions A5.1, the law of the noise is no more supposed to be known. Assumptions will only be made upon its two first moments.

Assumptions A5.2 (i) IR uξ ε (u) du = 0, (ii) IR u 2 ξ ε (u) du < +∞, (iii) f ∈ C 2,1 (b).
The following theorem gives the level of noise under which optimal rates still remain.

Theorem 5.1 U nder assumptions A4.2, A5.1 and A5.2, if furthermore β n = O n -4/5 then the choice h n = n -1/5 implies that

E (f n (x) -f (x)) 2 = O n -4 5 + O c n n 2
as n → +∞.

For higher order level noise (i.e. when n 4/5 β n → +∞), optimal rates may still be obtained but under stronger conditions upon c n .

Theorem 5.2 U nder assumptions A4.2, A5.1 and A5.2, 1) if c n is such that c n = O n 4/5 and h n = n -1/5 , then for

β n = O(h n ) we get E (f n (x) -f (x)) 2 = O n -4 5 n → +∞; 2) if n -4/5 c n → +∞, then for h n = (c n /n 2 ) 1/6 and β n = O(h n ) we get E (f n (x) -f (x)) 2 = O c n n 2 2/3 n → +∞.
Note that if assumptions of theorem 5.1 (with c n = O n 6/5 ) or theorem 5.2-(1) hold then we find the optimal rates of convergence for nonparametric density estimation in the i.i.d. case (see e.g. [START_REF] Farrel | On the best obtainable asymptotic rates of convergence in estimation of a density function at a point[END_REF]. Furthermore for processes with c n of order n (such as the r-adic ones) the worst asymptotic rates are of order n -4/5 for small β n :

β n = O n -4/5
and of order n -2/3 for larger β n given by β n = O n -1/6 .

Model with propagation of errors

We now consider the model close to (2.3) given by:

Y t = ϕ(Y t-1 ) + σε t .
(5.12)

We denote by f the invariant density (when it exists) associated to the no-noise model Y t = ϕ(Y t-1 ). Our goal is to estimate f when observations are given by (5.12) with σ = β n → 0 as n → +∞. Assumptions over the noise are still the same as in section 5.1. Furthermore we will suppose that ϕ is a a.e. differentiable function. For such models, we give some levels of noise under which optimal rates of convergence are reached.

Theorem 5.3

Suppose that assumptions A4.2 and A5.2 hold, if furthermore (i) ϕ ′ ∞ = 1 and β n = O n -13/10 or (ii) 1 < ϕ ′ ∞ < +∞ and β n = O (e -an ) with a > ln ϕ ′ ∞ , then the choice h n = n -1/5 implies that

E (f n (x) -f (x)) 2 = O n -4 5 + O c n n 2 as n → +∞.
Remark that in usual cases, we have 1 < ϕ ′ ∞ < +∞. Theorem 5.3 indicates that for such models, the noise should have very low level in order to reach optimal rates of convergence.

Proofs

Proof of Proposition 3.1

We use a similar of the one developed by Lasota-Mackey (1994, p. 417-419). Let us set:

M n = 1 n n-1 i=0 Π i (µ 0 ) = 1 n n i=1 µ i
and let (h k ) be a sequence which is dense in the space C 0 (E) of continuous bounded functions over E, then

|h k (M n )| = h k dM n ≤ h k ∞ , k ≥ 1 .
Then, for each k, there exists (M n k ) ⊂ (M n ) such that the sequence (h k (M n k )) converges. By diagonalization we may claim that there exists (M nn ) such that (h k (M nn )) converges for each k ≥ 1.

Therefore there exists an unique measure µ * ∈ P(E) such that we have the weak convergence of M nn to µ * , (we refer to Theorem 12.2.2 and Remark 12.2.2 in Lasota-Mackey, 1994).

Let us now verify that µ * is invariant for Π. Since we may set that the set B is compact, E -B is then open and using a well-known property (see [START_REF] Billingsley | Convergence of Probability Measures[END_REF], we get:

µ * (E -B) ≤ lim inf M nn (E -B) ≤ 1 -inf n µ n (B) ≤ 1 -(1 -ε) = ε.
Now we can write:

M nn = 1 k n k n-1 i=0 Π i µ 0 , k n ր ∞ . Thus, ΠM nn -M nn = 1 k n (Π kn µ 0 -µ 0 ) ,
and if h ∈ C 0 (E) |ΠM nn (h) -M nn (h)| = 1 k n (Π kn µ 0 (h) -µ 0 (h)) ≤ 2 h ∞ k n .
Hence, when h > 0, using (A 2 ) and the convergence of M nn towards µ * , when k n → ∞, we get:

(Πµ * )(h) = µ * (h) , h ∈ C 0 , thus Πµ * = µ * .

