Pierre Vallois

Charles S Tapiero

Volatility Estimators and the Inverse Range Process in a Random Volatility Random Walk and Wiener Processes

Introduction

Empirical evidence has shown that financial time series are not always "well behaved". They may have an unpredictable variance, underscoring departures from the "random walk hypothesis". These effects have been recognized and have been the subject of considerable research under the heading of ARCH and GARCH related models [START_REF] Engle | Autoregressive Conditional Heteroskedasticity with estimates of the variance of U.K. inflation[END_REF][START_REF] Bollerslev | Generalized autoregressive conditional heteroscedasticity[END_REF][START_REF] Bollerslev | ARCH models[END_REF] focusing on the estimation of an underlying process variance. A related effort based on samples range of a constant volatility random walk was pointed out by [START_REF] Parkinson | The extreme value method for estimating the variance of the rate of return[END_REF] however. Explicitly, using Feller's result (1957) Parkinson's volatility estimator is given by:

i i i i I i I R X t X t ∈ ∈ = -
is the ith sample range, n is the number of intervals i I , i=1,2…,n over which the range is estimated, T is the length of the interval while X(t) is assumed to be a normal process with volatility σ . An adjustment to this estimate, based on the transformation of Parkinson's samples is suggested by Kunitomo (1992), providing thereby an estimate which is equivalent to the estimates based on samples variances. Range based estimates in econometric have been studied

and applied further by [START_REF] Martens | Measuring volatility with the realized range[END_REF] as well as [START_REF] Alizadeh | Range based estimation of stochastic volatility models[END_REF] and [START_REF] Brandt | A no-arbitrage approach to range-based estimation of return covariances and correlations[END_REF] who claim that the daily range in financial data is a more robust estimator against the effects of microstructure noise than the realized variance. By contrast, in the study of data with long memory, the Hurst index based on a range to standard deviation statistic provide the means to estimate a latent long run memory in long time series (for example, [START_REF] Tapiero | Run length statistics and the Hurst exponent in random and birth-death random walks[END_REF].

The purpose of this paper is to provide volatility estimators in stationary random volatility models using instead an inverse range process statistic which will be defined here. The intent of the paper is to indicate the importance of such a (range) process to estimate directly the functional parameters of an unknown underlying volatility and provide statistical estimates for this volatility. Unlike the ARCH-GARCH approach which is used to counter the effects of heteroscedasticity in linear regressions (where volatility is random), and uses specific models of volatility to negate their statistical effects, our approach is based on the information that a range process can provide to estimate the underlying process volatility. Our approach therefore is not a substitute to ARCH and GARCH estimation techniques but complements it when the volatility is unknown or is indirectly observable. Of course, if the process can be directly observed and measured, such an estimate can be used in addition to the standard maximum likelihood techniques applied in estimating directly the volatility. For example, random (stationary) volatility (and variance) distributions such as the exponential, the lognormal, Cauchy (fat tail) and other distributions can be used as hypotheses regarding an underlying volatility process and the parameters estimated by an observation of the range.

Unlike papers by [START_REF] Feller | The asymptotic distribution of the range of sums of independent random variables[END_REF][START_REF] Vallois | The range of a simple random walk on Z[END_REF][START_REF] Vallois | Moments of an amplitude process in a random walk[END_REF], 1997a, 1997b, we assume that the random walk process volatility is characterized by a known probability distribution and calculate its essential statistics (although approximations can be reached when such a distribution is not specified). Both discrete and continuous time observation of a given range may be used in our calculations. As a result, both random walk and random Wiener processes are considered here.

Problems related to volatility (variance) estimation are particularly important in finance. For example, an investor-a buyer of options, may have only a probabilistic assessment of the underlying stock volatility, which is essential for option's pricing.

Random volatility tests are also an indicator of markets' incompleteness and therefore important to detect arbitrage opportunities. Similarly, volatility estimates are used in control charts to determine control limits (often based as well on sample ranges which assume that samples are iid). In such cases, process range statistics might be used to test the validity of a given set of control limits. We begin by developing estimators for a random volatility random walk and obtain an explicit expression to estimate the parameters of the volatility distribution. Subsequently, a random volatility Wiener process is considered and specific cases (volatility distributions and estimators) are resolved.

A Random Volatility Random Walk

Consider the symmetric random walk, initiated at zero:

(1) 0 1 0, , 1 n n i i X X n ε = = = ≥ ∑ where ( ) i i n
ε ≥ is a series of independently distributed random variables with ( )

1 1/ 2 i P ε = ± =
. We define as well the random (volatility) parameter, 0 α > independent of the random walk ( ) 0 n n X ≥ and consider the following process:

(2)

( ) ; 0. n n X X n α α = ≥
Such a process implies that the underlying random walk (a price for example) increases or decreases by increments of size α . When α is unknown, we then have a constant volatility process-albeit its parameters are unknown and therefore presumed to be defined by a probability distribution. When α is a stochastic process, say a Bernoulli process assuming values a with probability p and zero otherwise, the underlying process (2) becomes a trinomial random walk with unknown parameters (a,p) that may be determined in the same manner that α is estimated in this paper.

However, such extensions and some of their difficulties are discussed in the appendix and provide an area for further research. In particular, while the trinomial random walk is the sum of independent random variables, the random walk with a Bernoulli mixture results in a mixture distribution where the random increments are dependent.

For our current purpose, we rewrite (2) as follows:

(3)

n n n X X αε α α + = - ) ( 1 ) ( or (3') n n n z X X + = - ) ( 1 ) ( α α
where n n z αε = Where 0 α > is assumed to be a square integrable random variable. A priori, the process ( )

1 ; ) ( ≥ n X n α
is not a random walk however, except when 0 α > is constant, in which case it assumes the discrete values:.., 3 , 2 , , 0, , 2 , 3 ,....

α α α α α α - - - .as stated above. Due to the independence of 0 α > and ( ) 0 n n X ≥ , it is easy to show that ( ) 0 ) ( = α n X E and ( ) ( ) ( ) 2 ; 1 n Var X nE n α α = ≥ .
Thus, an unknown volatility, or non-perfectly observable process over a given period of time, can lead to uncertainty regarding a process evolution. Of course, if 0 α > is defined by a known probability distribution, the range derived statistics can then be used to test hypotheses regarding the underlying process and estimate the process volatility. The intent of this paper is to provide some results that can be used to such ends.

Consider first the inverse range process ( )

(4) ( ) { } 0 0 inf 0; max min , 0 i i i n i n a n X X a a θ ≤ ≤ ≤ ≤ = ≥ - ≥ ≥ .
In an analogous manner we define:

(5)

( ) { } ( ) ( ) 0 0 inf 0; max min , 0. i i i n i n a n X X a a α α α θ ≤ ≤ ≤ ≤ = ≥ - ≥ ≥ α being positive, then ( ) 0 0 max max i i i n i n X X α α ≤ ≤ ≤ ≤ = and ( ) 0 0 min min i i i n i n X X α α ≤ ≤ ≤ ≤ = . Since ( ) 0 ≥ i i X
is a process with values in » , we have:

(6) ( ) 0 0 0 0 inf 0; max min ,if / inf 0; max min 1 , if / i i i n i n i i i n i n a n X X a a a n X X a α α α θ α α ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤    ≥ - ≥ ∈       =       ≥ - ≥ + ∉           » »
where [ ] .. denotes the entry whole number. Thus,

(7) ( ) , if 1 , a a a a otherwise α θ α α θ θ α    ∈       =       +           » Explicitly we can also write: (8) ( ) ( 1) ; ( 1), 0,1, 2,3,... m a m a m m α α α θ θ < ≤ + = + =
With these definitions on hand, we calculate some of the properties of the perturbed random walk inverse range process. We consider first the mean and the variance.

Proposition 1:

Let the amplitude 0 a > , 0 α > be the volatility (independent of ( ) 1 n n X ≥ ) , and (.) F be the function ( )

( ) ; 0 F x P x x α = < ≥ . Then: (9) ( ) 1 1 ( 1) 2 1 n a a E a n n F F n n α θ ≥       = + -           -       ∑ (10) ( ) 2 2 1 1 ( 1)(2 2 1) 6 1 n a a E a n n n n F F n n α θ ≥         = + + - -         -       ∑
with the convention that / 0 a = ∞ and ( ) 1 F +∞ = . Further:

(11) ( ) ( ) 1 ( ) ( ) 1 n a a P a k P n k F F n n α θ θ ≥       = = = -       -       ∑ , (12) ( ) 1, 1 , ( ) n k n k P n n k θ + - = + = Λ -Λ with: (13) ( ) ( ) ( ) ( ) 1 1 , , 1 , , 1 , 1 , , , , , 0 / 2 0 / 2 1 ( 1) 1 ( 1) 1 ( 1) 1 ( 1) 1 1 (cos( )) , ( 1
) (cos( )) , n k n k k k n k n k n k n k n k n k n k m n k n m n k n m n m m n m n m n n n π ξ ξ ξ + + + + - - - - + + - + < < < < Λ = + - Γ + + - Γ + + - Γ + + - Γ Γ = Γ = - =

∑ ∑

Proof: (see Proposition 13, [START_REF] Vallois | The range of a simple random walk on Z[END_REF]Vallois, , p. 1026) ) This proposition clearly defines the random (unknown) volatility distribution ( )

( ) P a k α θ =
in terms of the non-random volatility distribution ( )

( ) P n k θ =
calculated by Vallois and Tapiero in several papers [START_REF] Vallois | The range of a simple random walk on Z[END_REF][START_REF] Vallois | Moments of an amplitude process in a random walk[END_REF], 1997a, 1997b) and the a-priori defined probability distribution of the volatility ( )

* F
. First, note that calculations of the first two moments can be simplified if we consider instead the distribution of the reciprocal volatility 1

β α = with distribution function ( ) ( ) G x P x β = ≤ .
In this case, we have instead of ( 9) and ( 10):

(14) ( ) 1 1 1 ( 1) 2 n n n E a n n G G a a α θ ≥  -     = + -                 ∑ (15) ( ) 2 2 1 1 1 ( 1)(2 2 1) 6 n n n E a n n n n G G a a α θ ≥  -       = + + - -               ∑ .
And using Proposition 1, we can write as well:

(16) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 n n a a P a k P n k F F n n n n P n k G G a a α θ θ θ ≥ ≥       = = = -       -        -     = = -             ∑ ∑
For approximation purposes, equations ( 14) and ( 15) provide an initial estimate for confidence intervals. For example, using Tchebycheff's inequality, we have:

(17) ( ) ( ) ( ) ( ) 2 var a P a E a T T α α α θ θ θ     - ≥ ≤    
providing an approximate test periods for the unknown volatility random walk.

Further, let the maximum likelihood is defined by the product probabilities of the amount of time that an amplitude a has been reached for the first time, at time , ( ) j a α θ which equals the observed sample j k , with j, denoting the sample number.

