J M P Albin
email: palbin@math.chalmers.se

A continuous non-Brownian motion martingale with Brownian motion marginal distributions

Keywords: AMS subject classification: Primary: 60G44, 60J65. Secondary: 33-99, 35K57, 60E99, 60G18, 60J60 Brownian motion, diffusion process, martingale, martingale problem

We construct a continuous martingale that has the same univariate marginal distributions as Brownian motion, but that is not Brownian motion.

A c c e p t e d m a n u s c r i p t 1 Introduction

Recently Fima Klebaner brought to our attention that it is an open problem whether there exists a continuous martingale that has the same univariate marginal distributions as standard Brownian motion, but which is not Brownian motion. In this note we make an explicit construction of such a martingale.

Let {X 1 (t)} t≥0 and {X 2 (t)} t≥0 be independent stochastic processes that are weak solutions to the time homogeneous diffusion type stochastic differential equation

dX(t) = dB(t) 2 X(t) , X(0) = 0, ( 1
)
where B is standard Brownian motion. Let Y be a random variable that is independent of X 1 and X 2 , and that has probability density function

f Y (y) = 4 (Γ( 3 4 )) 2 2 π 3/2 1 -(y/ √ 2) 4
for y ∈ (0, √ 2).

(

) 2
Our main result is the following affirmative answer to the above mentioned problem.

Theorem 1. The process M (t) = X 1 (t)X 2 (t)Y , t ≥ 0, is a continuous martingale such that M (t) = d B(t)

Proof of Theorem 1

As the diffusion coefficient σ(y) = 1/(2 y) of ( 1) is non-zero with

ε -ε dy σ(x + y) 2 = 4 ((x+ε) 3 -(x-ε) 3 ) 3 < ∞ for x ∈ R and ε > 0,
the Engelbert-Schmidt theory shows that (1) has a unique weak solution X (see e.g, Karatzas and Shreve, 1991, Theorem 5.5.7). To find the transition density

p t (x, y) = P X(t+s) ∈ (y, y+dy) X(s) = x dy
of X, we consider the transformed process Y = X 4 , which by Itô's formula satisfies

dY (t) = 2 Y (t) dB(t) + 3 2 dt, Y (0) = 0. ( 3
)
The process Y is usually called the CIR process, after [START_REF] Cox | A theory of the term structure of interest rates[END_REF], but was considered already by [START_REF] Feller | Two singular diffusion problems[END_REF]. According to these authors, Y has transition density

P Y (t+s) ∈ (y, y+dy) Y (s) = x dy =          x 1/8 2 t y 1/8 exp - x + y 2 t I -1/4 √ xy t for x > 0, 1 (2 t) 3/4 y 1/4 Γ( 3 4 ) exp - y 2 t for x = 0,
where I is the modified Bessel function of the first kind [recall that I -1/4 (x) ∼ (2/x) . By the lower of these formulas, together with symmetry of the solution to (1), we get

p t (0, y) = 2 1/4 y 2 Γ( 3 4 ) t 3/4 exp - y 4 2 t for y ∈ R and t > 0. (4)
The solution X to (1) is a continuous local martingale (see e.g, Karatzas and Shreve, 1991, Proposition 3.2.24). Therefore X is a continuous martingale if

E sup t∈[0,T ] |X(t)| n = E sup t∈[0,T ] Y (t) n/4 < ∞ for T > 0, (5)
for n = 1 (see e.g, Karatzas and Shreve, 1991, Theorem 3.3.28). Since

I -1/4 (x) ∼ e x / √ 2πx
Y (t) n/4 > u ∼ 2 P Y (T ) > u 4/n = 2 ∞ u 4/n 1 (2T ) 3/4 y 1/4 Γ( 3 4 ) exp - y 2T dy
as u → ∞. From this it is obvious that (5) holds for n ≥ 0, so that X is a martingale.

As X 1 and X 2 are martingales it is an elementary fact that they are martingales with resepect to the filtrations generated by themselves F 1 t = σ(X 1 (s) : 0 ≤ s ≤ t) and F 2 t = σ(X 2 (s) : 0 ≤ s ≤ t), respectively. Using the independence of X 1 , X 2 and Y , it is an elementary exercise to see that X 1 X 2 and M (t) = X 1 (t)X 2 (t)Y are martingales with resepect to the filtrations

{F 1 t ∨ F 2 t } t≥0 and {F 1 t ∨ F 2 t ∨ σ(Y )} t≥0
, respectively [where (5) together with elementary arguments ensure sufficient integrability properties]. Now, if M is Brownian motion, then M has quadratic variation process M t = t, so that [START_REF] Bibby | Diffusion-type models with given marginal distribution and autocorrelation function[END_REF]. This is impossible as

X 1 X 2 t = t/Y 2 , giving E{ X 1 X 2 t } = t E{1/Y 2 } = ∞, by
E X 1 X 2 t ≤ K E sup s∈[0,t] |X 1 (s)| 2 |X 2 (s)| 2 ≤ K E sup s∈[0,t] |X 1 (s)| 4 < ∞
for some constant K > 0, by the Burkholder-Davis-Gundy and Cauchy-Schwarz inequalities (see e.g, Karatzas and Shreve, 1991, Proposition 3.3.28), together with [START_REF] Carr | A note on sufficient conditions for no arbitrage[END_REF].

It remains to show that M (t) = d B(t) for t ≥ 0: As p t (0, y) = p 1 (0, y/t 1/4 )/t 1/4 by (4), we have 1), it follows that it is enough to prove that X 1 (1)X 2 (1)Y = d B [START_REF] Albin | Extremes of diffusions over fixed intervals[END_REF]. By symmetry, this follows if the Mellin transform for s ≥ 1.

X 1 (t) = d t 1/4 X 1 (1). As B(t) = d t 1/2 B(
E (|X 1 (1)||X 2 (1)| Y ) s-1 = (E{|X 1 (1)| s-1 }) 2 E{Y s-1 } of |X 1 (1)||X 2 (1)| Y agrees with E |B(1)| s-1 = ∞ 0 y s-1 2 √ 2π exp - y 2 2 dy = 2 s/2 Γ(s/2) √ 2π for s ≥ 1. (6)
By [START_REF] Cox | A theory of the term structure of interest rates[END_REF] it is therefore enough to prove that Y has Mellin transform

2 s/2 Γ(s/2) √ 2π 2 s/4 Γ((s+2)/4) 2 1/4 Γ( 3 4 ) 2 = (Γ( 3 4 )) 2 Γ(s/2) √ π (Γ((s+2)/4)) 2 = Γ( 3 4 )) 2 2 s/2 Γ(s/4) 2π Γ((s+2)/4) , ( 7
)
where the second equality follows from the Gauss-Legendre multiplication formula , so that M is a Gaussian process with stationary and independent increments, thereby contradicting the assumption that M is not Brownian motion.

Γ(s/4)Γ((s+2)/4) = 2 √ π 2 -s/2
A non-Brownian motion martingale M with Brownian motion marginal distributions is not a Gaussian process, because M has uncorrelated increments (being a martingale), that are independent if M is Gaussian, which is impossible by the previous paragraph.

Our construction of a martingale M (t) = X 1 (t)X 2 (t)Y with Brownian motion marginal distributions crucially relied on the self-similarity p t (0, y) = p 1 (0, y/t H )/t H for some H ∈ (0, 1 2 ], which made it enough to prove that X 1 (1)X 2 (1)Y = d B [START_REF] Albin | Extremes of diffusions over fixed intervals[END_REF]. Using this selfsimilarity together with the Kolmogorov forward equation

∂ 2 ∂y 2 σ(y) 2 2 p t (x, y) - ∂ ∂t p t (x, y) = ∂ 2 ∂y 2 1 8 y 2 p t (x, y) - ∂ ∂t p t (x, y) = 0,
elementary calculations and the change of variable z = y/t H give

∂ 2 ∂z 2 σ(t H z) 2 2 t 2H-1 p 1 (0, z) + ∂ ∂z H z p 1 (0, z) = 0. (8)
This makes necessary that σ(z) 2 /z 2-1/H is a constant (where in our case H = 1 4 ). With such a choice σ(z) 2 = 1 4 z 2-1/H of σ, [START_REF] Erdélyi | Higher Transcendental Functions[END_REF] is the equation for the stationary density of a diffusion with this diffusion coefficient and drift coefficient µ(z) = -H z, so that

p 1 (0, z) = 2 C σ(z) 2 exp z 0 2µ(y) σ(y) 2 dy = 8C z 1/H-2 exp -8H 2 z 1/H for z ∈ R,
for a suitable normalizing constant C > 0 (see e.g, Karatzas and Shreve, 1991, Section 5.5), which gives (4) using that p t (0, y) = p 1 (0, y/t 1/4 )/t 1/4 .

As the self-similarity imposed in the previous paragraph severely limits the possible diffusion coefficients σ of diffusions X that can feature in our construction of M , one might look for non-diffusion type self-similar martingales. In order for such a martingale X to feature in our construction it has to have tail probabililities at least as light as Gaussian tails, as X(1)Z cannot have lighter tails than X(1), except for a scaling factor, when Z is a non-degenerate random variable independent of X. Examples of

A c c e p t e d m a n u s c r i p t 3

 3 Further remarks A non-Brownian motion martingale M with Brownian motion marginal distributions does not have independent increments, because if M (t+s) -M (s) and M (s) = d B(s) are independent with sum M (t+s) = d B(t+s), then M (t+s)-M (s) must be Gaussian by Cramer's theorem (see e.g, Bondesson, 1995), which implies that M (t+s)-M (s) = d B(t+s)-B(s) = d B(t)

A c c e p t e d m a n u s c r i p t 1 - 2

 12 non-diffusion type light-tailed self-similar martingales includes, for example, X(t) = t 0 g B(s)/ √ s dB(t) for a bounded function g.However, for this X it seems near impossible to find the distribution of X(1), to check whether there exists a Z such that X(1)Z = d B(1). With the above notation, X(1)Z = d B(1) holds if and only if ln(|B(1)|) = d ln(|X (1)|) + ln(|Z|). This is the much researched factorization problem for probability distributions (see e.g, Bondesson, 1995), for which it is well-known that non-trivial factorizations do not always exist, especially not when one of the factors X(1) is given. The family of non-continuous martingales with Brownian motion marginal distributions constructed by Hamza and Klebaner (2006b) is rather non-explicit. However, Olle Häggström has communicated a simple explicit example of such a martingale to us: Let M (t) = √ t B(1)(-1) N (t) with the filtrationF t = σ(B(1)) ∨ σ(N (s) : s ≤ t),where {N (t)} t>0 is a non-homogeneous Poisson process independent of B with intensityλ(r) = 1/(4 r).As M is not continuous, but has the univariate marginal distributions of Brownian motion, we only have to show that M is a martingale. AsE{M (t)|F s } = √ t ξ (-1) N (s) E (-1) N (t)-N (s) = M (s) t s P{N (t)-N(s) odd} , this follows from the fact that (by basic properties of Poisson processes) P{N (t) -N (s) odd} = Bondesson, L. (1995), Factorization theory for probability distributions, Scand.