Proof of theorem 4.1

We have to show that the bias is independent of the noise distribution.

E f n (x) = 1 nh d n n-1 j=0 E W h x -Y j h n = 1 n 1 2πh n d n-1 j=0 E IR d e -i<t, x-Y j hn > φ K (t) φ ε (t/h n ) dt .
For characteristic functions such that φ K (t)/ φ ε (t/h) ∈ L 1 (IR) for all h, we get by Fubini's theorem

E f n (x) = 1 n 1 2πh n d n-1 j=0 IR d e -i<t, x hn > φ Y j (t/h n ) φ K (t) φ ε (t/h n ) dt. Independence implies that φ Y j (t/h n ) = φ X j (t/h n )φ ε (t/h n ) so if we set t/h n = s, we get E f n (x) = 1 n 1 2π d n-1 j=0 IR d e -i<s,x> φ X j (s)φ K (sh n ) ds = 1 n 1 2π d n-1 j=0 IR d e -i<s,x> φ K h * Π j f 0 (s) ds = 1 n n-1 j=0 IR d K h (x -y) Π j f 0 (y) dy = 1 nh d n n-1 j=0 E K x -X j h n . (6.13)
1) Assume first that X 0 has the invariant density f , i.e. f 0 = f , this implies that f j = Π j f 0 = f for all j. Then we get

1 h d n E W h x -Y j h n = 1 h d n IR d K x -y h n f (y) dy.
For f belonging to C 2,d (b) we get by Taylor's formula

E f f n (x) -f (x) 2 = O h 4 n (6.14)
where E f denotes expectation under f .

2) Let E ft be the expectation under f t , then for f 0 = f we get

E ft f n (x) -E f f n (x) = 1 nh d n n-1 j=0 IR d K x -y h n Π j f 0 (y) -f (y) dy ≤ 1 n n-1 j=0 Π j f 0 -f ∞ . Assumption A4.2(ii) implies that ∞ j=0 Π j f 0 -f ∞ < +∞ so E ft f n (x) -E f f n (x) 2 = O 1
n 2 which becomes neglictible as soon as nh 2 n → +∞, then theorem 4.1 is proved with majoration

E ft f n (x) -f (x) 2 ≤ 2 E ft f n (x) -E f f n (x) 2 + 2 E f f n (x) -f (x) 2 .

Proof of theorem 4.2

The variance of our estimator can be decomposed into two terms V n and C n given by

Var f n (x) = V n + C n (6.15)
where

V n = 1 n 2 h 2d n n-1 i=0 Var W h x -Y i h n , (6.16
)

C n = 1 n 2 h 2d n i =j Cov W h x -Y i h n , W h x -Y j h n . ( 6
.17)

6.3.1 Study of V n 1)
Assume firstly that f 0 = f (this implies g t = g 0 for all t). We get

1 n 2 h 2d n n-1 i=0 Var W h x -Y i h n = 1 nh 2d n Var W h x -Y 0 h n ≤ 1 nh 2d n IR d W 2 h x -y h n g 0 (y) dy ≤ ξ ε ∞ W h 2 2 nh d n = O 1 nh (2β+1)d n
since the density of the noise is bounded and W h 2 is given by lemma 4.1.

2) Assume now that f 0 = f , we have g i = f i * ξ ε . Let Var g i and Var g be the variances under densities g i and g = f * ξ ε . We now write

1 n 2 h 2d n n-1 i=0 Var g i W h x -Y i h n -Var g W h x -Y i h n := A + B
where A and B are given by

A = 1 n 2 h 2d n n-1 i=0 IR d W 2 h x -y h n g i (y) dy - IR d W 2 h
x -y h n g(y) dy (6.18)

B = 1 n 2 h 2d n n-1 i=0 IR d K x -y h n f (y) dy 2 - IR d K x -y h n f i (y) dy 2 .(6.19) Study of term A A = 1 n 2 h 2d n n-1 i=0 IR d W 2 h x -y h n (g i (y) -g(y)) dy = 1 n 2 h 2d n n-1 i=0 IR d W 2 h x -y h n IR d ξ ε (u) [f i (y -u) -f (y -u)] du dy ≤ 1 n 2 h 2d n n-1 i=0 IR d W 2 h x -y h n f i -f ∞ dy ≤ +∞ i=0 Π i f -f ∞ W h 2 2 n 2 h d n = O 1 n 2 h (2β+1)d n = o 1 nh (2β+1)d n
by assumption A4.2(ii) and lemma 4.1.