These probabilities are then given by the product of ( )

( ) j P n k θ =
(where a nonvolatile random walk will attain the amplitude n for the first time at time j k , multiplied by the probabilities 1

a a F F n n       -       -       .
In other words, the likelihood of a random volatility set of samples observed 1 2 3 , , ,..., M k k k k is given as follows:

(18)

( ) ( ) , 1 1 1 ( ) ( ) 1 M M j j j n j j a a P a k P n k F F n n α θ θ ≥ = =       = = = -       -       ∑ ∏ ∏
Given, 1 2 3 , , ,..., M k k k k as well as the theoretically derived probabilities ( )

( ) j P n k θ =
, the parameters implied in the volatility distribution can then be derived by maximum likelihood. In other words, maximizing the ML below provides alternative estimators to the volatility of the random walk.

(19) ( )

(.) 1 1 ( ) 1 M j F n j a a Max P n k F F n n θ ≥ =       = -       -       ∑ ∏
Of course, this is most likely a difficult expression to optimize. Simple cases can be handled easily while approximations based on moments estimators can also be reached. Using equation ( 16), we have by have:

(20) ( ) ( ) ( ) ( ) 1 1 j j n n n P a k P n k G G a a α θ θ ≥  -     = = = -             ∑ Where ( ) ( ) j P n k θ =
is given by ( 12) where j k n ≥ :

(21) ( )

1, 1 , ( ) j j j n k n n k n P n k θ + -- - = = Λ -Λ
Example 1:

Let β assume an exponential distribution with parameter µ . Then, ( )

( ) 1 x G x P x e µ β - = ≤ = -
After some tedious manipulations (verified as well by Maple), we have:

( ) ( ) 2 / 1 1 a E a e α µ θ - =     - ; ( ) ( ) ( ) / / 4 / 6 1 a a a e e Var a e µ µ α µ θ - - - + =     - . Of course, if ( ) ( ) ( ) ,1 ,2 , , ,..., n a a a ( ) ( ) ( ) , ( ) , ( )
a k P a k P a k α α α θ λ θ = = ≥ where ( ) ( ) ( ) ( ) ( ) 1 1 1 1 ( ) ( ) 1 1 ( ) ( ) 1 (
)

n n n n a a n n a a n a a P a k P n k F F n n n n P n k G G a a P n k e e θ θ θ ≥ ≥ - - - ≥ - -       = = = -       -        -     = = -               = = -       = - =     ∑ ∑ ∑ 1 ≥ ∑ We have : ( ) { } { } 1 1 , 1 1 ( ) 1 1 k k k k k R n n a a a n R R n R R n n P n k e E e E e µ µ µ θ - - - - - = < > ≥ ≥     = = =         ∑ ∑ Thus, ( ) { } 1 ( ) 1 k k k R a R R P a k E e µ α θ - - >   = =     Furthermore, ( ) ( ) { } 1 1 ( ) ( ) 1 i i i R a R R n i k i k P a k P a i E e µ α α θ θ - - > ≥ ≥ ≥   ≥ = = =     ∑ ∑ ∑
In these expressions, note that (see Figure 1 as well):

{ } { } { } 1 1 1 1 1 1 1
and 1 and 1

k k k k k k k k k k R R X S X X X I X X - - - - - - - > = = = + ∪ = = - With max , min k i k i i k i k S X I X ≤ ≤ = = n n-1 + i R 1 k - k 1 ( ) , k i k n k R n R n θ - = ⇔ = < i Figure 1
Example 2:

Say that the volatility is distributed between [ ]

n n n If b G a a ab n n If b G a a   ≤ =       > =     Note that 1 1 ; 1 n n G G n ab a a ab -     - = ≤ ≤         .
It can be verified that:

( )

1 1 1 ( 1)( 2) ( 1) 2 6 ab n n n ab ab E a n n ab ab α θ = - + +   = + - =         ∑
Similarly for the variance, we have (verified by Maple as well):

( )

θ - + + + =     .
Reasoning as in Example 1, it can be proved that:

( ) ( ) 1 1 ( ) k k P a k P R R ab ab α θ - = = ≤ ≤
Which can be used to calculate as shown previously, we can also verify that : If the volatility parameter assumes k values, our analysis will be more complex, however if we let the amplitude a varies, we can obtain a linear system of equations by using the probabilities

( ) { } ( ) { } ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 2 1 2 1 2 1 2 ( 1) (2 1) 1 2 2 E a E a E a E a P E a P a a a a P P α α α θ θ θ θ α θ α α α = =     = + =         = = + =         + = + = + = Set ( ) 1 1 p P α = = for convenience. Since ( ) ( ) 1 2 1 1 1 P P p α α = = - = = -, we have : [ ] ( ) 1 1 1 ( 1) (2 ) (2 1) 1 2 ( 1) ( 3
( ) ( ) ( ) ( ) 1

The Random Volatility Wiener Process

Extension to a Wiener processes can be pursued in a similar manner. Imhof (1985) has calculated the probability generating function ( )

( ) a E e λθ -
of the range process explicitly. Thus, if we consider a random volatility Wiener process, the inverse range process is defined by ( ) ( ) a a α θ θ β = . Noting that in distribution [START_REF] Vallois | The range of a simple random walk on Z[END_REF]:

(21) ( ) 2 ( ) ( ) d a a θ β β θ =
We also note that the Laplace transform is (Imhof, 1985):

(22) ( ) ( ) 2 ( ) ( ) 2 1 = , 0, ( / 2) a a E e E e ch a λθ β λβ θ λ β λ - - = ≥     and ( ) ( ) 2 4 ( ) ; var ( ) 2 12 a a E a a θ θ = = .
As a result, the mean and the variance of the time to attain a range a, is:

(23) ( ) ( ) 2 2 ( ) 2 a E a E α θ β = .
Further, since ( ) (

)

2 4 2 ( ) = ( ) E a E a α θ β θ
, additional manipulations lead to:

(24) ( ) ( ) ( ) 4 2 4 2 ( ( )) 4 3 12 a Var a E E α θ β β   = -    
The cumulative distribution given by:

(25) ( ) ( ) α θ µ - ≤ =     =             ∑
We may assume of course other distributions. For example, volatility might be lognormally distributed, Weibull, Cauchy (with infinite variance) etc. The resulting problems will be technically more difficult but these problems problem can be handled numerically.

Appendix 1: Proof of Proposition 1 α being positive, relation (7) implies:

( )

1 1 1 ( ) 1 ( ) 1 a a n n n n n E a E n E n α α α θ θ θ     = -< <     ≥ ≥         = +                 ∑ ∑
Since α is independent of ( ) 1 n n X ≥ , then:

( ) [ ] [ ] 1 1 ( ) ( ) 1 n n a a E a E n P n E n P n n α θ θ θ α α ≥ ≥     = = + -< <             ∑ ∑ Or ( ) [ ] 1 ( ) 1 n a E a E n P n n α θ θ α ≥   = -< ≤         ∑ But 1 ; 1 1 a a a n n n n n α α     -< ≤ = ≤ < ≥     -     { } 0 1 a a α α   < ≤ = ≥    
Let F be the function ( )

1 1 ( 1) 2 1 n a a E a n n F F n n α θ ≥       = + -           -       ∑
By the same token, ( ) ( )

2 2 1 1 n a a E a E n F F n n α θ θ ≥           = -           -       ∑ But, { } 2 2
( 1)(2 2 1) ( ) 6

n n n n E n θ + + -= and therefore:

  first time the range for a random walk of the type (3) is greater than a given level "a". The volatility being a-priori unknown. Both the first two moments and the probability distributions are determined. Let ( ) a θ be the first time the amplitude of process ( ) 0 n n X ≥ crosses the value a>0;

Discussion and Applications

In this paper we have calculated the statistics of an unknown volatility random walk, but with known probability distribution. The volatility albeit unknown is determined when the process can be observed. If the random volatility has a discrete probability distribution, this is equivalent to saying that the underling process may be a binomial or multinomial process of various orders. Then our equation ( 3) is reduced to:

where n

ζ is a trinomial random walk defined by :

Such a process and its inverse range process has been studied extensively by [START_REF] Vallois | The range of a simple random walk on Z[END_REF] and [START_REF] Vallois | The Range Inter-Event Process in a Symmetric Birth Death Random Walk[END_REF] and its results carry over to the case treated here (namely of unknown volatility). Such a case requires however further research due to the dependence which is introduced by the unknown volatility.

When α has a continuous probability distribution, the number of potential and underlying processes is very large. In this sense, the inverse-range process can be used as a statistic to reveal the process departure from an underlying process with a specified volatility.

( )

The distribution of ( )

is similarly given by:

Recall (Vallois, 1996, Proposition 15 and[START_REF] Vallois | The range of a simple random walk on Z[END_REF]Tapiero (1997, p.331):

ABSTRACT

The purpose of this paper is to study the mean, the variance, the probability distribution and the hazard rate of the inverse range process of an a-priori unknown volatility random walk. Motivation for this process arises when it is necessary to obtain statistics that pertain to a process volatility in addition to the usual variance statistics. As a result, range process statistics are indicated as an additional source of information in the study of processes' volatility. Examples and applications are considered.


Classify students' conceptions about the mutation's site to anticipate their knowings' mobilization before designing an experimental procedure

Gwendaël Chapel, Patricia Marzin-Janvier, Muriel

Classify students' conceptions about the mutations' site to anticipate their knowings' mobilization before designing an experimental procedure

Gwendaël Chapel, Patricia Marzin, Muriel Ney.

Type of presentation preferred: oral presentation

Abstract:

In high school, students have difficulties to understand and to learn genetics in general and DNA structure and mutations in particular. These difficulties are partly due to the different biological levels. Several studies showed that students have difficulties to differentiate these levels and that they are not able to order them in the right biological hierarchy. We studied students' conceptions on a specific question to get knowledge on their misunderstanding of these levels. To do that we asked students the following question: where (at what level) in the body, mutagen agents act. After a literature review, we defined six biological levels (organism, cell, nucleus, chromosome and DNA). Students were separated in three groups depending on whether they had studied the DNA structure and mutations or not (both notions, just the DNA structure or none). Analysis of students' productions confirmed our six predefined levels, but with an unequal distribution, some levels being ignored by students. Moreover, distributions were different for the different groups of students. The goal of this study is to know how students could mobilize their knowings about DNA and mutation, and use this knowledge to design a laboratory work session scaffolded by an intelligent learning environment.

1) Background, Framework, and Purpose Our aim was to identify and to discriminate students' conceptions about the genetic mutations' site. The ultimate goal of this research is to anticipate the mobilization of students' knowings when they design an experimental procedure in a laboratory work session (self citation, 2008). The knowledge domain at stake is genetics, i.e. genetic mutations and the structure of DNA that is the support of the genetic information.