A c c e p t e d m a n u s c r i p t that

  it remains an open problem whether there exists a non-Brownian motion continuous martingale with Brownian motion marginals.

	Based on the fundamental work of Kellerer (1972), who gave necessary and suffi-
	cient conditions for the existence of Markovian martingales with prescribed marginal
	distributions, Madan and Yor (2002) gave three constructive solutions to that problem.
	Hamza and Klebaner (2006b) noted that, in the case of Brownian motion marginals, two
	of these solutions reduced to Brownian motion itself, while the third resulted in a non-
	continuous process. Hamza and Klebaner proceeded to construct a whole family of non-
	continuous martingales with Brownian motion marginal distributions. They also noted

for t ≥ 0, where = d denotes equality in distribution, but which is not Brownian motion. The problem to construct martingales with prescribed univariate marginal distributions has recived attention recently, see e.g, Bibby et al. (2005), Campi (2004), Carr and Madan (2005), Hamza and Klebaner (2006a, 2006b), and Madan and Yor (2002).

In part, this is due to the significance the problem has in modelling in mathematical finance. In particular, martingales with Brownian motion marginals are solutions to the Bachelier model (see e.g,

[START_REF] Schachermayer | How close are the option pricing formulas of Bachelier and Black-Merton-Scholes?[END_REF]

, on this model).

Acknowledgements

We are grateful to Fima Klebaner for telling us about the problem, to Olle Häggström for the above mentioned communication, and to Mattias Sundén for a valuable suggestion.

 [START_REF] Albin | Extremes of diffusions over fixed intervals[END_REF]Research supported by The Swedish Research Council contract no. 621-2003-5214.


Jan-Frederik Mai
email: mai@tum.de

Matthias Scherer
email: scherer@tum.de

The Pickands representation of survival Marshall-Olkin copulas

Keywords: Extreme-value copula, Marshall-Olkin copula, Pickands representation AMS classification: 62H20, 62H05

The Pickands representation of an arbitrary survival Marshall-Olkin copula is computed. In dimension d ≥ 2, the corresponding dependence measure is discrete with support consisting of 2 d -1 atoms on the d-dimensional unit simplex.

Introduction

The d-dimensional Marshall-Olkin distribution is constructed as follows. On a probability space (Ω, F , P) let {E I | ∅ = I ⊂ {1, . . . , d}} be a collection of 2 d -1 independent exponential random variables with means E[E I ] = 1/λ I > 0, respectively. The random vector (τ 1 , . . . , τ d ), defined via τ k := min I : k∈I, I⊂{1,...,d}

{E I }, k = 1, . . . , d,
is said to exhibit the Marshall-Olkin distribution with parameters λ I , ∅ = I ⊂ {1, . . . , d}. [Marshall, Olkin (1967)] introduce this multivariate distribution and emphasize that it can be considered as a multivariate extension of the univariate exponential distribution, if one characterizes the latter in terms of the so-called lack of memory-property, see also [Galambos, Kotz (1978), p. 103-132] for a well-written exposé on this topic. By the so-called minstability of the exponential distribution, it follows that the random variable τ k is exponentially distributed with parameter Λ k := I:k∈I, I⊂{1,...,d} λ I , for k = 1, . . . , d. Moreover, the dependence structure of (τ 1 , . . . , τ d ) can be studied conveniently by means of its survival copula Ĉ. It is computed explicitly in [Li (2008), Proposition 1]. However, for the purpose of this article an alternative form is more appropriate, which is established as follows. The random vectors (τ 1 , . . . , τ d ) and (Λ 1 τ 1 , . . . , Λ d τ d ) have the same survival copulas. The latter follows a min-stable multivariate exponential distribution with standard exponential margins. It is derived from [Li (2009), Equation (4.3)] that the survival function of (Λ 1 τ 1 , . . . , Λ d τ d ) is given by

P(Λ 1 τ 1 > t 1 , . . . , Λ d τ d > t d ) = exp - I : ∅ =I⊂{1,...,d} λ I max z∈I {Λ -1 z t z } , (1)
for t 1 , . . . , t d ≥ 0. The survival copula of (Λ 1 τ 1 , . . . , Λ d τ d ), and hence of (τ 1 , . . . , τ d ), is obtained from (1) by setting

t i = -log u i , u i ∈ [0, 1], for i = 1, . . . , d. It is hence given by Ĉ(u 1 , . . . , u d ) = I : ∅ =I⊂{1,...,d} min z∈I u λ I /Λz z , u 1 , . . . , u d ∈ [0, 1]. (2)
It is straightforward to verify that Ĉ is an extreme-value copula, i.e. for all t > 0 it holds that Ĉ(u t 1 , . . . , u t d ) = Ĉ(u 1 , . . . , u d ) t . [START_REF] Pickands | Multivariate extreme value distributions[END_REF]] proves that each extreme-value copula is uniquely characterized by a finite measure δ on the d-dimensional unit simplex

S d = {(w 1 , . . . , w d ) ∈ [0, 1] d | w 1 +. . .+w d = 1}, which satisfies the boundary conditions S d u k dδ(u 1 , . . . , u d ) = 1, k = 1, . . . , d.
(3)

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It is shown that a copula C is of extreme-value kind if and only if there exists a measure δ on S d satisfying (3) such that

C(u 1 , . . . , u d ) = d i=1 u i P log u 1 log d i=1 u i ,..., log u d log d i=1 u i
, where the so-called Pickands dependence function P is defined on S d by

P (w 1 , . . . , w d ) = S d max{w 1 u 1 , . . . , w d u d } dδ(u 1 , . . . , u d ).
The measure δ is called Pickands dependence measure. Given the Pickands dependence function of an extreme-value copula, it is possible to conveniently compute coefficients of tail dependencies, see e.g. [Li (2008), Li (2009)] for computations of tail dependence coefficients in the Marshall-Olkin case and formulas for general extreme-value copulas. In order to study the dependence at sub-extreme levels, which is of interest in the field of extreme-value analysis, it is not enough to know the Pickands dependence function P . For this application, the associated measure δ is required.

A combination of the results in [Mai, Scherer (2009a), Mai, Scherer (2009b)] yields the Pickands dependence measure of a survival Marshall-Olkin copula in the special case when the d sets {λ I | |I| = k}, k = 1, . . . , d, are singletons, i.e. the distribution of (τ 1 , . . . , τ d ) is exchangeable. Even though the Marshall-Olkin distribution is widely used in the academic literature, we could not find a reference which states the Pickands dependence measure δ in the general case. Theorem 3.1 fills this gap and provides δ for an arbitrary survival Marshall-Olkin copula, given as in (2).

The Bivariate Case

The bivariate case is considered first, since it is easily verified and illustrates the d-dimensional result. With parameters λ 1 , λ 2 , λ 1,2 > 0 the bivariate survival Marshall-Olkin copula is given by

Ĉ(u 1 , u 2 ) = min u λ 1 λ 1 +λ 1,2 1 • u 2 , u 1 • u λ 2 λ 2 +λ 1,2 2 , u 1 , u 2 ∈ [0, 1]. Lemma 2.1 (Pickands Representation in dimension d = 2)
The Pickands measure δ on the two-dimensional unit simplex S 2 is a discrete

ACCEPTED MANUSCRIPT

measure whose support consists of three points. More precisely, it is given by

δ (1, 0) = λ 1 λ 1 + λ 1,2 , δ (0, 1) = λ 2 λ 2 + λ 1,2 , δ   λ 2 + λ 1,2 λ 1 + λ 2 + 2 λ 1,2 , λ 1 + λ 1,2 λ 1 + λ 2 + 2 λ 1,2   = λ 1,2 λ 1 + λ 1,2 + λ 1,2 λ 2 + λ 1,2
.

The corresponding Pickands dependence function is given by

P (w 1 , w 2 ) = max w 1 + w 2 λ 2 λ 2 + λ 1,2 , w 2 + w 1 λ 1 λ 1 + λ 1,2
.

Proof

For notational simplicity, apply the reparameterization

α := λ 1 λ 1 + λ 1,2 ∈ (0, 1), β := λ 2 λ 2 + λ 1,2 ∈ (0, 1).
It is easily verified that the measure δ defines a Pickands dependence measure, i.e. the boundary conditions (3) hold, since

S 2 u 1 dδ(u 1 , u 2 ) = α + (2 -α -β) 1 -α 2 -α -β = 1, S 2 u 2 dδ(u 1 , u 2 ) = β + (2 -α -β) 1 -β 2 -α -β = 1.
The Pickands dependence function is then computed as

P (w 1 , w 2 ) = S 2 max{w 1 u 1 , w 2 u 2 } dδ(u 1 , u 2 ) = α w 1 + β w 2 + (2 -α -β) max 1 -α 2 -α -β w 1 , 1 -β 2 -α -β w 2 = max w 1 + w 2 β, w 2 + α w 1 .
Finally, one may compute that this dependence function corresponds to the claimed Marshall-Olkin survival copula, i.e.

(u 1 • u 2 )

P log u 1 log u 1 +log u 2 , log u 2 log u 1 +log u 2 = min u α 1 • u 2 , u 1 • u β 2 . 2
3 The General Case

  Theorem 3.1 (Pickands Representation in dimension d ≥ 2) The Pickands dependence measure δ of a d-dimensional survival Marshall-Olkin copula with parameters λ I > 0, ∅ = I ⊂ {1, . . . , d}, is a discrete measure ACCEPTED MANUSCRIPT on S d with 2 d -1 atoms. More precisely, it is determined by δ = I ⊂ {1, . . . , d}, where e 1 , . . . , e d denotes the canonical basis of R d . The corresponding Pickands dependence function is given by P (w 1 , . . . , w d ) = I : ∅ =I⊂{1,...,d} λ I max j∈I w j Λ j . Proof First observe that for I ⊂ {1, . . . , d} fixed, with j ∈ I, it holds that z∈I With δ as given in the assertion, one computes using (4) in the last equality P (w 1 , . . . , w d ) = S d max{w 1 u 1 , . . . , w d u d } dδ(u 1 , . . . , u d ) = I : ∅ =I⊂{1,...,d} claimed Pickands dependence function is established. Moreover, it is straightforward to verify that P ( e k ) = 1 for all k = 1, . . . , d, which is equivalent to (3). Thus, P defines a Pickands dependence function, i.e. δ is a Pickands dependence measure. Next, one computes with u 1 , . . . , u d ∈ (0, 1) that P log u 1 log d i=1 u i , . . . , log u d log d i=1 u i = 1 d i=1 log u i I : ∅ =I⊂{1,...,d} λ I min z∈I log u z Λ z ,

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and one accordingly verifies that This establishes the claim. 2


Cláudia Neves

António Pereira

Detecting finiteness in the right endpoint of light tailed distributions

Keywords: Extreme Value Theory, right endpoint, test of hypothesis

In Extreme Value statistics we often encounter testing procedures for assessing the presence of the Gumbel domain, attached to the simple null hypothesis of shape parameter γ = 0, thus praising the selection of extreme domains of attraction. However, the problem of assessing for light tailed distributions with finite or infinite right endpoint is seldom referred. The latter is an impending problem of practical importance, particularly at the enrollment of subsequent estimation of extremal features such as small exceedance probabilities. In this paper, we present two test statistics whose asymptotic behavior, albeit under some restrictive yet reasonable conditions, enables to distinguish light tailed distribution functions with finite right endpoint from those with infinite endpoint lying in the Gumbel domain. An illustrative example is provided via application to significant wave height data recorded at Figueira da Foz, Portugal, from 1958 until 2001.