Study of term

B B = 1 n 2 h 2d n n-1 i=0 IR d K x -y h n (f (y) + f i (y)) dy IR d K x -y h n (f (y) -f i (y)) dy ≤ K ∞ n 2 h d n ∞ i=0 f i -f ∞ = O 1 n 2 h d n = o 1 nh (2β+1)d n
by assumption A4.2(ii).

Finally assumptions A4.1, A4.2(i)-(ii) and A4.3 imply since n → +∞

1 n 2 h 2d n n-1 i=0 Var W h x -Y i h n = O 1 nh (2β+1)d n .
(6.20)

Study of C n

We have

C n = 2 n 2 h 2d n i=0<j≤n-1 E g i,j W h x -Y i h n W h x -Y j h n -E g i W h x -Y i h n E g j W h x -Y j h n ≤ I + J (6.21)
where

I = 2 n 2 h 2d n i=0<j≤n-1 E g i,j W h x -Y i h n W h x -Y j h n -E g W h x -Y i h n E g W h x -Y j h n (6.22) J = 2 n 2 h 2d n i=0<j≤n-1 E g W h x -Y i h n E g W h x -Y j h n -E g i W h x -Y i h n E g j W h x -Y j h n (6.23) Study of term I I = 2 n 2 h 2d n i=0<j≤n-1 IR 2d W h x -y h n W h x -z h n g (Y i ,Y j ) (y, z) -g(y)g(z) dydz .(6.24)
The sum being finite, we get by independence of the ε i 's

I = 2 n 2 h 2d n IR 2d W h x -y h n W h x -z h n   IR 2d ξ ε (y -s)ξ ε (z -t) i=0<j≤n-1 dP (X i ,X j ) (s, t) -dµ(s)dµ(t)   dydz ≤ 2 n 2 ξ ε 2 ∞ W h 2 1 E×E i=0<j≤n-1 dP (X i ,X j ) (s, t) -dµ(s)dµ(t) = O c n n 2 h 2βd n (6.25)
with assumption A4.2(iii) and lemma 4.1 since the density of the noise is supposed to be bounded.

Study of term J

The bias is independent of the noise distribution, so (6.13) implies that

J ≤ 2 n 2 h 2d n i=0<j≤n-1 E g W h x -Y i h n E g W h x -Y j h n -E g i W h x -Y i h n E g j W h x -Y j h n = 2 n 2 i=0<j≤n-1 ((K h * f )(x)) 2 -((K h * f i )(x)) ((K h * f j )(x)) ,
where we have set K h (.) = h -d K(./h).

By using inequality |a

1 a 2 -b 1 b 2 | ≤ |a 2 ||a 1 -b 1 | + |b 1 ||a 2 -b 2 |, we get J ≤ 2 n 2
i=0<j≤n-1

IR d K h (x -z)f (z) dz IR d K h (x -y) (f (y) -f i (y)) dy + IR d K h (x -y)f i (y) dy IR d K h (x -z) (f (z) -f j (z)) dz ≤ 4 K ∞ nh d n +∞ i=0 f -Π i f 0 ∞ = O 1 nh d n = o 1 nh (2β+1)d n , (6.26)
with assumption A4.2(ii). Equations (6.20), (6.21)-(6.26) now yield to theorem 4.2.

Proof of theorem 4.3

We make use of same notations as in the proof of theorem 4.2.

6.4.1 Study of term V n 1) Assume that f 0 = f , then we have

1 n 2 h 2d n n-1 i=0 Var g W h x -Y i h n ≤ 1 nh 2d n W 2 h x -y h n g(y) dy ≤ W h 2 ∞ nh 2d n = O (Λ n )
by using lemma 4.2.

2) If f 0 = f , the decomposition A + B given by (6.18) and (6.19) implies

A = O Λ n h (1-β)d n n
with lemma 4.2 and

B = O 1 n 2 h d n .
Finally under assumptions A4.2(ii) and A4.4 we get when n → +∞

1 n 2 h 2d n n-1 i=0 Var W h x -Y i h n = O (Λ n ) + O Λ n h (1-β)d n n + O 1 n 2 h d n . (6.27)

Study of term C n

Term J in (6.23) is still the same since it does not depend on the noise distribution, so we get

J = O 1 nh d n . (6.28)
For term I given by (6.22), Fubini's theorem implies that

I ≤ 2 W h 2 ∞ n 2 h 2d n IR 2d i<j dP (X i ,X j ) (t, s) -dµ(t)dµ(s) = O Λ n c n n (6.29)
under assumption A4.2(iii) and lemma 4.2. Theorem 4.3 is then proved by using relations (6.21) and (6.27) to (6.29).