Several studies showed that genetics is one of the most difficult topics in biology both for teaching and learning (Johnstone, Mahmoud, 1980;Bahar, Johnstone, Hansell, 1999). Bahar (1999) explained that one of the difficulties of this domain is the different biological levels. Marbach-Ad and Stavy (2000) indicated also this difficulty and characterized three of these levels: organism, cell and molecule. They explained that students have difficulties to differentiate the levels, which led to erroneous interpretations of the interactions between these levels. The DNA structure and genetic mutations lead to additional difficulties for students. These notions are defined at the molecular level and their effects can be seen at other levels. Pfriedrichsen and Stone (2004) found that students confused DNA molecule with cells and chromosomes. Some students had no ideas about the structure of DNA molecule, and other knew that DNA is like a ladder with rungs, but they had no ideas about the structure with bases, phosphate and ribose. Students had difficulties to differentiate the biological objects and levels, and to organize them into a hierarchy. Moreover, students make some confusions with the term mutation. The general meaning of the term is "change". However, it is a polysemic term that depends on the context. Its meaning is different in everyday context (in French: changes at adolescence, work transfer) and in scientific context (mutation is sometimes wrongly used instead of metamorphosis) (Albaladejo, 1988).

2) Rationale

All of these international studies (but no studies in France) explained that students' difficulties in genetics are partly due to confusions about the different biological levels. To our knowledge no studies provide a classification and identification of students' conceptions. Furthermore, some studies tried to help students to understand the hierarchical organization of biological levels, but never in a laboratory work session. In our research, we tried to identify and discriminate students' conceptions about the site where a mutagen agent acts. The categorization of students' answers will help us in the next part of our study, that is, to anticipate the mobilization of students' knowings when they design an experimental procedure (self citation, 2007).

3) Methods

From the literature review and after a knowledge analysis (not shown here), we focussed on six biological levels: organism, cell, nucleus, chromosome, gene and DNA. We proposed students an exercise that allows us to discriminate, classify and quantify their answers into these six levels. In this exercise, students completed a diagram (a circle representing an animal cell) and indicated where a ray can act. They were asked to complete their answer in written text. The participants were 102 students, aged 16-17. They were separated into three groups (A-B-C). In group A (27 students), students had never studied the DNA structure and mutations notions. In group B (49 students), they had studied the DNA structure the week before, and in group C (26 students), they had studied all notions (one month before).

To differentiate students' answers, the most meaningful data were their written justifications, where they explained their drawing. With these data, we were able to classify students' answers into six biological levels. For each level, we grouped answers that talked about different components of the level (e.g. for cell: cell's membrane, "inside of the cell"). For example, student answer "Rays modify cell's membrane" was classified in the Cell level. Some student cited several levels. If they indicated an order between these levels, we considered only the lowest level. In " Rays act on the chromosome which carries a gene. Rays modify the DNA sequence and the gene is modified." the level was set as DNA because there was a clear indication of order. However, if students cited two levels without ordering them or giving any relationship between them, we put this answer in two levels. For instance, the answer: "Rays act on nucleus and on organelles." was classified in Nucleus and in the level Other (organelle was not set as a separated level). For this reason, there were more answers than the number of students.

4) Results

After the classification of all students' answers, we calculated a percentage for each level (Table 1). Adding answers from the three groups, the most frequent levels were: Nucleus, DNA and Cell. One can see that the percentage of answers for Nucleus level is almost divided by two when changing group: A>B>C. For DNA level, there is the opposite phenomenon: A<<B<C. Furthermore, one can look at the distribution of answers in each group. Group B gave answers in each of the six levels. Answers of group C were in four levels. The least knowledgeable group, group A, provided answers in three levels and mostly in two.

5) Conclusions and Implications

Our situation allowed us to categorize students' answers based on their textual and drawing descriptions and explanations about the site in an unspecified organism where rays (a mutagen agent) act. There were only four students on 102 who did not answer. An important point of this study was the classification of students' answers. All of the six levels defined in our analysis and based on the literature review appeared in the data. The answers' distribution among levels was far from uniform. Two levels had only one answer: Organism and Chromosome. The diagram to complete was a cell, which could have discouraged students to give an answer about a higher level. As for the Chromosome level, this was a rather unexpected result since this level was introduced to them the year before. There were also differences between groups for a given level. This difference could be due to the moment where the exercise was given: before, during and after the teaching of the notions. Students' answers at the molecular level (Gene or DNA levels) are more numerous in groups B and C than in group A where most students talked about cellular levels (Cell or Nucleus). The origin of this difference could be the recent study of DNA structure. The study of mutations had also an effect (comparing groups B and C). Students of group C gave more answers at the DNA level (which could be due to the fact that students in this group studied nucleotides' mutations), at the Gene level (they studied genetic diseases), and at the Cell level (they did a laboratory work where they observed cell mutations after an UV radiation). To confirm these results, we plan to use this situation with a larger number of students.

For the next step of our study, we make hypothesis about students' reasoning when they chose one or other level. For example, a student described the ray's action on cell as follows: "Membrane is used to protect cell, and if rays act on it, it will be destroyed". This notion of membrane's destruction was a reference to a mutation cause: loss of something and at the Cell level. For the Nucleus level, one justification was often given: "Rays act on nucleus and divide it in two parts". This explanation referred to the nucleus' doubling as the mutation's origin, and showed a confusion with two other notions, mitosis and meiosis. Two other origins were present in answers: substitution (crossing-over like rays' action on chromosomes) and addition (one gene more). Our aim is now to understand how (under what conditions) students can have a conceptual change from the cellular (submicroscopic) paradigm to the molecular (microscopic) paradigm. In other word, introduce the fact that genetic mutations are modifications in the succession of nucleotides bases that are organised in sequences. The main notion needed to understand this is the genetic code organization. In a new study, we will propose a problem-based learning situation designed to allow conceptual change from a cellular to a molecular level. This situation will involve a laboratory work session and will be scaffolded by an intelligent learning environment. Albaladejo, C. (1988). Pupils' meaning for 'mutation '. Journal of biological education, vol. 22 (3), 215-219. Bahar, M., Johnstone, A.H., & Hansell, M.H., (1999). Revisiting learning difficulties in biology, Journal of Biological Education, vol. 33 (2), 84-86. Johnstone, A.H., & Mahmoud, N.A. ( 1980). Isolating topics of high perceived difficulty in school biology, Journal of Biological Education, vol. 14 (2), 163-166. Marbach-Ad, G., & Stavy, R. (2000). Students' cellular and molecular explanations of genetic phenomena, Journal of Biological Education, vol. 34 (4), 200-205. Pfriedrichsen, P., Stone, B. (2004). Examining Students' Conceptions of Molecular Genetics in an Introductory Biology Course for Non-Science Majors: A Self Study. NARST, 2004. Added 2 self citations.

6) Bibliography

Table 1 :

 1

	Students' answers (A: notions never studied, B: DNA structure studied, C: DNA
	structure and mutations studied).
	Group	A		B		C		Total
		n = 28	%	n =56	%	n = 26	%	n =110	%
	Organism	0	0	1	1,79	0	0	1	0,91
	Cell	10	35,71	9	16,07	7	26,92	26	23,63


Keywords:

STATISTICS OF GEOMETRIC RANDOM SIMPLICIAL COMPLEXES E. FERRAZ AND A. VERGNE

Abstract. Given a Poisson process on a d-dimensional torus, its random geometric simplicial complex is the complex whose vertices are the points of the Poisson process and simplices are given by the Cech complex associated to the coverage of each point. We compute explicitly the variance of number of k-simplices as well as the variance of the Euler's characteristic. The solution strategy used to compute the second moment can be used to compute analytically the third moment and allows to stablish a conjecture for the nth moment. We apply concentration inequalities on the results of homology and the moments of the Euler's characteristics to find bounds for the for the coverage probability

Motivation

As technology goes on [START_REF] Kahn | Mobile networking for smart dust[END_REF][START_REF] Lewis | Wireless Sensor Networks[END_REF][START_REF] Pottie | Wireless integrated network sensors[END_REF], one can expect a wide expansion of the so-called sensor networks. Such networks represent the next evolutionary step in building, utilities, industrial, home, agriculture, defense and many other contexts [START_REF] Chong | Sensor networks: Evolution, opportunities, and challenges[END_REF].

These networks are built upon a multitude of small and cheap sensors which are devices with limited transmission capabilities and power. Each sensor monitors a region around itself by measuring some environmental quantities (e.g., temperature, humidity), detecting intrusion, etc, and broadcasts its collected informations to other sensors or to a central node. The question of whether information can be shared among the whole network often is of crucial importance.

Many researches have recently been dedicated to this problem considering a variety of situations. It is possible to categorize three main scenarios: those where it is possible to choose the position of each sensor, those where sensors are arbitrarily deployed in the target region with the control of a central station and those where the sensor locations are random in a decentralized system.

The problem of the first scenario is that, in many cases, placing the sensors is impossible or implies a high cost. Sometimes this impossibility comes from the fact that the cost of placing each sensor is too large and sometimes the network has an inherent random behavior (like in the ad hoc case, where users move). In addition, this policy cannot take into account the configuration of the network in the case of failure of some sensor.

The drawback of the second scenario is a higher unity cost of sensors, since each one has to communicate with the central station. Besides, the central station itself increases the cost of the whole system. Moreover, if sensors are supposed to know their positions, an absolute positioning system has to be included in each sensor, making their hardware even more complex and then more expensive.

It is thus important to investigate the third scenario: randomly located sensors, no central station. Actually, if we can predict some characteristics of the topology of a random network, the number of sensors (or, as well, the power supply of them) can be a priori determined such that a given network may operate with high a) b) c)

Figure 1. a) Sensors and their coverage; b) simplicial complex representation when sensors are monitoring the region; c) simplicial complex representation when sensors are communicating amoung them.

probability. For instance, we can choose the mean number of sensors such that, if they are randomly deployed, there is more than 99% of probability the target region to be completely covered. Usually, sensors are deployed in the plane or in the ambient space, thus mathematically speaking, one has to deal with configurations in R 2 , R 3 or a mainfold. The recent works of Ghrist and his collaborators [START_REF] Ghrist | Coverage and hole-detection in sensor networks via homology[END_REF][START_REF] Silva | Coordinate-free coverage in sensor networks with controlled boundaries via homology[END_REF] show how, in any dimension, algebraic topology can be used to compute the coverage of a given configuration of sensors. Trying to pursue their work for random settings, [START_REF] Decreusefond | On the poisson random geometric graph: the one dimensional case[END_REF] has completely solved this problem in one dimension, without using, however, the sofisticated tools of algebraic topology. Due the fact that we cannot order the points in R d , it is not possible to find as much results as found for the one-dimensional case, but the results in this work holds for any dimension d.