Introduction

In statistical analysis of rare events, the Generalized Extreme Value distribution stems from the fundamental Fisher and Tippett theorem [START_REF] Fisher | Limiting forms of the frequency distribution of the largest and smallest member of a sample[END_REF] as a unified version of the only three possible limits for the distribution of the maximum from a random sample with parent distribution function F , provided suitable normalization in scale and location, for a large enough sample size n. That is, assume there exist normalizing constants a n > 0 and b n ∈ R such that

lim n→∞ P a -1 n max(X 1 , . . . , X n ) -b n ≤ x = lim n→∞ F n (a n x + b n ) = G(x), (1)
for all x, with some non-degenerate distribution function G. Then, G must be one of only three possible extreme value distributions -Fréchet, Gumbel or Weibull -while these, in turn, can be nested as a one parameter family of distributions [START_REF] Von Mises | La distribution de la plus grande de n valeurs[END_REF]): the Generalized Extreme Value distribution,

G γ (x) :=    exp(-(1 + γx) -1/γ ) , 1 + γx > 0 if γ = 0 exp(-exp(-x)) , x ∈ R if γ = 0. (2)
We say that the distribution function F of the independent random variables X 1 , X 2 , . . . , X n , . . . is of particular interest in many applied sciences where extremes come into play, not only because of the simplicity of inference pertaining to γ = 0, but also for the great variety of distributions possessing an exponential tail, ranging from light-tailed distributions whether having finite or infinite right endpoint like the Normal and Gamma distributions, to moderately heavy tails such as the Lognormal distribution. In the context of extreme value analysis, where we wish to extrapolate beyond the sampled data, a screening process for the most likely underlying tail type may thus be considered an utter advantage.

Statistical procedures on the selection of extreme domains are extensively used in order to have a grasp of the degree of tail heaviness. For instance, [START_REF] Neves | Testing extreme value conditions -an overview and recent approaches[END_REF] and [START_REF] Hüsler | Review of testing issues in extremes: in honor of Professor Laurens de Haan[END_REF] offer a catalog of numerous procedures for assessing extreme value conditions, the first focusing different approaches within the univariate case and the second going through univariate and multivariate settings.

Less attention has yet been paid to the problem of assessing the presence of a distribution function F with finite endpoint. Nevertheless, this is a striking problem of practical importance, for instance, when we wish to perform transformations to the sample observations that might hinder tail heaviness.

In this paper we confine our attention to the Weibull and Gumbel domains, that is, we are addressing distribution functions in D(G γ ) γ≤0 . Our proposal is to devise statistical tools, at the hypothesis testing level, with the ultimate aim of distinguishing between light-tailed distributions with infinite and finite right endpoint. The latter not only embraces the problem of statistical choice between Weibull and log-Normal distributions as recently treated by [START_REF] Kim | Selection between Weibull and lognormal distributions: a comparative simulation study[END_REF] although under a simulation perspective, but also accounts for ramifications on the possibility of applying data transforms, particularly in the case of an underlying distribution F with infinite right endpoint x F . A noteworthy example corresponds to changing data from the normal parent through an exponential function, which translates as a leap from an underlying light-tailed (normal) distribution to the moderately heavy lognormal parent.

Under a semi-parametric approach, we are only interested in the shape of the underlying distribution at high quantiles, meaning that the extreme value index γ is the only parameter to keep under scrutiny.

The following extended regular variation property [START_REF] De Haan | Slow variation and characterization of domains of attraction[END_REF]) is a well known necessary and sufficient condition for F to belong to an extreme domain of attraction, i.e., F ∈ D(G γ ) if and only if

lim t→∞ U (tx) -U (t) a(t) =    x γ -1 γ , γ = 0 log x, γ = 0, (3)
for every x > 0 and some positive measurable function a, with U standing for a quantile type function pertaining to F defined by the generalized inverse

U (t) := 1 1 -F ← (t) = inf x : F (x) ≥ 1 - 1 t .
Observe that the limit function (x γ -1)/γ is the tail quantile function of the Generalized Pareto distribution (GPd),

F γ (x) := 1 + log G γ (x) = 1 -(1 + γx) -1 γ for    x ≥ 0 if γ ≥ 0 0 ≤ x ≤ -1 γ if γ < 0. (4)
This fact reflects its exceptional role in extreme value theory: while restricting attention to a top portion of the original sample, the GPd stems from the seminal works of [START_REF] Balkema | Residual life time at great age[END_REF] and [START_REF] Pickands | Statistical inference using extreme order statistics[END_REF] as the limiting distribution for the excesses W i = X i -u|X i > u, i = 1, . . . , k u over a sufficiently high threshold u. The GPd will be taken into account in Section 3 for the performance evaluation, via simulation, of the new test statistics proposed in Section 2, whereas the exponential distribution function F γ=0 will be taken as a prototype of the subclass comprising light-tailed distributions with infinite right endpoint. The remainder of the paper regards an example of application in Section 4.

Two tests

Let X 1 , X 2 , . . . , X n be independent random variables with the same distribution function F and let X 1,n ≤ X 2,n ≤ . . . ≤ X n,n denote their ascending order statistics. On account of simplicity, we assume throughout that the right endpoint

x F := U (∞) = lim t→∞ U (t) is in (0, ∞].
In order to deal with the problem of casting out distributions with finite right endpoint lying in the Gumbel domain and therefore attached with γ = 0, we shall take k upper order statistics in a way that these might present a satisfactory picture of the tail of 1 -F . Hence, let k = k(n) be an intermediate sequence such that k → ∞ and k/n → 0 as n → ∞ while X n-k,n thus refers to the k-th intermediate level which constitutes the designated eventually high (random) threshold (see text below the GPd function ( 4)).

In this context, the main concern goes to casting out the presence of light-tailed distributions with infinite right endpoint x F := U (∞), in the sense that we are particularly interested in testing the null hypothesis H 0 : F ∈ D(G 0 ) and x F = ∞ against the alternative H 1 : F ∈ D(G γ ) γ≤ 0 and x F < ∞ . To tackle this testing problem, we introduce two test statistics:

T (1) k,n := 1 k k i=1 X n-i,n -X n-k,n -â(n/k) X n,n -X n-k,n (5)
T (2) k,n := 1 k k-1 i=1 i X n-i+1,n -X n-i,n X n-k,n , (6)
where â is a suitable estimator for the scale function a introduced in (3). These two tests are bound to be used as complement of each other under the general guidelines provided in Section 3 regarding finite samples. Convenient normalized versions of T

(1) k,n and T

(2) k,n are provided in the main Theorem below which expounds the asymptotic normality of the referred test statistics under suitable yet reasonable conditions involving the following second order behavior of U .

Apart from the first order condition (3), we shall need a second order condition, specifying the inherent rate of convergence. Hence, we shall assume the existence of a function A, not changing in sign and tending to zero as t → ∞ such that

lim t→∞ U (tx)-U (t) a(t) -x γ -1 γ A(t) = H γ,ρ (x) := 1 ρ x γ+ρ -1 γ + ρ - x γ -1 γ , (7)
for all x > 0, where ρ ≤ 0 is the second order parameter governing the speed of convergence in (3). We then say that the function U is of second order extended regular variation (notation: U ∈ 2ERV (γ, ρ)). In case ρ = 0 and/or γ + ρ = 0, note that the function (x a -1)/a defined for all x > 0, a ∈ R reads as log x for a = 0. We also remark that lim t→∞ A(tx)/A(t) = x ρ , for every x > 0, i.e., |A| is of regular variation at infinity with index ρ (cf. de [START_REF] De Haan | Generalized regular variation of second order[END_REF]), hence the notation |A| ∈ RV ρ .

Theorem 1 Let X 1 , X 2 , . . . be i.i.d. random variables with the same distribution function

F . Let x F = ∞ and F ∈ D(G 0 ).
1. Furthermore, assume the second order condition (7

) holds. If k = k(n) is an intermediate sequence, i.e., k → ∞ and k = o(n), as n → ∞, such that √ kA(n/k) → 0, as n → ∞, and
√ k 1 k k i=1 X n-i,n -X n-k,n a(n/k) -1, â(n/k) a(n/k) -1 d -→ (B 1 , B 2 ), (8)
A C C E P T E D M A N U S C R I P T

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with (B 1 , B 2 ) bivariate normal with zero mean vector and covariance matrix not depending on ρ, then

√ k log k T (1) k,n d -→ B 1 + B 2 . (9) 2. For any intermediate sequence k = k(n), √ k log(n/k) T (2) k,n - √ k d -→ N (0, 1), (10)
Before proceeding with the proof we call on Lemma 2 below.

Lemma 2 (Theorem B.2.2 and Corollary B.2.13 of de Haan and Ferreira (2006))

Suppose the function U is such that (3) holds with some γ ≤ 0. Then, the auxiliary function a is in RV γ and

(i) if γ = 0 and U (∞) = ∞, then U is of slow variation in the sense that U ∈ RV 0 , i.e., lim t→∞ U (tx)/U (t) =
1, for every x > 0, and a(t) = o U (t) , as t → ∞.

(ii) If γ = 0 and U (∞) < ∞, then lim t→∞ a(t)/ U (∞) -U (t) = 0 and U (∞) -U (t) ∈ RV 0 . (iii) If γ < 0, then U (∞) < ∞, lim t→∞ a(t)/ U (∞) -U (t) = -γ and U (∞) -U (t) ∈ RV γ .