Proof of theorem 5.1

Suppose that we have observed Z 0 , . . . , Z n-1 given by the "ideal" model:

Z t = ϕ(Z t-1 ) Z 0 = X 0 .
One may construct the associated estimator

f n (x) = 1 nh n n-1 i=0 K x -Z i h n .
Now the mean-square error of our estimator f n is bounded by

E f n (x) -f (x) 2 ≤ 2E f n (x) -f n (x) 2 + 2E f n (x) -f (x) 2 .
It's easy to show (see the remark following corollary 4.1) that under assumptions A4.2 and for h n = n -1/5 we have

E f n (x) -f (x) 2 = O n -4/5 + O c n n 2 .
Thus we have only to consider the term E f n (x) -f n (x) 2 . Since K is supposed to be Cauchy-Schwarz's inequality implies

E f n (x) -f n (x) 2 ≤ C nh 4 n n-1 i=0 E (Y i -Z i ) 2 ,
where C is a positive generic constant. We have

Y i = ϕ (i) (X 0 ) + β n ε i and Z i = ϕ (i) (Z 0 ) = ϕ (i) (X 0 ), so E f n (x) -f n (x) 2 ≤ Cβ 2 n h 4 n .
Thus the choice h n = n -1/5 and

β n = O n -4/5 implies E f n (x) -f n (x) 2 = O n -4/5
and finally

E f n (x) -f (x) 2 = O n -4/5 + O c n n 2 .
6.6 Proof of theorem 5.2 6.6.1 Study of the bias term

(1) When f 0 = f , the bias of the estimator is given by

E n (x) -f (x) = 1 h n IR K x -y h n IR f (y -u) 1 β n ξ ε u β n du -f (x) dy = IR K(v) IR f (x -vh n -u) 1 β n ξ ε u β n du -f (x) dv. (6.30)
Assumptions A5.2 and Taylor's formula imply for 0 < θ < 1,

IR f (x -vh n -tβ n )ξ ε (t) dt -f (x) = IR [f (x -vh n -tβ n ) -f (x)] ξ ε (t) dt = IR (-vh n -tβ n ) f ′ (x) + (vh n + tβ n ) 2 2 f ′′ (x -θ(vh n + tβ n )) ξ ε (t) dt = J 1 (v) + J 2 (v) + J 3 (v) + J 4 (v) + J 5 (v) (6.31)
where the J i (v) are respectively given by

J 1 (v) = -vh n f ′ (x), J 2 (v) = -β n f ′ (x) IR tξ ε (t) dt, J 3 (v) = IR v 2 h n 2 f ′′ (x -θ(vh n + tβ n )) ξ ε (t) dt, J 4 (v) = β 2 n 2 IR t 2 f ′′ (x -θ(vh n + tβ n )) ξ ε (t) dt, J 5 (v) = h n β n IR vtf ′′ (x -θ(vh n + tβ n )) ξ ε (t) dt.
By assumptions A5.2 and since the kernel K is symetric we get:

IR K(v)J 1 (v) dv = 0 (6.32) IR K(v)J 2 (v) dv = 0. (6.33)
Furthermore, we have successively,

IR K(v)J 3 (v) dv = h 2 n 2 IR 2 v 2 K(v)f ′′ (x -θ (v h n + t β n )) ξ ε (t) dtdv, IR K(v)J 4 (v) dv = β 2 n 2 IR 2 t 2 f ′′ (x -θ (h n v + β n t)) ξ ε (t)K(v) dtdv, IR K(v)J 5 (v) dv = h n β n IR 2 tvK(v)f ′′ (x -θ (h n v + β n t)) ξ ε (t) dtdv.
Since f ′′ is supposed to be continuous and bounded, Lebesgue's theorem implies when

n → +∞ lim n→+∞ h -2 n IR K(v)J 3 (v) dv = 1 2 f ′′ (x) IR v 2 K(v) dv, (6.34) lim n→+∞ β -2 n IR K(v)J 4 (v) dv = 1 2 f ′′ (x) IR t 2 ξ ε (t) dt, (6.35) lim n→+∞ β -1 n h -1 n IR K(v)J 5 (v) dv = 0. (6.36)
We finally get by relations (6.32)-(6.36):