The principal idea of the problem is that each sensor can control some environmental information (such as temperature, pression, presence of an intruder, etc) around them. The homology of the coverage of this network, as shown in [START_REF] Ghrist | Coverage and hole-detection in sensor networks via homology[END_REF], can be represented by a simplicial complex. A simplicial complex is a generalization of a graph, so, while we represent a graph with points and edges, a simplicial complex can be represented by points, edges, filled triangles, filled tetraedrons and so on. Almost all the work considers the radius of monitoring ǫ but a different interpretation can be done if the sensors are communicating amoung them. In this case, we suppose that sensors have a power suply allowing them to transmit theirs ID's and, at the same time, sensors have receivers which can identify the transmitted ID's of other sensors above a threshold power. The sensors, knowing mutually the ID's of the close neighbors, are considered connected, creating an information network. The problem remains analogous as the previous one, except that we substitute the coverage radius ǫ by a communication one of ǫ/2. We can see examples of simplicial complexes representations given by sensors communicating amoung them or monitoring a region in Fig. 1.

In this work we consider that sensors are points of a Poisson point processus. This assumption reflects the fact that, due the lack of control of the sensors positioning, only a random fraction of the availables sensors will actually lie in the target region or some sensors may shut down by running out of energy, moreover, the position of each sensor, a priori, does not interact with the positions other sensors. Instead of using the Euclidean norm, we use the maximum norm along this paper. We consider this for three reasons: this norm represents a superior and an inferior limits for the euclidean norm (we can inscribe and circunscribe a circle with two squares); due to the random interactions with the environment (causing shadowing and fading), even the euclidean norm cannot capture with precision the real behavior of this kind of sensor networks, so we choose the norm that allows us to simplify the calculations; using the maximum norm, the Cech complex become equal to the Rips-Vietoris complexes, which allows us to apply directly some results of the algebraic topology. Finally, we assume that sensors lie over d-torus. We justify this choice for three reasons: it avoids the border effects, it helps to determine weather or not a sensor network in the d-box is completely covered and if ǫ is small compared to a, the calculations for all parameters in the d-torus are a good approximation for the [0, a] d box. The coverage of a point and the region where points can lie in are illustrated in Fig. 1, representing the case where a point is deployed over a plan. One of the main results of this paper is the explicit expression for the variance of the number of k-simplices, N k , the covariance between N k and N l and the variance of the Eulers's characteristic, χ, in such complex, which allows us to apply concentrations inequalities. For d ≥ 2, χ is expressed by a power serie and if d = 1, it is possible to find its closed-form expression. A complex closed-form expression for the third order central moment of N k is explicitely calculated and using the same strategy solution of this case, it is possible to find an expression that allows the computation of the moments of any order of the number of simplices.

The paper is organized in the following way: Section 2 presents the preliminaires of the topological tools used in the paper as well as the tools used from the Malliavin calculus; the calculations of the second order moments as well an application of the concentration inequlities are presented in the Section 3; in section 4, we present generalization of the procedure to find the second order moments allowing the computation of the n-th moment of the number of k-simplices.

Algebraic Topology

For further reading on topology, see [START_REF] Hatcher | Algebraic Topology[END_REF][START_REF] Armstrong | Basic Topology[END_REF][START_REF] Munkres | Elements of Algebraic Topology[END_REF]. Graphs can be generalized to more generic topological objects known as simplicial complexes. While graphs model binary relations, simplicial complexes represent higher order relations. Given a set of points V , a k-simplex is an unordered subset {v

0 , v 1 , • • • , v k } where v i ∈ V and v i = v j for all i = j. The faces of the k-simplex {v 0 , v 1 , • • • , v k } are defined as all the (k -1)-simplices of the form {v 0 , • • • , v j-1 , v j+1 , • • • , v k } with 0 ≤ j ≤ k.
A simplicial complex is a collection of simplices which is closed with respect to the inclusion of faces, i.e., if {v 0 , v 1 , • • • , v k } is a k-simplex then all its faces are in the set of (k -1)-simplices.

Given U = (U v , v ∈ T) a collection of open sets, the Cech complex of U denoted by C(U), is the abstract simplicial complex whose k-simplices correspond to (k + 1)-tuples of distinct elements of U that have non empty intersection, so

{v 0 , v 1 , • • • , v k } is a k-simplex if and only if k i=0 U v k = ∅.
One can define an orientation for a simplicial complex by defining an order on vertices. A change in the orientation corresponds to a change in the sign of the coefficient as

[v 0 , • • • , v i , • • • , v j , • • • , v k ] = -[ v 0 , • • • , v j , • • • , v i , • • • , v k ].
For each integer k, C k (X) is the vector space spanned by the set of oriented ksimplices of X. The boundary map ∂ k is defined to be the linear transformations

∂ k : C k → C k-1 which acts on basis elements [v 0 , • • • , v k ] via ∂ k [v 0 , • • • , v k ] = k i=0 (-1) k [v 0 , • • • , v i-1 , v i+1 , • • • , v k ].
Examples of such operations are given in Fig. 3. This map gives rise to a chain complex: a sequence of vector spaces and linear transformations

v0 v1 v2 v0 - v2 + [v0, v1] + [v1, v2] ∂ -→ [v2] -[v0] v0 v1 v2 v0 v1 v2 [v0, v1, v2] ∂ -→ [v1, v2] -[v0, v2] +[v0, v1] v0 v1 v2 v3 Filled Empty v0 v1 v2 v3 [v0, v1, v2, v3] ∂ -→ +[v1, v2, v3] -[v0, v2, v3] +[v0, v1, v3] -[v0, v1, v2]
• • • ∂ k+2 -→ C k+1 (X) ∂ k+1 -→ C k (X) ∂ k -→ C k-1 (X) • • • ∂2 -→ C 1 (X) ∂1 -→ C 0 (X).
A standard result then asserts that for any integer k,

∂ k • ∂ k+1 = 0. If one defines Z k = ker ∂ k and B k = im∂ k+1 , this induces that B k ⊂ Z k . 0 0 0 C k Z k B k C k+1 C k-1 ∂ -→ ∂ -→ Figure 4. A chain complex showing the sets C k , Z k and B k .
The k-dimensional homology of X, denoted H k (X) is the quotient vector space,

H k (X) = Z k (X) B k (X) •
and the k-th Betti number of X is its dimension:

β k = dim H k = dim Z k -dim B k .
Let s k be the number of k-simplices in a simplicial complex X. The well known topological invariant named Euler characteristic for X, denoted by χ(X), is an integer defined by:

χ(X) = ∞ i=0 (-1) i β i .
A well known theorem states that this is also given by:

χ(X) = ∞ i=0 (-1) i s i .
The simplicial complexes we consider are of a special type. They can be considered as a generalization of geometric random graphs.

Definition 1. Given ω a finite set of points on the torus. For ǫ > 0, we define

U ǫ (ω) = {B d∞ (v, ǫ), v ∈ ω} and C ǫ (ω) = C(U ǫ (ω)), where B d∞ (x, r) = {y ∈ T a d , x -y ∞ < r}. Theorem 1. Suppose ǫ < a/4. Then C ǫ (ω) has the same homotopy type as U ǫ (ω).
In particular they have the same Betti numbers.

Proof. This will follow from the so-called nerve lemma of Leray, as stated in [START_REF] Rotman | An Introduction to Algebraic Topology[END_REF]Theorem 7.26] or [START_REF] Björner | Topological methods[END_REF]Theorem 10.7]. One only needs to check that any non-empty intersection of sets B d∞ (v, ǫ) is contractible.

Consider such a non-empty intersection, and let x be a point contained in it. Then, since ǫ < a/4, the ball B d∞ (x, 4ǫ) can be identified with a cube in the Euclidean space. Then each B d∞ (v, ǫ) containing x is contained in B d∞ (x, 4ǫ), hence also becomes a cube with this identification, hence convex. Then the intersection of these convex sets is convex, hence contractible. Definition 2. Let ω be a finite set of points in T a d . For any ǫ > 0, the Rips-Vietoris complex of ω, R ǫ (ω), is the abstract complex whose k-simplices correspond to unordered (k + 1)-tuples of points in ω which are pairwise within distance less than ǫ of each other.

Lemma 2. For the torus

T d a equipped with the product distance d ∞ , R ǫ (ω) has the homotopy type of the Cech complex C 2ǫ (ω)
The proof is given in [START_REF] Ghrist | Coverage and hole-detection in sensor networks via homology[END_REF] in a slightly different context, but it is easy to check that it works here as well. It must be pointed out that Cech and Rips-Vietoris simplicial complexes can be defined similarly for any distance on T d a but it is only for the product distance that the homotopy type of both complexes coincides.

By Lemma 2, k points are forming a (k -1)-simplex whenever they are twoby-two closer than 2ǫ from each other. We define along the paper h

(v 1 , • • • , v k ) as h(v 1 , • • • , v k ) = h k (v 1 , • • • , v k ) = 1≤i<j≤k 1 [ vi-vj <2ǫ] ,
which determines if a set of k distinct ordered points generates a (k -1)-simplex.

Proposition 3. Let ω ∈ T d
a be a set of points, generating the simplicial complex

C ǫ (ω). Then, if i > d, β i (ω) = 0.
Proof. By Theorem 1, C ǫ (ω) has the same homology as U ǫ (ω). But U ǫ (ω) is an open manifold of dimension d, so its Betti numbers β i (ω) vanish for i > d, see for example [START_REF] Greenberg | Algebraic Topology: a First Course[END_REF]Theorem 22.24].

Proposition 4. Let ω ∈ T d
a be a set of points, generating the simplicial complex C ǫ (ω). There are only two possible values for the d-th Betti number of C ǫ (ω):

i) β d = 0, or ii) β d = 1.
If the second holds, then we also have χ(C ǫ (ω)) = 0.

Proof. By Theorem 1, C ǫ (ω) has the same homology as U ǫ (ω). Now, U ǫ (ω) is an open submanifold of the torus, so there are only two possibilities:

i) U ǫ (ω) is a strict open submanifold, hence non-compact ii) U ǫ (ω) = T d a .
In the first case, β d (ω) = 0 by [START_REF] Greenberg | Algebraic Topology: a First Course[END_REF]Corollary 22.25]. In the second case C ǫ (ω) has same homology as the torus, hence β d (ω) = 1 and χ(ω) = 0.

2.1. Application to sensor networks. We now interpret the topological properties of simplicial complexes in terms of connectivity and coverage. In terms of coverage in a network, a 0-simplex represents a single sensor and the existence of a k-simplex means that the (k + 1) points of this simplex are covering the convex hull containing those points. We can see in Figure 1, examples of some simplices and their interpretation in terms of sensor networks.

In a very intuitive fashion, the number of k-simplices itself shows some tendency in the network: if in two networks with identical number of sensors, one of them has more 1-simplices than the other, this first one has a tendency to be more connected; by the same reason, if a network has more 2-simplices than another one, the region on the first case tends to be more strongly covered.

In a more sophisticated way, Theorem 1 formalizes that, in order to determine coverage of sensors, it suffices interpret them as Cech complexes. Unfortunately, a moment of thought shows that constructing the Cech complex cannot be done by pairwise only communications between sensors. Thus, the only complex that can be computed this way is the Rips-Vietoris complex.