Proof of Theorem 1:

We start by noting that given a random variable X with distribution function F and pertaining tail quantile function U , we have the equality in distribution

X d = U (Y )
with Y a standard Pareto random variable with distribution function 1 -(y ∨ 1) -1 . Naturally, if we consider a sample {X i } n i=1 of independent and identically distributed random variables from the same parent F and let X 1,n ≤ X 2,n ≤ . . . ≤ X n,n be their ascending order statistics, we get the equality in distribution

X n-i,n n-1 i=0 d = U (Y n-i,n ) n-1 i=0 . (11)
In order to properly apply (7) and subsequent uniform inequalities in the sense of [START_REF] Drees | On smooth statistical tail functionals[END_REF]

a(n/k) X n,n -X n-k,n d = a(Y n-k,n ) U (Y n,n ) -U (Y n-k,n ) a(n/k) a(Y n-k,n ) . ( 12
)
Since for γ = 0 condition (3) rephrases as

a(t) U (tx) -U (t) = 1 log x - 1 log 2 x U (tx) -U (t) a(t) -log x 1 + o(1) , (t → ∞) A C C E P T E D M A N U S C R I P T

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all x > 0, eq. ( 12) leads to

a(n/k) X n,n -X n-k,n d = a(Y n-k,n ) U Y n,n Y n-k,n Y n-k,n -U (Y n-k,n ) a(n/k) a(Y n-k,n ) = 1 log Y n,n -log Y n-k,n -A(Y n-k,n ) H 0,ρ Y n,n Y n-k,n (log Y n,n -log Y n-k,n ) 2 1 + o p (1) a(n/k) a(Y n-k,n ) , (13)
as

n → ∞. Define Q k = (log Y k,k -log k)/ log k.
Owing to Rényi's representation (cf. Lemma 3.2.3 from de [START_REF] De Haan | Extreme Value Theory: An Introduction[END_REF], we see that with any measurable measurable function g,

g Y n-i+1,n Y n-k,n k i=1 d = g(Y k-i+1,k ) k i=1 , ( 14
)
where Y 1 , Y 2 , . . . , Y n are independent standard Pareto random variables with distribution function 1 -

(y ∨ 1) -1 . Furthermore, lim k→∞ Q k = 0 a.s.
(see page 194 of de [START_REF] De Haan | Extreme Value Theory: An Introduction[END_REF] and this takes care of the first term in the normalized version of T

(1) k,n written as follows:

√ k log k T (1) k,n = log k â(n/k) X n,n -X n-k,n 1 0 √ k X n-[ks],n -X n-k,n â(n/k) -log s ds = log k a(n/k) X n,n -X n-k,n a 0 (n/k) a(n/k) 1 0 √ k X n-[ks],n -X n-k,n a 0 (n/k) -log s ds - √ k â(n/k) a 0 (n/k) -1 1 0 log s ds =: I × (II -III).
Here [ks] denotes the highest integer not greater than ks. The new auxiliary function a 0 is such that a 0 (t) ∼ a(t), as t → ∞, and a 0 (t)/a(t) -1 = o A(t) . It now intervenes because we wish to apply Corollary 2.4.6 of de [START_REF] De Haan | Extreme Value Theory: An Introduction[END_REF] regarding the empirical tail quantile process. From ( 13) and ( 14) we have seen so far that

I d = log k a(Y n-k,n ) U (Y n,n ) -U (Y n-k,n ) a 0 (n/k) a(Y n-k,n ) = 1 Q k + 1 - A(n/k) log k H 0,ρ (Y k,k ) Q 2 k + 2Q k + log k a 0 (n/k) a(n/k) 1 + o p (1) . ( 15
)
Hence I P -→ 1, as n → ∞. Note that A(t) tends to zero as t goes to infinity even if ρ = 0. In (15) the random quantities a(Y n-k,n ) and A(Y n-k,n ) were replaced by their deterministic counterparts, sequences a(n/k) and A(n/k) respectively. This is sustained by the fact that both a and |A| are of regular variation

leading thereby to a(Y n-k,n )/a(n/k) P -→ 1 and A(Y n-k,n )/A(n/k) P -→ 1 as n → ∞.
Finally we have from

II -III = 1 0 W n (s) s -W n (1) ds -α(W n ) 1 0 log s ds + √ kA n k 1 0 ρ H 0,ρ 1 s -H 0,ρ (1) ds + o p (1) 1 0 s -1/2-ε ds,
where {W n (s)} s>0 is a sequence of standard Brownian motions and α denotes a measurable functional (not depending on s) of this same sequence. Hence result in (9).

The second part of the Theorem is supported on the extended regular variation of U for γ = 0 encapsulated in (3). Define A(t) := a(t)/U (t). Using similar arguments as in the proof of 1., by virtue of the uniform inequalities in the sense of Drees (1998) (see also Proposition B.1.10 in de [START_REF] De Haan | Extreme Value Theory: An Introduction[END_REF] together with (i) of Lemma 2, we get

T (2) k,n d = 1 k k-1 i=1 i U (Y n-i+1,n ) U (Y n-k,n ) -1 - U (Y n-i,n ) U (Y n-k,n ) -1 = A k n 1 k k-1 i=1 i log Y k-i+1,k -log Y k-i,k + o p A k n . (16)
Now, Theorem B.3.6 of de [START_REF] De Haan | Extreme Value Theory: An Introduction[END_REF] ensures A(t) ∼ (log t) -1 , as t → ∞, since the (slow) regular variation of second order we get for γ ≤ 0 amounts to the extended regular variation statement in (3) (cf. Theorem 1 in [START_REF] Neves | From extended regular variation to regular variation with application in extreme value statistics[END_REF]. Moreover, Rényi's representation theorem entails that Shorack and Wellner, 1986, p.336). Hence,

k log(Y 1,k ) d = log(Y 1 ), independent of i log(Y i+1,k ) -log(Y i,k ) d = log(Y k-i+1 ), i ≤ k -1 (see e.g.
√ k log(n/k) T (2) k,n = √ k 1 k k i=1 log Y i -1 - √ k 1 k log Y 1 -1 1 + o p (1) .
The result thus follows from the Central Limit theorem. J Remark 3 Condition (8) holds for suitable estimators of a(n/k). We shall use the estimator â(n/k) related with the Moment estimator for the extreme value index γ. In this respect we refer to [START_REF] Dekkers | A moment estimator for the index of an extremevalue distribution[END_REF] and Section 2.1 of de [START_REF] De Haan | Extreme Value Theory: An Introduction[END_REF]. We define

â n k := X n-k,n M (1) k,n 2 1 - M (1) k,n 2 M (2) k,n -1 , ( 17
)
pertaining to the moment statistics

M (j) k,n := 1 k k i=1 log X n-i+1,n -log X n-k,n j , j = 1, 2. (18)
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Under the conditions in Theorem 1, it is not difficult to see that with â(n/k) defined above we have

T * 1 := √ k log k T (1) k,n d -→ N (0, 1).
Consistency of the tests is ensured by similar arguments to the ones in the proof of Theorem 1 coupled with Lemma 2. In particular, if x F < ∞ and under the conditions of Theorem 1, T

(1) k,n P -→ ∞ if γ = 0, and

T (1) k,n P -→ -∞ if γ < 0. With respect to T (2)
k,n , we obtain the following representation, valid for γ ≤ 0, ρ < 0 and x F = U (∞) < ∞, and akin to (16):

T (2) k,n = A n k 1 k k i=1 (Y i ) γ -1 γ - e γ log(Y 1 )/k -1 γ 1 + o p (1) = A n k I -II 1 + o p (1) . ( 19
)
The Central Limit theorem ascertains k,n shifts towards left for light-tailed distributions associated with negative γ and towards right for distributions in the Gumbel domain but with finite endpoint.

√ k I -1/(1 -γ) = O p (1

Simulation Results

According to Theorem 1 and subsequent Remark 3, the testing procedures that statistics T (j) k,n , j = 1, 2, yield unfold as follows. With view towards testing the hypothesis

H 0 : F ∈ D(G 0 ) and x F = ∞ versus H 1 : F ∈ D(G γ ) γ≤ 0 and x F < ∞,
take the normalized versions

T * 1 = √ k log k T (1) k,n and T * 2 = √ k log n k T (2) k,n -1 .
For the estimation of the scale required in T * 1 we settle with the one related to the Moment estimator (see Remark 3). Under the null hypothesis, provided mild restrictions upon the growth of the intermediate sequence k = k(n), Theorem 1 ascertains that T * j behave as standard normal random variables, eventually. Corresponding rejection regions at a significance level α ∈ (0, 1) are thus allocated by fulfilling |T * j | > z 1-α/2 , for j = 1, 2, where z α = Φ ← (α) denotes the α-quantile pertaining to the standard normal distribution function Φ. difficulty in application of the tests by retaining only the k largest daily maxima above the threshold 5.

This leaves us with a sample comprising n = 1272 independent observations. Application of the testing procedures under study yield the sample paths presented in Figure 4 which suggest the presence of a distribution detaining finite right endpoint. The test supported on T * 2 returns values in the negative direction leaning fast towards a light-tailed distribution in the Weibull domain (γ < 0). The latter comes as no surprise from the high frequency of rejections that T * 2 has accounted for in the simulations presented in Section 3: at a significance level α = 0.05 T * 1 -test is usually a bit more reluctant in rejection of the null hypothesis in favor of the one-sided alternative pertaining to the Weibull domain. Now it yields a sample path near the lower the critical level z 0.025 = -1.96 for several values of k lying in the most stable region of the graph that can be taken as intermediate and thus we can lean on our findings from the T * 2 -test .

  is in the domain of attraction of G γ if (1) holds with G = G γ (notation: F ∈ D(G γ )). The extreme value index γ ∈ R (featuring in (2)) is often regarded as a gauge of tail heaviness of the underlying distribution function F . For positive γ, the power-law behavior in the tail of F has important implications since it may suggest, for instance, the presence of infinite moments. The Fréchet domain of attraction D(G γ ) γ>0 encloses heavy-tailed distribution functions with polynomially decaying tails such as Pareto, Cauchy, Student's-t and the Fréchet distribution itself. All distribution functions in the Weibull domain of attraction are light-tailed with finite right endpoint. Beta distributions and, in particular, the Uniform distribution belong to the Weibull domain D(G γ ) γ<0 . The intermediate case of the Gumbel domain D(G 0 )

Figure 1 :

 1 Figure 1: Empirical power of T * 1 and T * 2 with k = 100, at a significance level α = 0.05, on the basis of 5000 samples of size n = 1000 from the GPd(γ), plotted against |γ|, γ ≤ 0.

Figure 2 :Figure 4 :

 24 Figure 2: Estimated type I error probability of T * 1 (left) and T * 2 (right) plotted against k = 5, 6, . . . , 300. The solid straight line stands for the significance level α = 0.05.