(i) When β n = h n , lim n→+∞ h -2 n (E f n (x) -f (x)) = 1 2 f ′′ (x) IR v 2 K(v) dv + IR t 2 ξ ε (t) dt . (ii) When β n = o(h n ), lim n→+∞ h -2 n (E f n (x) -f (x)) = 1 2 f ′′ (x) IR v 2 K(v) dv.
2) Assume now that f 0 = f , we have

E ft f n (x) -E f f n (x) = 1 nh n n-1 j=0 IR K x -y h n Π j f 0 (y) -f (y) dy ≤ 1 n n-1 j=0 Π j f o -f ∞ which with assumption A4.2(ii) implies E ft f n (x) -E f f n (x) = O 1 n
which is neglictible as soon as nh 2 n → +∞.

Study of variance term

The proof is quite similar to those of theorem 4.2. We make use of same notations and only main changes are outlined. Note that kernel K is now use instead of W h and d equals to 1. We consider decomposition (6.15) with C n and V n respectively given by (6.16) and (6.17).

Study of V n 1) Assume firstly that f 0 = f , we get

1 n 2 h 2 n n-1 i=0 Var K x -Y i h n ≤ 1 nh 2 n IR 2 K 2 x -y h n g(y) dy
Since f is supposed to be continuous, the same property holds for g and Bochner's lemma implies that lim sup

n→+∞ nh n n 2 h 2 n n-1 i=0 Var K x -Y i h n ≤ g(x) IR K 2 (v) dv (6.37)
2) Assume now that f 0 = f , then we get g i = f i * 1 βn ξ ε (./β n ) and terms A and B are bounded by

A ≤ K 2 2 n 2 h n n-1 i=0 Π i f -f ∞ = O 1 n 2 h n = o 1 nh n
with assumption A4.2(ii). In the same way,

B ≤ K ∞ n 2 h n ∞ i=0 f i -f ∞ = o 1 nh n .
Finally assumptions A4. by assumption A4.2(iii).

For term J of relation (6.23), we have for n → +∞.

J ≤ 4 K ∞ nh n n-1 i=0 IR K h (x -y) IR 1 β n ξ ε (t/β n ) f (y -t) -Π i f 0 (y -t) dtdy ≤ 4 K ∞ nh n ∞ i=0 f -Π i f 0 ∞ = O 1 nh n . ( 6

Proof of theorem 5.3

The proof is quite similar to those of theorem 5.1. We consider the "ideal" model (i.e. without noise):

Z t = ϕ(Z t-1 ) Z 0 = Y 0 .
We suppose that one has n observations Z 0 , . . . , Z n-1 and we construct the estimator of the invariant density associated to this model:

f n (x) = 1 nh n n-1 i=0 K x -Z i h n .
Using the same decomposition as in the proof of theorem 5.1 we get

E f n (x) -f (x) 2 ≤ 2E f n (x) -f n (x) 2 + 2E f n (x) -f (x) 2 .
For the second term, we have under assumptions A4.2 and with h n = n -1/5 that

E f n (x) -f (x) 2 = O n -4/5 + O c n n 2 .
For the first term, since K is supposed to be Lipschitzian, Cauchy-Schwarz's inequality implies

E f n (x) -f n (x) 2 ≤ C nh 4 n n-1 i=0 E (Y i -Z i ) 2 .
With the help of Taylor's formula, it can be established by recurrence that:

Y i = ϕ (i) (Y 0 ) + β n θ i ∆ i
where ∆ i is given by: ∆ i = ε i + ∆ i-1 ϕ ′ (ϕ (i-1) (Y 0 ) + β n θ i-1 ∆ i-1 ) ∆ 0 = 0 with 0 < θ i < 1 for any i.

Note that for any i, ∆ i is independent of ε i+1 , thus one obtains successively

E ∆ 2 i ≤ E ε 2 i + ϕ ′ 2 ∞ E ∆ 2 i-1 E (Y i -Z i ) 2 ≤ β 2 n E ∆ 2 i E (Y i -Z i ) 2 ≤ β 2 n E (ε 2 ) i-1 j=0 ϕ ′ 2j ∞ .
These results imply that

E f n (x) -f n (x) 2 ≤ Cβ 2 n E (ε 2 ) nh 4 n n-1 i=1 i-1 j=0 ϕ ′ 2j ∞ .
The behaviour of ϕ ′ 2 ∞ now yields to theorem 5.3.