An interpretation to Euler characteristic is given by Proposition 4, where we see that χ = 0 is a necessary condition to have a complete coverage of the torus, and β d = 1 is a necessary and sufficient condition. This could in turn translate into conditions for coverage in [0, a] d when considered as embedded in Euclidean space (i.e., not as a torus), but then one needs to be careful about border effects. For example, one can say that

β d = 1 is a sufficient condition for coverage of [ǫ, a -ǫ] d .

Stochastic Model

Poisson point process.

To characterize the randomness of the system, we consider that the set of points is represented by a Poisson point process ω with intensity λ in a Polish space Y . The space of configurations on Y , is the set of locally finite simple point measures (cf [START_REF] Privault | Stochastic Analysis in Discrete and Continuous Settings[END_REF]):

Ω Y = ω = n k=0 δ(x k ) : (x k ) k=n k=0 ⊂ Y, n ∈ N ∪ {∞} ,
where δ(x) denotes the Dirac measure at x ∈ Y . Simple measure means that ω({x}) ≤ 1 for any x ∈ Y . Locally finite means that ω(K) < ∞ for any compact K

Sensor network coverage Cech complex representation

S 1 v 1 S 1 S 2 v 1 v 2 S 1 S 2 S 3 v 1 v 2 v 3 S 1 S 2 S 2 v 1 v 2 v 3 S1 S2 S3 S4 ǫ ǫ ǫ ǫ v 1 v 2 v 3 v 4
Table 1. Topological representation of the coverage of a sensor network. Each node v represents a sensor. From top to bottom, the highest order simplex is a vertex, an edge, a triangle, three edges, a tetrahedron. of Y . It is often convenient to identify an element ω of Ω Y with the set corresponding to its support, i.e.,

n k=0 δ(x k ) is identified with the unordered set {x 1 , • • • , x n }. For A ∈ B(Y ), we have δ(x k )(A) = 1 [x k ∈A] , so ω(A) = x k ∈ω 1 [x k ∈A] = A dω(x),
counts the number of atoms in A. The configuration space Ω Y is endowed with the vague topology and its associated σ-algebra denoted by F Y . Since ω is a Poisson point process of intensity λ: i) For any A, ω(A) is a random variable of parameter λS(A), i.e.,

P(ω(A) = k) = e -λS(A) (λS(A)) k k! • ii) For A ′ ∈ B(Y )
, for any disjoints A, A ′ , the random variables ω(A) and ω(A ′ ) are independent.

Define ∆ n = {(x 1 , • • • , x n ) ∈ Y n | x i = x j , ∀i = j}. Let f (x 1 , • • • , x n
) be a measurable function and let F (ω) be a random variable given by

F (ω) = xi∈ω∩A,1≤i≤n xi =xj if i =j f (x 1 , • • • , x n ) = A∩∆n f (x 1 , • • • , x n ) dω(x 1 ) • • • dω(x n ),
A well known property of the Poisson point processes [START_REF] Daley | An introduction to the theory of point processes[END_REF] states that

E λ [F (ω)] = A f (x 1 , • • • , x n ) dλ(x 1 ) • • • dλ(x n ). A real function f : Y n → R is called symmetric if f (x σ(1) , • • • , x σ(n) ) = f (x 1 , • • • , x n )
for all permutations σ of S n .The space of symmetric square integrable random variables is denoted by

L 2 (λ) •n . For f ∈ L 2 (λ) •n , the multiple Poisson stochastic integral I n (f n ) is then defined as I n (f n )(ω) = ∆ n f n (x 1 , • • • , x n )( dω(x 1 ) -dλ(x 1 )) • • • ( dω(x n ) -dλ(x n )). If f n ∈ L 2 (λ) •n and g m ∈ L 2 (λ) •m , the isometry formula (1) E λ [I n (f n )I m (g m )] = n!1 [m=n] f n , g m L 2 (λ) •n
holds true (see [START_REF] Privault | Stochastic Analysis in Discrete and Continuous Settings[END_REF]). Furthermore, we have:

Theorem 5. Every random variable F ∈ L 2 (Ω Y , P) admits a (unique) Wiener-Poisson decomposition of the type

F = E λ [F ] + ∞ n=1 I n (f n ),
where the series converges in L 2 (P) and, for each n ≥ 1, the kernel f n is an element of L 2 (λ) •n . Moreover, we have the isometry

(2) F -E λ [F ] 2 L 2 (λ) •n = ∞ n=1 n! f n 2 L 2 (R+) •n .
For

f n ∈ L 2 (λ) •n and g m ∈ L 2 (λ) •m , we define f n ⊗ l k g m , 0 ≤ l ≤ k, to be the function: (3) (y l+1 , • • • , y n , x k+1 , • • • , x m ) -→ Y l f n (y 1 , • • • , y n )g m (y 1 , • • • , y k , x k+1 , • • • , x m ) dλ(y 1 ) . . . dλ(y l ).
We denote by

f n • l k g m the symmetrization in n + m -k -l variables of f n ⊗ l k g m , 0 ≤ l ≤ k.
This leads us to the next proposition, shown in [START_REF] Privault | Stochastic Analysis in Discrete and Continuous Settings[END_REF]:

Proposition 6. For f n ∈ L 2 (λ) •n and g m ∈ L 2 (λ) •m , we have I n (f n )I m (g m ) = 2(n∧m) s=0 I n+m-s (h n,m,s ), where h n,m,s = s≤2i≤2(s∧n∧m) i! n i m i i s -i f n • s-i i g m belongs to L 2 (λ) •n+m-s , 0 ≤ s ≤ 2(m ∧ n).
In what follows, given f ∈ L 2 (λ) •q (q ≥ 2) and t ∈ Y , we denote by f ( * , t) the function on Y q-1 given by (

x 1 , • • • , x q-1 ) -→ f (x 1 , • • • , x q-1 , t).
Definition 3. Let Dom D be the the set of random variables F ∈ L 2 (P ) admitting a chaotic decomposition such that

∞ n=1 qq! f n 2 < ∞.
Let D be defined by

D : Dom D → L 2 (Ω Y × Y, P × λ) F = E λ [F ] + n≥1 I n (f n ) -→ D t F = n≥1 nI n-1 (f n ( * , t)).
It is known, cf. [START_REF] Ito | Generalized poisson functionals[END_REF], that we also have

D t F (ω) = F (ω ∪ {t}) -F (ω), dP × dt a.e..
Definition 4. The Ornstein-Uhlenbeck operator L is given by

LF = - ∞ n=1 nI n (f n ),
whenever F ∈ Dom L, given by those F ∈ L 2 P such that their chaotic expansion verifies

∞ n=1 q 2 q! f n 2 < ∞.
Note that E λ [LF ] = 0, by definition and (1).

Definition 5. For F ∈ L 2 (P) such that E λ [F ] = 0, we may define L -1 by

L -1 F = - ∞ n=1 1 n I n (f n ).
Combining Stein's method and Malliavin calculus yields the following theorem, see [START_REF] Peccati | Stein's method and normal approximation of poisson functionals[END_REF]:

Theorem 7. Let F ∈ Dom D be such that E λ [F ] = 0 and Var(F ) = 1. Then, d W (F, N (0, 1)) ≤ E λ 1 - Y [D t F × D t L -1 F ] dλ(t) + Y E λ |D t F | 2 D t L -1 F dλ(t).
Another result from the Malliavin calculus used in this work is the following one, quoted from [START_REF] Privault | Stochastic Analysis in Discrete and Continuous Settings[END_REF]:

Theorem 8. Let F ∈ Dom D be such that DF ≤ K, a.s., for some K ≥ 0 and DF L ∞ (Ω,L 2 (Y )) < ∞. Then (4) P(F -E λ [F ] ≥ x) ≤ exp - x 2K log 1 + xK DF L ∞ (Ω,L 2 (Y ))
.

Proposition 9. Let X a compact subset of R d and consider the map τ : X → Y as x i = ky i for x i ∈ X, y i ∈ Y and k a positive real constant. Denote by τ * ω the image measure of ω by τ , i.e., τ * :

Ω X → Ω Y maps ω = ∞ i=1 δ(x i ) to τ * ∞ i=1 δ(kx i ).
The application τ * : Ω are the same for any k if λ τ = λ/k d and ǫ τ = kǫ.

Proof. A slightly changing on Propositions 6.1.7 and 6.1.8 of [START_REF] Privault | Stochastic Analysis in Discrete and Continuous Settings[END_REF] is enough to show that τ * maps the Poisson measure λ on Ω X to the Poisson measure λ τ = λ/k d on Ω Y . Then, it suffices to realize that for x i ∈ X and for y i ∈ Y :

h(x 1 , • • • , x k ) = 1≤i<j≤k 1 [ xi-xj <2ǫ] = 1≤i<j≤k 1 [ kxi-kxj <2kǫ] , hence h(y 1 , • • • , y k ) = 1≤i<j≤k 1 [ yi-yj <2ǫτ ] ,
which concludes the proof.

Second order moments

4.1. Number of k-simplices. We use all the definitions of the previous section.. The number of (k -1)-simplices can be counted by the expression:

N k = v1,...,v k ∈ω vi =vj if i =j h(v 1 , • • • , v k ) k! = ∆ k h(v 1 , • • • , v k ) k! dω(x 1 ) • • • dω(x k ) (5)
Lemma 10. We can rewrite N k as

N k = 1 k! k i=0 k i λ k-i I i (T d a ) i h(x 1 , . . . , x k ) dx 1 . . . dx k-i .
Proof. We have that

∆ k h(x 1 , • • • , x k )( dω(x 1 ) -λ dx 1 ) . . . ( dω(x i ) -λ dx i )λ dx i+1 . . . λ dx k = i j=0 (-1) j i j ∆ k h(x 1 , • • • , x k ) dω(x 1 ) . . . dω(x j )λ dx j+1 . . . λ dx k .
Thus, after some algebrism with the binomial factors, we have

1 k! k i=0 k i i j=0 (-1) j i j ∆ k h(x 1 , • • • , x k ) dω(x 1 ) . . . dω(x j )λ dx j+1 . . . λ dx k = k! ∆ k h(x 1 , • • • , x k ) dω(x 1 ) . . . dω(x k ) = N k ,
concluding the proof. Definition 6. Let C 1 and C 2 be two simplices with common vertices. For L ∈ P({1, 2}), let us denote m L the number of vertices belonging exactly to the list L of simplices.

Then M = m 12 + m 1 + m 2 is the total number of vertices and J 2 represents the integral on these two simplices:

J 2 (m 12 , m 1 , m 2 ) = ∆ m 12 +m 1 ∆ m 12 +m 2 h m12+m1 h m12+m2 dx 1 . . . dx M . with x 1 , • • • , x M being the M vertices. -1 0 1 x 2 x m12 . . . 1 x m12+1
x m12+m1 . . .