  ). By straightforward calculations we can see that II = o p (1/ √ k). Hence, we only have to take into account that | log t A(t)| ∈ RV γ , meaning thatit converges to zero as long as γ < 0, while for γ = 0 and x F = U (∞) ∈ (0, ∞) we have from Theorem

	B.3.6 of de Haan and Ferreira (2006) that A(t)/ c/U (∞) → 1, c > 0. Altogether, we can ascertain that √ k log(n/k) T (2)

Acknowledgments

Cláudia Neves' research was partially supported by the FCT Projects EXES-PTDC/MAT/64924/2006 and EXTREMA-PTDC/MAT/101736/2008. António Pereira's research was supported by Center for Research on Optimization and Control (CEOC), FCT/POCI 2010.

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The present section aims at illustrating the finite sample behavior of the proposed asymptotic test procedure by conducting Monte-Carlo simulations. In order to estimate the empirical rejection probabilities (type I error and power) we generated 5000 samples (replicates) of size n = 1000 from several distributions satisfying condition (7). Hence we shall be interested in the relative frequency of replicates which the null hypothesis was rejected for. Inevitably, the extreme value index γ is the most important design factor here but the second order parameter ρ has also a relevant part since it determines the speed at which T * 1 and T * 2 converge to their normal limits. The following models are taken as key examples:

• GPd (γ ≤ 0), with distribution function F γ given in (4).

• Negative Fréchet distribution (γ = ρ = 0): we say that a random variable X has a negative Fréchet

given by

• Gumbel distribution (γ = 0, ρ = -1).

• Logistic distribution (γ = 0, ρ = -1), with distribution function

• Fréchet (γ > 0, ρ = -1), with distribution function

Figure 1 displays the empirical power of the aforementioned tests, at the nominal size α = 0.05, for the underlying GPd by retaining the k +1 = 101 most extreme observations. The GPd, including the particular case of the exponential distribution for γ = 0, satisfies the extreme value condition (3) but not the second order relation ( 7). The GPd has tail quantile function (t γ -1)/γ and we may read the convergence in (7) as being so fast that ρ = -∞ can be assumed. On the light of the information shared by Figure 1, we can ascertain superiority of the T * 1 -test as far as testing exponential-type against Beta-type models is concerned mainly because T * 2 yields too many rejections for underlying distributions other than the (exact) exponential. The latter in verified by Figure 2.

In Figure 2 attraction corresponding to small deviations from the exponential distribution. In the particular case of underlying lognormal distribution, the T * 2 -test performs rather poorly for almost all values of k, mainly because this is already a moderately heavy tailed distribution, further away from the ideal exponential model.

Figure 3 displays empirical power obtained by generating from the GPd with small negative values of γ thus closer to the null hypothesis. According to both tests, the rejection of the null hypothesis is a highly frequent event in case of the negative Fréchet (with x 0 = β = 1), just like a distribution with finite endpoint belonging to the Gumbel domain should determine. The Fréchet distribution is also included for the purpose of illustration embracing the possibility of an underlying heavy-tailed distribution. We have considered γ = 0.3 in this case which corresponds to a not so heavy-tailed distribution with finite variance. For small values of k, the T * 2 -test leads to a prompt rejection of the hypothesis of infinite right endpoint for all distributions but offers an overall erratic behavior for the Fréchet(0.3) distribution. On the other hand, T * 1 places the Fréchet distribution where it should be in the graph: below all the others by yielding the smallest relative number of rejections. However, the T * 1 -test is not very sharp in detecting very small negative values of γ while the statistic T * 2 seems to perform relatively well in this respect.

An illustrative example

From the enormous amount of data available, comprising 3-hourly significant wave heights (Hs), collected at a particular location in the coast off Portugal (Figueira da Foz) from 1958 up to 2001, arises the perception that the assumption of independent data may be compromised by the latent tendency for extreme conditions to linger over several observations. However, in the situation at hand, we may circumvent such


Povilas Banys

Youri Davydov

Vygantas Paulauskas
email: vygan-tas.paulauskas@mif.vu.lt

Remarks on the SLLN for linear random fields

Keywords: MSC 2000 subject classifications. Primary: 60F10, 60F15, 60G60 Limit theorems, random fields, ergodic theorems

We consider random linear fields on Z d generated by ergodic or mixing (in particular case, independent identically distributed (i.i.d.)) random variables. Our main results generalize classical Strong Law of Large Numbers (SLLN) for multi-indexed sums of i.i.d. random variables. These results are easily obtained using ergodic theory. Also we compare the results for SLLN obtained using ergodic theory and with the help of the Beveridge-Nelson decomposition.

Introduction

Consider a linear random field

X t = k 0 ϕ k ε t-k , t = (t 1 , . . . , t d ) ∈ Z d , k = (k 1 , . . . , k d ) ∈ Z d , (1)
where the coefficients {ϕ k , k 0, k ∈ Z d } and random variables {ε t , t ∈ Z d } are such that the random field {X t , t ∈ Z d } is well defined and is stationary. Here and in what follows bold letters stand for vectors (multidimensional or infinite-dimensional), linear operations and inequalities are component-wise, for example, 1 = (1, . . . , 1), for x, y ∈ R d x y means that x i y i , i = 1, . . . , d.

In the paper [START_REF] Phillips | Asymptotics for linear processes[END_REF] it was demonstrated that the so-called Beveridge-Nelson decomposition (BND) presents rather simple method for proving limit theorems (Central Limit Theorem (CLT), Strong Law of Large Numbers (SLLN), Law of Iterated Logarithm (LIL), and Invariance Principle (IP)) for sums of values of linear processes. In the recent paper [START_REF] Paulauskas | On Beveridge-Nelson decomposition and limit theorems for linear random fields[END_REF] it was demonstrated that the same BND for linear random fields is useful when proving limit theorems for sums t∈Dn X t , where D n is some subset of Z d . Namely, this representation (exact formulation in the case d = 2 see below) allows to write t∈Dn

X t = M 1 t∈Dn ε t + R n ,
where M 1 = k 0 ϕ k , and R n in the case d = 2 and for simple sets D n (rectangles or squares) has not complicated form. Thus, assuming that a linear field is generated by i.i.d. random variables {ε t , t ∈ Z d }, we reduce the investigation of the limit behavior of sum of dependent stationary random variables to the same problem for sums of i.i.d. random variables and the problem to show that R n (in some sense) converges to zero. In [START_REF] Paulauskas | On Beveridge-Nelson decomposition and limit theorems for linear random fields[END_REF] the BND for linear random fields was applied to prove CLT and SLLN, earlier in [START_REF] Marinucci | Asymptotics for linear random fields[END_REF] BND was used to prove IP. In [START_REF] Paulauskas | On Beveridge-Nelson decomposition and limit theorems for linear random fields[END_REF] it was stressed that in the case d = 2 and sets D n = [1, n] 2 ∩ Z 2 this approach leads to very simple proofs, but at the same time moment conditions for innovations ε t and conditions for coefficients ϕ k are not optimal (this is the price which we pay for simplicity of proofs). In this short note we consider SLLN on rectangles

D n = {t ∈ Z d : 1 t i n i , i = 1, 2, . . . , d}.
(2) and our goal is to obtain generalization of the classical SLLN for multiindexed sums of i.i.d. random variables (see [START_REF] Smythe | Strong law of large numbers for r-dimensional arrays of random variables[END_REF]). The main message of our note is that the application of the ergodic theory to prove SLLN for linear fields gives much more general and stronger results comparing with ones obtained by using BND and even the proofs, based on application of ergodic theorems, are very simple. Since the application of BND for IP also faces some difficulties (see [START_REF] Paulauskas | On Beveridge-Nelson decomposition and limit theorems for linear random fields[END_REF] and Marinucci and Poghosyan ( 2001)), it seems that the most successful application of BND is for the CLT in [START_REF] Paulauskas | On Beveridge-Nelson decomposition and limit theorems for linear random fields[END_REF]. Before formulation of our results we introduce some notions from the ergodic theory. Let Y = {Y t , t ∈ Z d } be a strictly stationary random field. Let H = R Z d denote a space of all real-valued functions on Z d with a σ-algebra H, generated by cylindrical sets. Denote by {U h , h ∈ Z d } the group of translations:

U h x t = x t-h , x ∈ H, t, h ∈ Z d .
Let P stand for a distribution of a random field Y in H. Strict stationarity of Y means that P is invariant with respect to translations: PU -1 h = P. Let T denote σ-algebra of invariant sets:

T = {A ∈ H : U h (A) = A, ∀ h ∈ Z d }.
A random field is ergodic if σ-algebra T is trivial:

∀ A ∈ T P(A) = 0 or 1.
(3)

From the ergodic theory it follows that relation (3) is equivalent to the following relation:

∀ A, B ∈ H n -d 0≤h≤n-1 P(A ∩ U -1 h (B)) → P(A)P(B), (4)
as n → ∞, here n = (n, . . . , n). For more information on ergodic theory see [START_REF] Georgii | Gibbs measures and phase transitions[END_REF] or [START_REF] Walters | An Introduction to Ergodic Theory[END_REF].

We say that a random field Y is mixing if

∀ A, B ∈ H P(A ∩ U -1 h (B)) → P(A)P(B), (5)
as ||h|| → ∞, here ||•|| stands for any norm in R d . It is clear that (5) implies (4), thus, mixing implies ergodicity of a random field. Such definitions for measure-preserving transformations (but in case d = 1) can be found in [START_REF] Walters | An Introduction to Ergodic Theory[END_REF].

While using BND approach we consider linear fields of the form (1), the ergodic theory approach allows us to consider more general linear random fields with summation extended not over positive quadrant bur over all Z d . Namely, let

Y t = k∈Z d ϕ k ε t-k , t ∈ Z d . ( 6
)
We consider sums

S n = t∈Dn Y t , (7)
and let us denote |n| := d i=1 n i . We say that SLLN holds for S n , if

|n| -1 S n a.s. → 0, (8)
when n tends to infinity. There are several interpretations of the growth of n. In our paper we shall use two possibilities. We shall write n → ∞ if

n i → ∞, i = 1, . . . , d. (9)
The second possibility of growth of n is to assume that

|n| → ∞. (10)
Evidently, (10) follows from ( 9), but not converse. Let us denote by L q,p condition

k∈Z d d i=1 (|k i | + 1) q |ϕ k | p < ∞ and L p := L p,p . If a random variable X satisfies E|X|(ln(1 + |X|)) d-1 < ∞, (11)
we shall write X ∈ L log L d-1 .

Theorem 1. Let ε t , t ∈ Z d be a strictly stationary ergodic random field with Eε 0 = 0 and ε 0 ∈ L log L d-1 . Suppose that condition L 0,1 holds and S n is defined in (7). Then, if n → ∞, the relation (8) holds.