Note that without approximations, one obtains:

X 3 = ϕ(X 2 ) + ε 3 = ϕ (ϕ(X 1 ) + ε 2 ) + ε 3 = ϕ [ϕ (ϕ(X 0 ) + ε 1 ) + ε 2 ] + ε 3 = ϕ [ϕ (ϕ(X 0 )) + ε 1 ϕ ′ (ϕ(X 0 ) + θε 1 ) + ε 2 ] + ε 3 = ϕ (3) (X 0 ) + γ 1 ϕ ′ [ϕ (ϕ(X 0 )) + θ 1 γ 1 ] + ε 3 (0.42)
where we have set γ 1 = ε 1 ϕ ′ (ϕ(X 0 ) + θε 1 + ε 2 and θ = θ(X 0 , ε 1 ), θ 1 = θ 1 (X 0 , ε 1 , ε 2 ) with θ, θ 1 ∈ [0, 1]. Now (0.41) can be deduced from (0.42) by setting θ and θ 1 equal to 0.

More generally, one gets the following approximation:

X k ∼ ϕ (k) (X 0 ) + ε k + ε k-1 ϕ ′ ϕ (k-1
) (X 0 ) + . . . +ε j ϕ ′ ϕ (k-1) (X 0 ) ϕ ′ ϕ (k-2) (X 0 ) • •ϕ ′ ϕ (j) (X 0 ) + . . . +ε 1 ϕ ′ ϕ (k-1) (X 0 ) . . . ϕ ′ (ϕ(X 0 )) .

Finally, if we set Γ k = ϕ ′ ϕ (k) (X 0 ) , we have

X k ∼ ϕ (k) (X 0 ) + ε k + Γ k-1 ε k-1 + Γ k-1 Γ k-2 ε k-2 + . . . + Γ k-1 Γ k-2 . . . Γ 1 ε 1 .

Corollary 4. 1 U

 1 nder assumptions of theorem 4.1 and theorem 4.2, as n → ∞ we get (1) for c n = O n 4+d(2β+2) 4+d(2β+1)

Corollary 4. 2 U

 2 nder assumptions of theorems 4.1 and 4.3, for c n = O n θ and h n = τ 2ad θ ′ 1/β (ln n) -1/β

  {X t , t ∈ Z Z} and {ε t , t ∈ Z Z} are independent.

	A4.1
	(i) {X t , t ∈ Z Z} is an E-valued process.
	(ii) {ε t , t ∈ Z Z} are i.i.d. random variables with known density ξ ε and with
	independent components.
	(iii)

  Deconvolution kernel estimators have been widely studied for i.i.d. observed variables see e.g. Carrol and Hall (1988), Liu and Taylor (1989), Stefanski and Carrol (1990), Fan (1991), or for mixing and stationary processes: Masry (1991)-(1993), Fan and Masry

  {X t , t ∈ Z Z} is a real process. (ii) {ε t , t ∈ Z Z} are i.i.d. real random variables with unknown density ξ ε . (iii) {X t , t ∈ Z Z} and {ε t , t ∈ Z Z} are independent.

	.11)
	with (X t ) satisfying relation (4.2). This model is similar to (2.1).
	Furthemore we consider the following assumptions:
	Assumptions A5.1
	(i)

  1, A4.2(i)-(ii) and A5.2 imply since n → +∞ Study of C nFor the term I given by relation (6.24), we obtain by Fubini's theorem and boundedness of K

	lim sup n→+∞	nh n n 2 h 2 n	n-1 i=0	Var K	x -Y i h n	≤ g(x)	IR	K 2 (v) dv.	(6.38)
	I ≤ dP (X = O n-1 2 n 2 h 2 n K 2 ∞ E×E i=0 c n n n 2 h 2				(6.39)

i ,X j ) (s, t) -dµ(s)dµ(t)

Appendix 1) Consider the linear autoregressive process

By reccurence, one obtains

and δ 0 = 0. For "small" δ t one gets the approximation

where Y 0 = X 0 . Then it's easy to see that lim

2) Now let us consider a (possibly) non linear autoregressive process such as

where ϕ is a measurable function and X 0 is a given random variable.

We intend to this model with a dynamical system such as

We can write

Let assume that ε 1 is small and that ϕ is an a.e. continuously differentiable function. We assume furthermore that X 1 has a density. Then, we have P X 1 ∈ Dϕ = 0 where Dϕ is the set of x such that ϕ ′ (x) is not defined.

Thus,

so we get approximatively

(0.41)