J 2 (m 12 , m 1 , m 2 ) = m 12 + m 1 + m 2 + 2m 1 m 2 m 12 + 1 . ( 6
)
Proof. Let us split the integration domain of J 2 in two domains S 1 and S 2 corresponding to the cases:

(1) All the vertices are connected with each other, thus there is only one simplex. The integral on S 1 is simply the number of points in the simplex:

M = m 12 + m 1 + m 2 .
(2) There are at least two vertices at distance d > 1, which leads to two simplices. By symetry we can choose to order the m L vertices for each L ∈ P({1, 2}) from lowest to greatest or the opposite and choose which simplex is on which side of the axis. Thus we have the integral on S 2 equal to 2m 12 !m 1 !m 2 !A, with A an integral whose calculation is detailed below. We choose to enumerate the vertices of the simplexes such that:

• x 1 , • • • , x m12 are the m 12 common vertices.

• x m12+1 , • • • , x m12+m1 are the m 1 vertices of only C 1 . • x m12+m1+1 , • • • , x M are the m 2 vertices of only C 2 .
Without loss of generality we can choose the origin to be x 1 . The vertices are now order as described in Fig. 5:

0 ≤ x m12 ≤ x m12-1 ≤ • • • ≤ x 2 ≤ 1, -1 ≤ x 2 -1 ≤ x m12+1 ≤ x m12+2 ≤ • • • ≤ x m12+m1 ≤ x m12+1 + 1, x m12+1 ≤ x m12+m1+1 -1 ≤ x M ≤ x M-1 ≤ • • • ≤ x m12+m1+1 ≤ 1, Let us denote J a (f )(x) = x a f (u) du then we write the composition J (2) a (f )(x) =
x a u a f (v) dv du. We also denote m = m 12 + 1 and n = m 12 + m 1 + 1, then we have:

A = 1 0 J (m12-2) 0 (1)(x 2 ) 0 x2-1 -J (m1-1) xm+1 (1)(x m ) 1 xm+1 J (m2-1) xn-1 (1)(x n ) dx n dx m dx 2 .
We easily find that:

J (m12-2) 0 (1)(x 2 ) = x m12-2 2 (m 12 -2)! , -J (m1-1) xm+1 (1)(x m ) = 1 (m 1 -1)! , J (m2-1) xn-1 (1)(x n ) = 1 (m 2 -1)! .
Thus we have:

A = 1 (m 12 -2)!(m 1 -1)!(m 2 -1)! 1 0 x m12-2 2 0 x2-1 -x m dx m dx 2 = 1 (m 1 -1)!(m 2 -1)!(m 12 + 1)! ,
concluding the proof.

Theorem 12. Let ǫ ≤ a/6. Then, the covariance between the number of (k -1)simplices, N k , and the number of (l -1)-simplices, N l , for l ≤ k is given by

(7) Cov λ [N k , N l ] = l-1 i=0 1 i!(k -l + i)!(l -i)! (λ(2ǫ) d ) k+i a 2ǫ d k + i + 2 i(k -l + i) l -i + 1 d . Proof. We want to evaluate E λ [(N k -E λ [N k ])(N l -E λ [N l ])]
. By Lemma 10, this can be written as

E λ 1 k! k i=1 k i λ k-i I i f k i 1 l! l i=1 l i λ l-i I i f l i ,
where

f n j = (T d a ) j h(v 1 , • • • , v n ) dv 1 . . . dv n-j .
Using the isometry formula, given by Eq. ( 1), we have

Cov λ [N k , N l ] = 1 k!l! l i=1 k i l i λ k+l-2i E λ I i f k i I i f l i = 1 k!l! l i=1 k i l i λ k+l-2i i! f k i f l i L 2 (λ) •i = l-1 i=0 1 i!(k -l + i)!(l -i)! λ k-l+2i f k l-i f l l-i L 2 (λ) •(l-i) . (8)
Hence, we are reduced to compute

f k j f l j L 2 (λ) •(j) = (T d a ) j (T d a ) l-j h(v 1 , • • • , v l ) dv j+1 . . . dv l (T d a ) k-j h(v 1 , • • • , v k ) dv j+1 . . . dv k λ dv 1 . . . λ dv j .
Since a > ǫ/6,the integration region is convex (see Fig. 6) and we have

f k j f l j L 2 (λ) •(j) = [0,a] d λ dv 1 ([0,a] d ) k-1 h(0, v 2 , • • • , v k ) × h(0, v 2 , • • • , v j , v ′ 1 , • • • , v ′ l-j ) dv ′ l-j . . . dv ′ 1 dv k . . . dv j+1 λ dv j . . . λ dv 2 . Moreover, if v i = (u i,1 , • • • , u i,d ) and v ′ i = (u ′ i,1 , • • • , u ′ i,d
) and we proceed to the following substitutions:

u i,1 = 2ǫx i if 2 ≤ i ≤ j, u i,1 = 2ǫy k-j if j + 1 ≤ i ≤ k, u ′ i,1 = 2ǫz i if 1 ≤ i ≤ l -j, v 0 v 1 v 2 v 0 v 1 v 2 a) b)
Figure 6. a) Maximum cover in T a and ǫ = a/6. The red region shows the cover of a point v 0 , the blue region is the cover of v 1 and the greer region is the cover of v 2 . b) Maximum cover in the same conditions of a) when ǫ = a/5. In this case, we the three covers intersect each other pairwise, but there is no intersection of the three covers.

This results in a Jacobian (2ǫ) k+l-2i-1 and we recognize the integral to be exactely J 2 (j, kj, lj) as defined in Definition 6. Thus, we have:

f k j f l j L 2 (λ) •(j) = λ i a d (2ǫ) k+l-2i-1 (J 2 (j, k -j, l -j)) d .
Finally, using Eq. ( 6) and Eq. ( 8) gives the result.

Remark. We remark that the possibility of writing Var(N k ) as Eq. ( 7) is due the fact that we use the maximum norm. This simplifies the calculations since we can treat each component individually. However, considering the Euclidean norm it is still possible to find analytically a closed-form expression for Var(N k ), but its calculation involves nasty integrals and a generic term cannot be found. When we consider the Rips-Vietoris complex in T 2 a , the variance of the number of 1-simplices and 2-simplices are given by:

V λ [N 2 ] = a 2ǫ 2 π 2 (4λǫ 2 ) 2 + π 2 (4λǫ 2 ) 3 ,
and

V λ [N 3 ] = a 2ǫ 2 (4λǫ) 3 π 6 π - 3 √ 3 4 + (4λǫ 2 ) 4 π π 2 2 - 5 12 - π √ 3 2 +(4λǫ 2 ) 5 π 2 4 π - 3 √ 3 4 2   • 4.2.
Euler's characteristic. Since we have an expression for the variance of the number of k-simplices, it is possible to calculate one for the Euler characteristic.

Theorem 13. Let ǫ ≤ a/6. Then, the variance of the Euler characteristic in a d torus is:

V λ [χ] = a 2ǫ d ∞ n=1 c d n (λ(2ǫ) d ) n ,
where

c d n = n j=⌈(n+1)/2⌉   2 j i=n-j+1 (-1) i+j (n -j)!(n -i)!(i + j -n)! n + 2(n -i)(n -j) 1 + i + j -n d - 1 (n -j)! 2 (2j -n)! n + 2(n -j) 2 1 + 2j -n d .
Proof. The variance of χ is given by:

V λ [χ] = E λ (χ -E λ [χ]) 2 = E λ   ∞ k=1 (-1) k N k - ∞ k=1 (-1) k E λ [N k ] 2   = E λ   ∞ k=1 (-1) k (N k -E λ [N k ] 2   = E λ   ∞ i=1 ∞ j=1 (-1) i+j (N i -E λ [N i ])(N j -E λ [N j ])   .
We remark that N i ≤

N i 1 i! , so there is a constant c such that E λ   ∞ i=1 ∞ j=1 |(N i -E λ [N i ])(N j -E λ [N j ])|   ≤ ∞ i=1 ∞ j=1 N i 1 i! N j 1 j! ≤ cE λ [e n1 ] 2 < ∞.
Thus the alternating serie converges absolutely allowing us to exchange the mean with the sums and we can write

V λ [χ] = ∞ i=1 (-1) i ∞ j=1 (-1) j Cov λ [N i , N j ] .
The result follows by Eq. ( 7) and some tedious but straightforward algebra.

Lemma 14. Let n be a positive integer, then

n j=1 n j j -1 n -j -1 - j -1 n -j = (-1) n .
Proof. We first simplify the expression:

n j=1 n j j -1 n -j -1 - j -1 n -j = n j=1 2n -3j j n j j n -j ,
Then, applying hypergeometric functions, we solve the sum:

n j=1 2n -3j j n j j n -j = (-1) n .
Theorem 15. In one dimension, the expression of the variance of the Euler characteristic is:

V λ [χ] = a λe -2λǫ -4λ 2 ǫe -4λǫ .
Proof. If d = 1, according to Theorem 13:

V λ [χ] = a 2ǫ ∞ n=1 c 1 n (2λǫ) n , (9)
and we define

α n = n j=⌈ n+1 2 ⌉   2 j i=n-j+1 (-1) i+j n (n -j)!(n -i)!(i + j -n)! - n (n -j)! 2 (2j -n)!   . and β n = c 1 n -α. It is well known that 2j-n i=0 (-1) i j i = (-1) 2j-n-1 j -1 2j -n ,
using Stiffel's relation, we obtain:

α n = (-1) n n n! n j=⌈ n+1 2 ⌉ n j 2 2j-n i=0 (-1) i j i + 2(-1) n n j = 1 (n -1)! n j=⌈ n+1 2 ⌉ 2 n j j -1 n -j -1 - n j j n -j -2(-1) n n j = 1 (n -1)! n j=⌈ n+1 2 ⌉ n j j -1 n -j - j -1 n -j -1 -2(-1) n n j . (10)
The identity n j = n n-j allows us to write that

n j=⌈(n+1)/2⌉ (-2(-1) n ) n j = n j=0 n j = 2 n , n odd, n j=⌈(n+1)/2⌉ (-2(-1) n ) n j = n n/2 + n j=0 - n j = -2 n + n n/2
, n even.

Since j-1 n-j = 0 for j < n+1

2

, we have

n j=⌈ n+1 2 ⌉ n j j -1 n -j - j -1 n -j -1 = n j=1 n j j -1 n -j - j -1 n -j -1 for n odd and n j=⌈ n+1 2 ⌉ n j j -1 n -j - j -1 n -j -1 = - n n/2 + n j=1 n j j -1 n -j - j -1 n -j -1 .
for n even. According to Lemma 14, we get:

n j=⌈(n+1)/2⌉ n j j -1 n -j -1 - j -1 n -j = -1, n odd, n j=⌈(n+1)/2⌉ n j j -1 n -j -1 - j -1 n -j = 1 - n n/2 , n even.
Then, we substitute these two last expressions in Eq. ( 10) to obtain

α n = (-1) n (1 -2 n )1 [n≥1] (n -1)! ,
and thus

∞ i=0 α n x n = -xe -x + 2xe -2x .
Proceeding along the same line, β n is given by

β n = n j=⌈ n+1 2 ⌉   2 j i=n-j+1 (-1) i+j 2(n -i)(n -j) (n -j)!(n -i)!(i + j -n + 1)! - 2(n -j) 2 (n -j)! 2 (2j -n + 1)! = (-1) n (-2 + 2 n )1 [n≥1] (n -1)! - 21 [i≥2] (i -2)! ,
and again we can simplify the power serie

∞ i=0 β n x n : ∞ i=0 β n x n = 2xe -x -2(x + x 2 )e -2x .
Then, substituting α n and β n in Eq. ( 9) yields the result.