In particular, the result holds for the case where ε t , t ∈ Z d are i.i.d. random variables satisfying the same moment conditions . Therefore in the case where Y t = ε t (this will be if ϕ 0 = 1, ϕ k = 0 for all k = 0) we get the generalization of the result of [START_REF] Smythe | Strong law of large numbers for r-dimensional arrays of random variables[END_REF] where it is shown that if ε t , t ∈ Z d are i.i.d. random variables, then SLLN for S n = t∈Dn ε t holds if an only if ( 11) is satisfied. If we consider the class of all strictly stationary ergodic random fields, which includes a random field ε t , t ∈ Z d with i.i.d. random variables ε t , therefore moment condition ε 0 ∈ L log L d-1 in our theorem is necessary, too. But as results of [START_REF] Rieders | The size of the average of strongly mixing random variables[END_REF] show, for particular types of dependent stationary sequences situation with necessary conditions can be different, the same can be said about random fields.

If we require stronger condition on initial random field ε t , t ∈ Z d , we get a stronger result.

Theorem 2. Let ε t , t ∈ Z d be a strictly stationary mixing random field with Eε 0 = 0 and ε

0 ∈ L log L d-1 . Suppose that condition L 0,1 holds. Then the relation (8) holds, if |n| → ∞.
Both formulated theorems rely on the classical ergodic theorem of Zygmund-Fava.

Theorem A. Let ε t , t ∈ Z d be a strictly stationary ergodic random field with Eε 0 = 0 and ε 0 ∈ L log L d-1 . Then

V n = |n| -1 t∈Dn ε t a.s. → 0, as n → ∞.
As it is formulated here, this result is an easy corollary from Theorem 1.1 in Krengel (1985), p. 196, where it is formulated in more general setting -for some operators (contractions), acting on finite measure space. Reduction to probability space and shift operators is standard, and that the limit is zero (in [START_REF] Krengel | Ergodic Theorems[END_REF] the existence of a limit is stated) follows from ergodicity of the random field under consideration.

Having these two results on SLLN for linear random fields it is clear that by using BND we can not get such general and strong results. Thus, application of BND to prove SLLN for linear random fields (the same remark can be applied for linear processes, too) can be justified only from methodological point of view: if one wants (for some reasons) to avoid ergodic theory, one can reduce the problem of SLLN for linear random fields to the case of SLLN for multi-indexed sums of i.i.d. random variables and use (comparatively elementary) [START_REF] Smythe | Strong law of large numbers for r-dimensional arrays of random variables[END_REF] result and elementary moment inequalities to estimate remainder term appearing in BND (as it was done in [START_REF] Paulauskas | On Beveridge-Nelson decomposition and limit theorems for linear random fields[END_REF]). We formulate one more such result in the case d = 2 in the next proposition. In all aspects this result is weaker comparing with theorems formulated above, only the moment condition is only very little stronger than ( 11), but it is necessary to note that to achieve such condition instead of the moment inequality we used a result from ergodic theory. The obtained result is stronger than results in [START_REF] Paulauskas | On Beveridge-Nelson decomposition and limit theorems for linear random fields[END_REF], and, in a sense, it demonstrates the limits of the BND approach in the problem. Now we return to a linear random field defined in (1) and we denote

S (1) n = t∈Dn X t . ( 12
)
Proposition 3. Let ε t , t ∈ Z 2 be i.i.d. random variables with Eε 0 = 0. Suppose that for some 1 < p 2 moment condition E|ε 0 | p < ∞ and condition L p (with summation only over positive quadrant) holds. Then, under (10), SLLN for S

(1) n holds.

Proofs

Before the proof of theorems we shall formulate BND, as it was stated in [START_REF] Paulauskas | On Beveridge-Nelson decomposition and limit theorems for linear random fields[END_REF]. Let L = (L 1 , L 2 ) be lag operators, defined by formulas:

L 1 ε t = ε t 1 -1,t 2 , L 2 ε t = ε t 1 ,t 2 -1 . Let us set Φ(L) = k 0 ϕ k L k 1 1 L k 2 2 .
To formulate BND we need more notation. Denote

µ 1 = Φ(1, 1) = k 0 ϕ k , A 2 (L) = Φ * (L)∆ 2 (L), ∆ 2 (L) = (1 -L 1 )(1 -L 2 ), Φ * (L) = k 0 ϕ * k L k 1 1 L k 2 2 , ϕ * k = l k+1 ϕ l , A 1 (L) = B(L 1 )∆ 1 (L 1 ) + D(L 2 )∆ 1 (L 2 ), ∆ 1 (L i ) = (1 -L i ), B(L 1 ) = l 0 b l L l 1 , b l = ϕ * l,-1 = k 1 l+1,k 2 0 ϕ k 1 ,k 2 , D(L 2 ) = l 0 d l L l 2 , d l = ϕ * -1,l = k 1 0,k 2 l+1 ϕ k 1 ,k 2 .
Now we can formulate BND as it was stated in [START_REF] Paulauskas | On Beveridge-Nelson decomposition and limit theorems for linear random fields[END_REF].

Proposition 4. The following identity holds

Φ(L) = µ 1 + A 2 (L) -A 1 (L). ( 13
)
The relations

k 0 |ϕ * k | p < ∞, l 0 |b l | p < ∞, l 0 |d l | p < ∞, µ 1 < ∞ (14) hold if either condition L p in case 1 p < ∞ or condition L 1,p in case p < 1 is satisfied.
Also we shall need the following simple result.

Lemma 5. Let Y t be a random field defined in (6), d ≥ 2, a random field ε t satisfies conditions of Theorem 1 and the condition L 0,1 holds. Then for all t ∈ Z d E|Y t |(ln(1

+ |Y t )) d-1 < ∞. (15)
Proof of Lemma 5. Let us denote g(x) = |x|(ln(1+|x|)) d-1 . This function is convex and increasing for x ≥ 0, therefore using obvious inequality (1 + cx) ≤ (1 + x)(1 + c) we have for all c > 0, x > 0

g(cx) ≤ 2 d-2 (cg(x) + xg(c)).
We must prove the boundedness of Eg(Y t ). Let us denote c = k∈Z d |ϕ k |, then, using the fact that g is convex and increasing, we have

Eg(Y t ) = Eg(| k∈Z d ϕ k ε t-k |) Eg(| k∈Z d |ϕ k ||ε t-k |) = Eg(c k∈Z d |ϕ k | c |ε t-k |) 2 d-2 cEg( k∈Z d |ϕ k | c |ε t-k |) + g(c)E k∈Z d |ϕ k | c |ε t-k | 2 d-2 cE k∈Z d |ϕ k | c g(|ε t-k |) + g(c) k∈Z d |ϕ k | c E|ε t-k | 2 d-2 cEg(|ε 0 |) + g(c)E|ε 0 | < ∞.

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

The lemma is proved.

Proof of Theorem 1. Theorem 1 directly follows from Theorem A. Namely, the random field {Y t } is strictly stationary and ergodic (as a function of a random field {ε t }), EY 0 = 0, from Lemma 5 it follows that Y 0 ∈ L log L d-1 . Hence we can apply Theorem A.

Proof of Theorem 2. Let, for simplicity, agree to write [0, n] c for a set of those k ∈ (Z + ) d , which satisfy k n. It is easy to see that the relation

T n := |n| -1 S n a.s.
→ 0, when |n| → ∞, is equivalent to the following condition: for each > 0 there exists n ( ) = (n

( ) 1 , . . . , n ( ) d ) such that P {|T n | < , ∀n ∈ [0, n ( ) ] c } = 1.
At first we prove the theorem in the case d = 2. From Theorem 1 it follows that

T n a.s. → 0, as n → ∞, that is, n i → ∞, i = 1, 2.
Let Ω 0 , P (Ω 0 ) = 1, be a set of those ω, for which this relation holds. (Here (Ω, F, P ) is a probability space on which all random variables under consideration are defined.) Let us fix > 0. Then ∀ ω ∈ Ω 0 , there exists n ( ) = (n

( ) 1 , n ( ) 2 ) such that, ∀ n ≥ n ( ) , |T n | < . Let us take some 1 ≤ m ≤ n ( )
2 -1 and consider sums

T (n 1 ,m) = 1 n 1 m k≤(n 1 ,m) Y k = 1 m m j=1 1 n 1 n 1 k=1 Y (k,j) .
The random field {Y n , n ∈ Z 2 }, as a function (under very mild conditions on a function; in our case it is linear function) of a mixing field {ε n , n ∈ Z 2 }, is mixing, too. Therefore for any fixed j a random process {Y (k,j) , k ∈ Z} is also mixing, hence it is ergodic, and we get

T (n 1 ,m) a.s. → 0, as n 1 → ∞.
Let Ω 1,m , P (Ω 1,m ) = 1 be a set of those ω for which this relation holds.

Taking the same > 0 we can find n ,m such that for n 1 ≥ n ,m and all

ω ∈ Ω (1,m) , we have |T (n 1 ,m) | < . Now for some 1 ≤ r ≤ n ( )
1 -1 in a similar way we consider sums T (r,n 2 ) , and (for the same ) we introduce a set Ω 2,r , P (Ω 2,r ) = 1, and number n r, such that, for all ω ∈ Ω (2,r) , and all

n 2 ≥ n r, , we have |T (r,n 2 ) | < . Let us take n( ) = (n ( ) 1 , n( ) 2 ) with n( ) 1 = max{n ( ) 1 , n ,m : m = 1, . . . , n ( ) 2 -1}, n( ) 2 = max{n ( ) 2 , n r, , r = 1, . . . , n ( ) 1 -1}. Let us denote Ω = Ω 0 ∩ ∩ n ( ) 2 -1 m=1 Ω 1,m ∩ ∩ n ( ) 1 -1 r=1 Ω 2,r .
It is clear that P ( Ω) = 1, and from construction it follows that for all ω ∈ Ω and for all n ∈ [0, n( ) ] c , |T n | < .

Thus, we have proved the theorem in the case d = 2. Now, using mathematical induction, we shall prove the general case d > 2. Let us assume that the statement of the theorem is true for all dimensions k ≤ d -1, d ≥ 3. We prove then that the statement of the theorem holds for dimension d. The proof is similar to that in the case d = 2, only notations are more complicated. Let us denote n(k) = (n 1 , . . . , n k-1 , n k+1 , . . . , n d ) and (n(k), m) = (n 1 , . . . , n k-1 , m, n k+1 , . . . , n d ), k = 1, . . . , d. Again, by using theorem 1, we have that SLLN is valid for S n as n → ∞. Let Ω 0 , P (Ω 0 ) = 1, be a set of those ω, for which this relation holds. Let us fix > 0. Then, ∀ ω ∈ Ω 0 , there exists n ( ) = (n

( ) 1 , . . . , n ( ) d ) such that, ∀ n ≥ n ( ) , |T n | < .
Fix some 1 ≤ k ≤ d and take some 1 ≤ m ≤ n

( ) k -1. Consider sums T (n(k),m) = 1 |n(k)|m r (n(k),m) Y r = 1 m m j=1 1 |n(k)| r(k) n(k) Y (r(k),j) ,
where |n(k)| = |n|/n k . Then, by using the same argument as in the proof of the case d = 2, we get that {Y (n(k),j) , n(k) ∈ Z d-1 }, for a fixed j, is also mixing. Therefore, from the induction assumption for a fixed m we get,

T (n(k),m) a.s. → 0, as |n(k)| → ∞.
Denote by Ω (k,m) , P (Ω (k,m) ) = 1, the set where this convergence for T (n(k),m) holds. Taking the same > 0, which we fixed at the beginning of the proof, we will find n(k) ( ,m) = (n

( ,m) 1 , . . . , n ( ,m) k-1 , n ( ,m) k+1 , . . . , n ( ,m) d
) such that for all n(k) ∈ [0, n(k) ( ,m) ] c and for all ω ∈ Ω (k,m) , |T (n(k),m) | < . Then by applying the same argument to the other coordinates we will get the set of pairs:

{(Ω (1,m 1 ) , n(1) ( ,m 1 ) ) . . . , (Ω (d,m d ) , n(d) ( ,m d ) ), 1 ≤ m k ≤ n ( ) k -1}.
To finish the proof we must find Ω, for which P ( Ω) = 1, and n such that, for all n ∈ [0, n ] c and ω ∈ Ω, |T n | < . This can be easily done, namely

Ω = Ω 0 ∩ d k=1 ∩ n k -1 m k =1 Ω (k,m k ) . and n = (n 1 , . . . , n d ), n k = max n k , max j =k n j,k , n j,k = max m≤n ( ) j -1 n ( ,m) j .
The theorem is proved.

Proof of Proposition 3. Applying (13) to the sum in (27) we get

S

(1)

n = µ 1 Z n + R n , Z n = t∈Dn ε t , (16)
where

R n = ξ n 1 ,n 2 -ξ n 1 ,0 -ξ 0,n 2 + ξ 0,0 + η n 1 ,n 2 -η 0,n 2 + ζ n 1 ,n 2 -ζ n 1 ,0 , (17)
ξ t = Φ * (L)ε t = k 0 ϕ * k ε t-k , η t 1 ,n 2 = n 2 t 1 =1 εt 1 ,t 2 , εt 1 ,t 2 = B(L 1 )ε t 1 ,t 2 = l 0 b l ε t 1 -l,t 2 , ζ n 1 ,t 2 = n 1 t 1 =1 εt 1 ,t 2 , εt 1 ,t 2 = D(L 2 )ε t 1 ,t 2 = l 0 d l ε t 1 ,t 2 -l .
Z n is a sum of i.i.d. mean zero random variables ε t with E|ε 0 | p < ∞, for some 1 < p 2, therefore SLLN for Z n holds. Then, in order to prove the proposition, we must prove that |n| -1 R n a.s. → 0, as |n| → ∞.

(18)

We have

R n = R 1 n + R 2 n + R 3 n , (19)
where R 1 n = ξ n 1 ,n 2 -ξ n 1 ,0 -ξ 0,n 2 +ξ 0,0 , R 2 n = η n 1 ,n 2 -η 0,n 2 , R 3 n = ζ n 1 ,n 2 -ζ n 1 ,0 , therefore we must prove the relation ( 18) for each R i n , i = 1, 2, 3. We start with R 1 n and prove that n 1

P (||n| -1 ξ n | > ) < ∞. ( 20
)
Since ε t are i.i.d. and Eε t = 0, the moment inequality for Clearly, the same relation holds for ξ n 1 ,0 , ξ 0,n 2 , ξ 0,0 , and we get that R 1 n a.s.

ξ t = k 0 ϕ * k ε t-k yields E|ξ t | p C k 0 |ϕ * k | p E|ε t-k | p C k 0 |ϕ * k | p < ∞. (21) Therefore,
→ 0.

(24)

Unfortunately, the moment inequality is too rough for other two terms R 2 n and R 3 n (this was noted in Paulauskas ( 2009)), since, using low order moments for ζ t and η t , we get divergent series in (20), or we must require existence of moment of the order 2 + δ, δ > 0. Therefore we shall use one result from ergodic theory. Let us consider random variable η t and let us denote

K n = 1 |n| η n = 1 n 1 n 2 n 2 t 2 =1
εn 1 ,t 2 .

(25)

n 1 P

 1 (||n| -1 ξ n | > ) E|ξ 1 | p

Acknowledgment

The authors are grateful to anonymous Referee, whose remarks and suggestions improved the presentation of the results.

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

Taking into account the definition of εt , we have

Therefore, denoting

Applying Proposition 50.2 from [START_REF] Parthasarathy | Introduction to probability and measure[END_REF]

Here it is worth to note that the same conclusion we can get under weaker moment condition ε 0 ∈ L log L using Exercise 50.4 from the same book, but this moment condition is insufficient in order to get ( 23) by means of the moment inequality. Note that

Sequence {X n , n ≥ 1} is a stationary, ergodic (since it is generated by a sequence of i.i.d. random variables) and with finite mean, thus we can write

Since we need to show that K n a.s.

→ 0, as |n| → ∞, it remains to consider the case, where n 2 → ∞ and n 1 stays bounded. If |n| → ∞ in a such way that n 1 is fixed and

k 0 is some fixed number and n 2,k → ∞, then the sequence K n k can be divided into k 0 subsequences in a such way, that in each subsequence the first index of summands is fixed. Then we apply the argument used above and get the convergence to zero a.s. of each subsequence, therefore the sequence itself also converges to zero a.s. Thus we have proved (30), if |n| → ∞.

It is easy to see that the convergence to zero a.s. of (n 1 n 2 ) -1 η 0,n 2 can be proved in the same way, therefore we have Collecting ( 19), ( 24), ( 31), (32) we get (18), and the proposition is proved.


Arantza Fariña-Sarasqueta

Gesina Lijnschoten

Harm J T Rutten

PhD Adriaan J C Van Den Brule
email: a.van.den.brule@pamm.nl

Gesina Van Lijnschoten

J T Harm

Rutten

Value of gene polymorphisms as markers of 5-FU therapy response in stage III colon carcinoma: a pilot study

Keywords: 5-FU, colon carcinoma, OPRT, DPYD, TYMS, TYMP, SNP

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In colon cancer, the role of pharmacogenetics for drug toxicity and efficacy is still under discussion [START_REF] Braun | Association of Molecular Markers With Toxicity Outcomes in a Randomized Trial of Chemotherapy for Advanced Colorectal Cancer: The FOCUS Trial[END_REF]. 5-Fluorouracil (5-FU) is the main drug of choice in the treatment of stage III colon carcinoma. Several proteins are involved in the metabolism of 5-FU and many of the genes coding for these proteins have been shown to be polymorphic.

Ororate phosphorybosil transferase (OPRT) and thymidine phosphorylase (TP) activate 5-FU by phosphorylation into active metabolites which respectively incorporate into RNA or inhibit thymidylate synthase (TS). Dihydropyrimidine dehydrogenase (DPD) inactivates 5-FU in the liver [START_REF] Longley | Drug resistance, predictive markers and pharmacogenomics in colorectal cancer[END_REF][START_REF] Longley | 5-fluorouracil: mechanisms of action and clinical strategies[END_REF]. The genes encoding for these proteins harbor functional polymorphisms.

The OPRT gene contains several polymorphisms, among those the G638C SNP causes a Gly 213 Ala substitution. This SNP has been associated with a higher expression and activity of the protein and with an increased toxicity of 5-FU therapy [START_REF] Ichikawa | Orotate phosphoribosyltransferase gene polymorphism predicts toxicity in patients treated with bolus 5-fluorouracil regimen[END_REF].

There are several polymorphisms described in the TP gene (TYMP), however there are no confirmed polymorphisms in coding regions causing changes in aminoacid sequence. The value of these polymorphisms as markers of response to 5-FU therapy is to our knowledge, unknown.

The polymorphisms in the enhancer region of the TS gene (TYMS) have been widely studied in their relation to response to 5-FU therapy and with protein expression and activity. The studied polymorphisms consist in a 28bp repeat at the 5' untranslated region of the gene and a G>C SNP in the second repeat of the three repeat allele. In the Caucasian population the variants with two (2R) or three (3R) repeats are the most frequent alleles found. On the basis of the effects of the SNP in the second repeat of the 3R allele on TS protein expression, patients could be classified as high TS protein producers when carrying the G allele and low TS protein producers when carrying the C allele [START_REF] Kawakami | Identification and functional analysis of single nucleotide polymorphism in the tandem repeat sequence of thymidylate synthase gene[END_REF][START_REF] Mandola | A novel single nucleotide polymorphism within 5' tandem repeat polymorphism of the thymidylate synthase gene abolishes USF-1 binding and alters transcriptional activity[END_REF]. However, up to date, the results concerning the effect of these polymorphisms in 5-FU response remain inconclusive [START_REF] Popat | Thymidylate synthase expression and prognosis in colorectal cancer: a systematic review and meta-analysis[END_REF].

Finally, the DPYD gene has been shown to play a very important role in toxicity of 5-FU. The polymorphism in the exon/intron boundary at exon 14 is responsible for severe toxicity in these patients [START_REF] Vallböhmer | DPD is a molecular determinant of capecitabine efficacy in colorectal cancer[END_REF]. However, little is known about the value of these and other polymorphisms as markers of response.

We aimed to study the value of known polymorphisms in the OPRT, TYMP, TYMS and DPYD genes as markers of response in stage III colon carcinoma patients treated with 5-FU chemotherapy in combination

Materials and methods

Patient material

Sixty stage III colon carcinoma patients treated with surgery and 5-FU chemotherapy following the Mayo regime were studied.

All diagnoses were made at a central laboratory for pathology between 2003 and 2004. Population data were obtained from the cancer registry database of the Comprehensive Cancer Centre South. Follow-up information was obtained from medical records.

The use of clinical material for this retrospective study was approved by the institutional review board according to the guidelines of the Dutch Federation of Research Associations.

DNA was isolated from normal colonic mucosa from formalin fixed paraffin embedded (FFPE) material after proteinase K digestion and purification using the HPTTP kit (Roche, Almere, the Netherlands).

Target genes and polymorphisms

Polymorphisms must be non synonymous and confirmed by independent research groups. However, in the case of the TYMP gene, there were no confirmed non synonymous polymorphisms and therefore we chose one confirmed synonymous polymorphism.