Indeed, Corollary 15 suggests the possibility of finding a simple expression for variance of the Euler's characteristic in higher dimensions. Applying it for d = 2, we have a sum in a squared term given by:

n + 2(n -i)(n -j) 1 + i + j -n 2 = n 2 + 4n(n -i)(n -j) 1 + i + j -n + 2(n -i)(n -j) 1 + i + j -n 2 •
With respect to the case in one dimension, no extra knowledge is needed to simplify the first two terms, and, if x = λ(2ǫ) 2 , they are given respectively by:

a 2ǫ 2 (-x + x 2 )e -x + 2(x -2x 2 )e -2x
and a 2ǫ 2 4(xx 2 )e -x + 4(-x + 2x 2 )e -2x .

Unfortunately, we are not able to find a way to express the third term without an infinite serie. This holds for any dimension: we can always find a closed-form for the terms depending on n d and n d-1 , but we cannot go any further.

Theorem 16. We have Dχ ≤ 2 and Dχ L ∞ (Ω,L 2 (T d a )) < ∞ and

P(χ -χ ≥ x) ≤ exp - x 4 log 1 + 2x V λ [χ] .
Proof. In two dimensions, the Euler characteristic is:

χ = β 0 -β 1 + β 2 .
Therefore we can bound Dχ by the variation of β 0β 1 added to the variation of β 2 when we add a vertex to a simplicial complex.

If we add a vertex on the torus, either the vertex is isolated or not. In the first case, it forms a new connected component incrementing β 0 by 1, and the number of holes that is β 1 is the same. Otherwise, as there is no new connected component, β 0 is the same, but the new vertex can at most fill a hole incrementing β 1 by 1. Therefore, the variation of β 0β 1 is at most 1. Now, let us look at the variation of β 2 when we add a vertex to a simplicial complex. According to Proposition 3 is at most 1, showing that Dχ ≤ 2. Then, we use Eq. ( 4) to complete the proof.

N th order moments

For this section, without loss of generality, using Proposition 9, we can choose k = 1/2ǫ, so λ τ = λ(2ǫ) d , ǫ τ = 1/2 and ak = a/2ǫ.

We are interested in the central moment, so we introduce the following notation for the centralized number of (k -1)-simplices: Ñk = N k -Nk .

Finally, let us denote that i j = 0 as soon as i ≤ 0 or j ≤ 0 or ij ≤ 0 for i and j integers. Definition 7. We extend the Definition 6 used in the second order moments calculations.

Let C 1 , C 2 and C 3 be three simplices with common vertices. For L ∈ P({1, 2, 3}), let us denote m L the number of vertices belonging exactly to the list L of simplices.

Then M = m 123 + m 12 + m 13 + m 23 + m 1 + m 2 + m 3 is the total number of vertices and J 3 represents the integral on these three simplices:

J 3 = ∆ p 1 ∆ p 2 ∆ p 3 h p1 h p2 h p3 dx 1 . . . dx M .
with p i being the number of vertices of simplex

C i for i = 1, • • • , 3, for instance p 1 = m 123 + m 12 + m 13 + m 1 , and x 1 , • • • , x M being the M vertices.
Definition 8. We denote J 3 (i, j, s, t) the integral defined above such that

• m 123 = 2t -i -j + s ∨ 0 • m 12 = i + j -s -t ∨ 0 • m 13 = i -t ∨ 0 • m 23 = j -t ∨ 0 • m 1 = k -i ∨ 0 • m 2 = k -j ∨ 0 • m 3 = k -s ∨ 0.
Theorem 17. The third moment of the number of (k -1)-simplices is given by:

E λ Ñk 3 = i,j,s,t λ 3k-i-j t! k i k j k s i t j t t i + j -s -t J 3 (i, j, s, t), with s ≥ |i -j|.
Proof. The chaos decomposition of the number of (k -1)-simplices is as shown in lemma 10:

Ñk = I 1 (f 1 ) + • • • + I k (f k ) = k i=1 I i (f i ), with f i (x 1 , • • • , x i ) = k i h(x 1 , • • • , x k )λ k-i dx k . . . dx i+1 , And
I i (f i ) = f i ( dω(x 1 ) -dλ(x 1 )) . . . ( dω(x i ) -dλ(x i )).
Then, we define g i,j,i+j-s = i+j-s∧i∧j t=⌈ i+j-s 2 ⌉ t! i t j t t i+j-s-t f i • u-t t f j and using the chaos expansion (cf Proposition 6):

Ñk 3 = (I 1 (f 1 ) + • • • + I k (f k )) 3 =   k i=1 k j=1 I i (f i )I j (f j )   (I 1 (f 1 ) + • • • + I k (f k )) = k i,j=1 i+j s=|i-j| I s (g i,j,i+j-s )(I 1 (f 1 ) + • • • + I k (f k )) = k i,j,l=1 i+j s=|i-j| I s (g i,j,i+j-s )I l (f l ).
When taking the expectation of Ñk , we use the isometry formula in Eq. ( 1). Denoting u = i + js, we obtain:

E λ Ñk 3 = E λ   k i,j=1 i+j∧k s=|i-j|∨1 I s (g i,j,u )I s (f s )   = k i,j=1 i+j∧k s=|i-j|∨1 g i,j,u f s λ s dx 1 . . . dx s = k i,j=1 i+j∧k s=|i-j|∨1 u∧i∧j t=⌈ u 2 ⌉ λ s t! i t j t t u -t (f i • u-t t f j )f s dx 1 . . . dx s .
Then we recognize the integral defined in Definition 8:

E λ Ñk 3 = k i,j,=1 i+j∧k s=|i-j|∨1 u∧i∧j t=⌈ u 2 ⌉ λ 3k-i-j t! k i k j k s i t j t t u -t J 3 (i, j, s, t).
Finally, relaxing the boundaries on the sums conclude the proof. Then M = L∈P({1,••• ,n}) m L is the total number of vertices and J n represents the integral on these n simplices:

J n = ∆ p 1 • • • ∆ pn h p1 . . . h pn dx 1 . . . dx M .
with p i being the number of vertices of simplex

C i for i = 1, • • • , n, and x 1 , • • • , x M being the M vertices.
Theorem 18. The expression of the nth power of the number of (k -1)-simplices is given by:

(11) Ñ n k = i1,••• ,in s1,•••sn-2 t1,•••tn-2   n-2 j=1 t j ! m j,1 t j m j,2 t j t j u j -t j   I a (• j∈A f ij )I b (• j∈ Āf ij ). With for j ∈ {1, • • • , n -2}: • 1 ≤ i 1 , • • • , i n ≤ k, • s j ≥ |m j,1 -m j,2 |, • m j,1 = i 2j-1 if 1 ≤ j ≤ ⌊ n 2 ⌋ and s 2(j-⌊ n 2 ⌋)-1 otherwise, • m j,2 = i 2j if 1 ≤ j ≤ ⌊ n 2 ⌋ and s 2(j-⌊ n 2 ⌋) otherwise, • u j = m j,1 + m j,2 -s j , • A ⊂ {1, • • • , n}, • If n is even, then a = s n-3 and b = s n-2 , • If n is odd, then a = s n-2 and b = i n .
Proof. The decomposition of the centralized number of (k -1)-simplices is:

Ñk = I 1 (f 1 ) + • • • + I k (f k ) = k i=1 I i (f i ).
Now, we raise Ñk to the nth power:

Ñk n = k i=1 I i (f i ) n .
First, we consider the case where n is even, we can group the factors by 2:

Ñ n k = k i1=1 I i1 (f i1 ) k i2=1 I i2 (f i2 ) . . .   k in-1=1 I in-1 (f in-1 ) k in=1 I in (f in )   .
We then use the chaos expansion of Proposition 6:

I i (f i )I j (f j ) = 2(i∧j) s=0 I i+j-s   s≤2t≤2(s∧i∧j) t! i t j t t s -t f i • s-t t f j   = i+j s=|i-j| I s   i+j-s≤2t≤2(i+j-s)∧i∧j) t! i t j t t i + j -s -t f i • i+j-s-t t f j   .
Let us denote g s = t! i t j t t i+j-s-t f i • i+j-s-t t f j , so we can re-write, relaxing the boundaries on the sums:

I i (f i )I j (f j ) = s≥|i-j| t I s (g s ).
Thus, we have:

Ñ n k = k i1,i2=1 s1≥|i1-i2| t1 I s1 (g s1 ) • • • k in-1,in=1 s n/2 ≥|in-1-in| t n/2 I s n/2 (g s n/2 ).
We go on grouping terms by 2 until we only have a product of 2 chaos left: First we made n/2 chaos expansions, leading to n/2 sums with indexes s

j , j = 1, • • • , n/2.
To reduce the number of chaos to 2, we have to make other chaos expansions. For j ≥ n 2 + 1, the sum indexed by s j represents the expansion of the chaos indexed s 2(j-n 2 )-1 and s 2(j-n 2 )-1 . We have 2 chaos remaining when j = 2(j -n 2 ) + 2, i.e. when j = n -2.