OPRT

The G638C SNP (rs1801019) causing a Gly213Ala substitution was tested by real time PCR with the following primers and probes forward 5' GCT GAG ACA GTT GGG AGA GTG A 3', reverse 5' TGA GTT CTT TGG GTG CTT CCT T 3', probe for G allele 6FAM 5'CGA ATC ATA ATG GTT C3'and probe for C allele 6FAM 5'AGC GAA TCA TAA TGC TT3'. Reactions were performed using Roche chemistry in a final volume of 20 µl in the light cycler v2 (Roche, Almere, the Netherlands).

TYMP

The rs470119 SNP was assessed by restriction fragment length polymorphism (RFLP). PCR was performed using the following primers forward 6FAM-5'TCC AGA GCC CAG GTA3' and reverse 5'CTG GCC AGG GTC TCC ATC A3'. The 71 bp long PCR product was then digested with MboI restriction enzyme (New England Biolabs, Ipswich, United Kingdom). After digestion, fragment length analysis was carried out by capillary electrophoresis. The following fragment length was expected for homozygous GG, 40bp and 30bp, for AA 71 bp and for heterozygous AG 71bp, 40bp and 30bp.

TYMS

The two polymorphisms in the enhancer region of this gene were typed using PCR and RFLP as described elsewhere [START_REF] Gosens | Thymidylate synthase genotyping is more predictive for therapy response than immunohistochemistry in patients with colon cancer[END_REF]. Briefly, the 28 bp repeat was typed by PCR means followed by electrophoresis on 2% agarose gel. The G>C SNP was typed by digestion of the PCR product with HaeIII restriction enzyme. The G to C substitution abrogates the restriction site for this enzyme. Subsequently, products were separated by agarose gel electrophoresis.

DYPD

The SNP A1627G (rs1801159) causing Ile543Val was tested by PCR followed by sequencing using the following primers, forward 5'GCA GTC ACA ATA TGG AGC3' and reverse 5'TTA CCT TAT CAA GAG AGA AAG TT3'. The expected length of the product was 225 bp. Subsequently, PCR products were purified using enzymatic purification with ExoSapIT (USB, USA) and the sequencing reaction was performed using the same primers as for the PCR reaction and Big Dye chemistry (Applied Biosystems, Nieuwerkerk aan den Ijssel, the Netherlands). Sequences were analyzed using the sequencing analysis 5. Categorical data were analyzed by means of a chi-square or Fischer's exact test. The end point of this study was progression free survival (PFS, time between surgery and disease progression).

Univariate survival analysis was performed by Kaplan Meier analysis and differences were analyzed using the Log Rank method. Hazard Ratios and multivariate analysis were calculated using the Cox Proportional Hazard model.

All tests are two tailed and a result was considered significant when p=0.05.

Results

Briefly, characteristics of the sixty patients studied were as follows, median age at diagnosis was 64 years (range 30-81), fifty two percent (n=31) of the tumors were located on the right side of the colon and 53% (n=32) of the patients were male. The majority 70% (n=42) had a T3 tumor. Median follow up was 39 months (range 2-57). 40% was still alive without evidence of disease at the end of the follow up, 24% had developed a local recidive or a distant metastasis, 31% was dead because of cancer related causes and 5.2% was dead because of non cancer related causes as specified in their medical records. Median time to progression was 15 months (range 6-47)

Frequencies of the different alleles are shown in table 1. All frequencies followed Hardy-Weinberg equilibrium and did not differ significantly from frequencies published on the hapmap database for the Caucasian population.

In this group of patients, there were no significant associations between any of the genotypes found and any of the clinical and histopathological variables tested including gender, tumor location, T stage and N stage.

Survival Analysis

For the survival analysis, TYMS SNP genotypes were grouped as putative high TS expression (genotypes 2R/3G, 3C/3G and 3G/3G) and putative low TS expression (2R/2R, 2R/3C. 3C/3C). No effect on progression free survival of the several genotypes found was seen in a univariate (figure 1) or in a multivariate survival analysis, containing other known prognostic variables for colon carcinoma such as T stage and N stage of the tumor.

Discussion

We aimed to study whether known SNPs in genes involved in 5-FU metabolism were good markers of therapy response in stage III colon carcinoma patients.

According to our results, we conclude that the SNPs tested in the OPRT, TYMP, TYMS and DPYD genes are not good markers of therapy response in the present cohort. No effect on survival of the different genotypes was seen, however, results should be considered with caution due to the small number of patients analyzed.

Although, increased expression of OPRT mRNA and protein activity have been related to a shorter survival of colorectal carcinoma patients treated with 5-FU [START_REF] Ishikawa | Clinical implications of thymidylate synthetase, dihydropyrimidine dehydrogenase and orotate phosphoribosyl transferase activity levels in colorectal carcinoma following radical resection and administration of adjuvant 5-FU chemotherapy[END_REF][START_REF] Ochiai | Prognostic impact of orotate phosphoribosyl transferase among 5-fluorouracil metabolic enzymes in resectable colorectal cancers treated by oral 5-fluorouracil-based adjuvant chemotherapy[END_REF][START_REF] Ochiai | Prognostic impact of ororate phosphoribosyl transferase activity in resectable colorectal cancers treated by 5-fluorouracil-based adjuvant chemotherapy[END_REF] together with the fact that the studied SNP has been proven to be associated to a higher protein expression and activity [13] [4], no effects of the different genotypes in DFS was seen in the present cohort. The OPRT CC variant is very rare in the general population. Our results reflect the low frequency of this genotype in the Caucasian population. If the CC variant would have an effect on survival, the fact that it is such a rare variant makes it very difficult to prove since numbers of patients needed would be very large.

The existing literature is more conclusive about TP protein expression which seems to have no influence in survival of colorectal patients [START_REF] Artinyan | Molecular predictors of lymph node metastasis in colon cancer: increased risk with decreased thymidylate synthase expression[END_REF][START_REF] Ciaparrone | Predictive role of thymidylate synthase, dihydropyrimidine dehydrogenase and thymidine phosphorylase expression in colorectal cancer patients receiving adjuvant 5-fluorouracil[END_REF][START_REF] Lassmann | Predictive molecular markers for colorectal cancer patients with resected liver metastasis and adjuvant chemotherapy[END_REF][START_REF] Liersch | Lymph node status and TS gene expression are prognostic markers in stage II/III rectal cancer after neoadjuvant fluorouracil-based chemoradiotherapy[END_REF][START_REF] Metzger | High basal level gene expression of thymidine phosphorylase (platelet-derived endothelial cell growth factor) in colorectal tumors is associated with nonresponse to 5-fluorouracil[END_REF][START_REF] Yanagisawa | Modified irinotecan/5-FU/Leucovorin therapy in advanced colorectal cancer and predicting therapeutic efficacy by expression of tumorrelated enzymes[END_REF]. Accordingly, the SNP in the TYMP gene was not a good marker for therapy response in our group.

The role of the typed polymorphisms on the TYMS gene remains controversial. It has been widely studied as protein and mRNA expression as well as DNA genotyping, still the results are inconclusive as reviewed by Popat et al [START_REF] Popat | Thymidylate synthase expression and prognosis in colorectal cancer: a systematic review and meta-analysis[END_REF]. These contradictory results are probably due to differences in methodology, technology and patients' selection. In the present patient group, TYMS polymorphisms in the enhancer region are not good markers of therapy response, even when grouping patient by category of putative TS expression no difference in survival was found between putative low and high producers. These results agree with previous results in a larger cohort of patients [START_REF] Farina-Sarasqueta | TS gene polymorphisms are not good markers of response to 5-FU therapy in stage III colon cancer patients[END_REF].

Finally, although DPYD has been widely studied in relation to 5-FU toxicity, it also seems to have a role in outcome [START_REF] Ciaparrone | Predictive role of thymidylate synthase, dihydropyrimidine dehydrogenase and thymidine phosphorylase expression in colorectal cancer patients receiving adjuvant 5-fluorouracil[END_REF][START_REF] Yanagisawa | Modified irinotecan/5-FU/Leucovorin therapy in advanced colorectal cancer and predicting therapeutic efficacy by expression of tumorrelated enzymes[END_REF][START_REF] Tokunaga | Clinical role of ororate phosphoribosyltransferase and dihydropyrimidine dehydrogenase in colorectal cancer treated with postoperative fluoropyrimidine[END_REF][START_REF] Tsuji | Tumor dihydropyrimidine dehydrogenase expression is a useful marker in adjuvant therapy with oral fluoropyrimidines after curative resection of colorectal cancer[END_REF][START_REF] Westra | Predictive value of thymidylate synthase and dihydropyrimidine dehydrogenase protein expression on survival in adjuvantly treated stage III colon cancer patients[END_REF]. However, whether this effect is due to the enzyme itself or due to high toxicity and subsequent therapy interruption is not clear. In the present cohort of patients, the studied SNP is not a good marker of therapy response.

In conclusion, polymorphisms in genes involved in 5-FU metabolism are not valuable as markers of response in the present cohort of colon cancer patients. Recently, the results of a clinical trial over the value of some polymorphisms in predicting toxicity in a large group of colon carcinoma patients failed to find any relation between the tested SNPs and toxicity [START_REF] Braun | Association of Molecular Markers With Toxicity Outcomes in a Randomized Trial of Chemotherapy for Advanced Colorectal Cancer: The FOCUS Trial[END_REF]. In that study the value of several SNPs involved in 5-FU metabolism was tested in relation to toxicity. In the same way, we cannot show any value of these SNPs as markers of response to 5-FU in our group of patients.

Tables and figures

  for Windows (Chicago, Il., U.S.A.) was used for statistical analysis.

Figure 1 :

 1 Figure 1: Kaplan Meier Curves for PFS according to all genotypes tested; a) TYMP genotypes, b) OPRT, c) DPYD, d)TYMS repeat e) TYMS expression category determined by SNP genotypes (Low expression 2R/2R, 2R/3C, 3C/3C; High expression 2R/3G and 3G/3G)

Table 1 :

 1 Frequencies of the different alleles

	SNPs	Frequencies N (%)
	OPRT
	GG	44 (75)
	GC	15 (25)
	CC	0 (0)
	TYMP
	GG	25 (42)
	AG	29 (48)
	AA	6 (10)
	TYMS
	2 repeats	10 (17)
	2 & 3 repeats	35 (58)
	3 repeats	15 (25)
	TYMS SNP
	2R/2R	10 (16.7)
	2R/3RC	16 (26.7)
	2R/3RG	19 (31.7)
	3RC/3RC	7 (11.7)
	3RG/3RC	4 (6.7)
	3RG/3RG	4 (6.7)
	DPYD
	AA	37 (67)
	AG	14 (26)
	GG	4 (7)

with leucovorin.