Moreover, there are as much sums indexed with t j as with s j , that is n -2. Thus we can write: • m j,1 = i 2j-1 if 1 ≤ j ≤ n 2 and s 2(j-n 2 )-1 otherwise,

Ñ n k = k i1,••• ,in=1 s1,•••sn-2 t1,•••tn-2
• m j,2 = i 2j if 1 ≤ j ≤ n 2 and s 2(j-n 2 ) otherwise. Then, denoting u j = m j,1 + m j,2s j and A the subset of {1, • • • , n} such that if j ∈ A then the chaos i j is expanded in the chaos s n-3 , we have: The notation • j∈A f ij represents the product defined in Eq. ( 3) of the functions f ij for j ∈ A, but whom variables depend on all the i 1 , • • • , i n , s 1 , • • • , s n-2 , and

t 1 , • • • , t n-2 .
Now, if n is odd, we consider n -1 which is even, therefore we have: • m j,1 = i 2j-1 if 1 ≤ j ≤ n-1 2 and s 2(j-n-1 2 )-1 otherwise, • m j,2 = i 2j if 1 ≤ j ≤ n-1 2 and s 2(j-n-1 2 ) otherwise. And with u j = m j,1 + m j,2s j , Theorem 19. The expression of the nth moment of the number of (k -1)-simplices is given by:

Ñ n k = k i1,••• ,in-1=1
E λ Ñk n = i1,••• ,in s1,••• ,sn-3 t1,••• ,tn-2 λ nk+c   n j=1 λ -ij k i j     n-2 j=1 t j ! m j,1 t j m j,2 t j t j u j -t j   J n (i 1 , • • • , i n , s 1 , • • • , s n-3 , t 1 , • • • , t n-2 ).
With for j ∈ {1, • • • , n -2}: Proof. The expression of the nth power of the number of (k -1)-simplices is given in Eq. ( 11):

• if j ≤ n -3, s j ≥ |m j,1 -m j,2 |, • m j,1 = i 2j-1 if 1 ≤ j ≤ ⌊ n
Ñ n k = k i1,••• ,in=1 s1,••• ,sn-2 t1,••• ,tn-2   n-2 j=1 t j ! m j,1 t j m j,2 t j t j u j -t j   I a (• j∈A f ij )I b (• j∈ Āf ij ).
If n is even, we have:

Ñ n k = k i1,••• ,in=1 s1,••• ,sn-2 t1,••• ,tn-2   n-2 j=1 t j ! m j,1 t j m j,2 t j t j u j -t j   I sn-3 (• j∈A f ij )I sn-2 (• j∈ Āf ij ).
So let us focus on the only part of the equation which is likely to change when we take the expected value, that we will denote:

K = sn-3 sn-2 I sn-3 (• j∈A f ij )I sn-2 (• j∈ Āf ij ).
We then use the property of Eq. ( 1) and recognize the integral from Definition 9: Then for n odd we directly write:

E λ [K] = sn-3   n j=1 λ k-ij k i j   λ sn-3 J n (i 1 , • • • , i n , s 1 , • • • , s n-3 , t 1 , • • • , t n-2 ) = sn-3 λ nk+sn-3   n j=1 λ -ij k i j   J n (i 1 , • • • , i n , s 1 , • • • , s n-3 , t 1 , • • • , t n-2 ),
K ′ = in sn-2 I in (• j∈I f ij )I sn-2 (• j∈ Ī f ij ), E λ [K ′ ] = in λ nk+in   n j=1 λ -ij k i j   J n (i 1 , • • • , i n , s 1 , • • • , s n-3 , t 1 , • • • , t n-2 ),
with i n ∈ {|m n-2,1m n-2,2 | ∨ 1, k}.

The binomials with the i j allow us to relax the boundaries on the sums on i j , concluding the proof.

2 aFigure 2 .

 22 Figure 2. Ilustration of the coverage of a point and the region where points can lie, in the 2 dimensional case

Figure 3 .

 3 Figure 3. Examples of boundary maps. From left to right. An application over 1-simplices. Over a 2-simplex. Over a 3-simplex, turning a filled tetrahedron to an empty one.

1 Figure 5 .

 15 Figure 5. Example of relative positions of the points

Definition 9 .

 9 Let C 1 , • • • , C n be n simplices with common vertices. For L ∈ P({1, • • • , n}), let us denote m L the number of vertices belonging exactly to the list L of simplices.

I sn- 3

 3 (φ sn-3 )I sn-2 (φ sn-2 ), With s j ≥ |m j,1m j,2 | for j ∈ {1, • • • , n -2} if we denote:

I

  sn-3 (φ sn-3 )I sn-2 (φ sn-2 ) = I sn-3 (• j∈A f ij )I sn-2 (• j∈ Āf ij ).

  s1,••• ,sn-3 t1,••• ,tn-3I sn-4 (φ sn-4 )I sn-3 (φ sn-3 ) k in=1 I in (f in ) = k i1,••• ,in=1 s1,••• ,sn-2 t1,••• ,tn-2 I sn-2 (φ sn-2 )I in (f in ), with s j ≥ |m j,1m j,2 | for j ∈ {1, • • • , n -2}using the same notations for n -1 instead of n:

I sn- 2 (

 2 I sn-2 (• j∈{1,••• ,n-1} f ij ),concluding the proof.

2 ⌋

 2 and s 2(j-⌊ n 2 ⌋)-1 otherwise,• m j,2 = i 2j if 1 ≤ j ≤ ⌊ n 2 ⌋ and s 2(j-⌊ n 2 ⌋) otherwise, • m j,3 = s j if 1 ≤ j ≤ n -3 and s n-3 otherwise, • u j = m j,1 + m j,2m j,3 , • If n is even, then c = s n-3 and s n-3 ≥ |m n-2,1m n-2,2 | ∨ |m n-3,1m n-3,2 |, • If n is odd, then c = i n and i n ≥ |m n-2,1m n-2,2 |.

with s n- 3 ≥

 3 |m n-2,1m n-2,2 | ∨ |m n-3,1m n-3,2 |.


Ayad Ghannam
email: aghannam@laas.fr

David Bourrier
email: bourrier@laas.fr

Christophe Viallon
email: ccviallon@laas.fr

Thierry Parra
email: dparra@laas.fr

Low Cost SU8 Based Above IC Process for High-Q RF Power Inductors Integration

Keywords: Above-IC, Copper, Electroplating, High-Q, Inductor, Process, Negative Photoresist, SU8

This paper presents a new process for integration of high-Q RF power inductors above low resistivity silicon substrates. The process uses the SU8 resin as a dielectric layer. The aim of using the SU8 is to form thick dielectric layer that can enhance the performance of the inductors. The flexibility of the process enables the possibility to realize complex shaped planar inductors with various dielectric and metal thicknesses to meet the requirements of the application. Q values of 55 at 5GHz has been demonstrated for an inductance value of 0.8nH using a 60 µm thick SU8 layer and 30 µm thick copper ribbons.

Introduction

Above-IC integration of passive devices is one of the most promising enabling technologies for developing high performance low-cost miniaturized RF components. During the last decade, several achievements were realized in terms of technological developments. However, these technologies are restricted to low or medium power devices because of several limitations such as, mainly, high mechanical strain and low achievable thickness for dielectric films. Thus, these above-IC technologies are not able to meet the requirements for high-Q inductor integration on top of low resistivity silicon substrates, such as high power LDMOS substrate.

In order to overcome these limitations and to fulfill the high-Q inductors requirements for RF power devices, we developed a low cost, SU8 based above-IC process. Up until recently, the SU8 resin was used as a photoresist in microelectronics industry, to provide a high-resolution mask for fabrication of semiconductor devices. In a previous work [START_REF] Ghannam | Dielectric microwave characterization of the SU-8 thick resin used in an above IC process[END_REF], we demonstrated the interest of this resin as a dielectric material for RF applications below 10GHz. In this paper, we will present the technological steps of an above-IC process and prove its effectiveness for RF power applications by designing and integrating High-Q above-IC inductors.

Above-IC process technological steps

In order to integrate high performance passive RF power devices like inductors, it has been advised to use a thick dielectric layer (> 50µm) to decrease magnetic loss into the substrate or the ground shield [START_REF] Ghannam | High-Q SU8 Based Above-IC Inductors for RF Power Devices[END_REF]. After investigating several dielectric photoresists (BCB, Intervia …), the SU8 was chosen for its great technological properties (high thicknesses, low mechanical stress, easy deposit …).

However, using high dielectric thicknesses results into technological challenges since vias must be used to connect the integrated passive device to the underlying ship's access pads. Thus, filling these vias cannot be done using conventional metal evaporation or sputtering techniques because of their limitations (low thicknesses, high price …). Instead, metal electroplating must be used.

Electroplating requires the deposit of a seed layer and the usage of molds to restrict the growth of metal to required areas/locations (Fig. 1b). These molds are formed using a thick photoresist which technological properties (viscosity, opening aspect ratio,...) set via characteristics such as height, shape and minimum sizing. Moreover, the photoresist must be cleaned out at the end of the step in order to be able to etch the seed layer. For this process, we used a negative sacrificial photoresist called the BPN.

The developed above-IC process is composed of here after major steps (Fig. 1):

1. Vias electroplating: This step requires the sputtering of a Ti/Cu seed layer (a) (500Å/1000Å), the deposition of the sacrificial photoresist BPN (b), the electroplating of Cu vias (c), the stripping of the BPN (d) and the etching of the seed layer (e). 2. Dielectric deposit: Deposit of the SU8 (3050) resin using spin-off (f). After this step, mechanical polishing can be necessary to level the surface (g). 3. Ribbons electroplating: This step requires the sputtering of a Ti/Cu seed layer (h) (500Å/1000Å), the deposition of the sacrificial photoresist BPN (g), the electroplating of Cu ribbons (i), the stripping of the BPN (j) and the etching of the seed layer (k).

The deposition of the BPN requires a prebake at 120°C, an UV (365nm) exposure and a development phase. The parameters of these steps depend on the desired thickness of the photoresist [START_REF] Bourrier | Comparison of new thick negative resist to Su8[END_REF].

The deposition of the SU8 requires a prebake at 95°C, an UV (365nm) exposure, a post exposure bake at 95°C, a development phase using PGMEA (Propylene Glycol Methyl Ether Acetate) developer and finally a hard bake (can be omitted). As for the BPN, the parameters of these steps depend on the desired thickness [START_REF]Microchem SU8 datasheet[END_REF].

(a) (e) (i) (b) (f) (j) (c) (g) (k) (d) (h) Substrate Metal 2 Seed layer

Neg. Resin SU8 Resin Copper

Fig. 1: Above-IC technological steps

Above-IC inductors: Experimental result

To demonstrate the effectiveness of the developed process, we have fabricated high-Q inductors (Fig. 2) above a low resistivity silicon substrate (8mΩ.cm). The structures are formed by firstly electroplating a 2µm thick copper Cu solid ground shield, then growing 60µm thick Cu vias followed by spin-coating a 60µm thick SU8 layer, and finally, electroplating 30µm thick Cu ribbons. Vias and ribbons can achieve a height /size aspect ratio of 16/1.

A vector network analyzer (Anritsu 37397C) and a probe station are used for microwave Sparameter measurements up to 20 GHz. A SOLT calibration is performed to locate the reference planes at the probe tips and thru/open/short standards have been implemented and are used to deembed the coplanar tapered probing pads.

Measurements of these inductors exhibit a Q value of 55 at 5 GHz for a total inductance value of 0.8 nH (Fig. 3). The good agreement between measurements and simulations must be pointed out, which validates both our inductor optimization and the technological process control.

Conclusion

In this work, we have detailed an SU8 based above-IC process that enables the integration of high-Q RF power inductors. The presented process uses only few easy technological steps, which make it low cost and suitable for manufacturing. The effectiveness of the process has been demonstrated by integrating high-Q (55@5GHz) power inductors on an 8mΩ.cm high power LDMOS silicon substrate. Its potentialities are quite important as well: since we processed SU8 thicknesses up to 400µm, this technology can fit a large range of applications like on chip antenna integration, packaging ...

Fig. 2 :Fig. 3 :

 23 Fig. 2: SEM micrograph of SU-8 based power inductors

Acknowledgments

This research work was supported by Freescale Semiconductors. The authors wish to thank Jean-Marie BOULAY and Gerard BOUISSE for their help and fruitful discussions.