icm 2006 short communications abstracts section 13 · (g2) partial derivative of gsatisﬁes local...

ICM 2006

Short Communications

Abstracts

Section 13Probability and Statistics

ICM 2006 – Short Communications. Abstracts. Section 13

Characterizations of the beta distribution on symmetric matrices

Abdelhamid Hassairi

Department of Mathematics, Sfax University, B.P. 802 , [email protected]

2000 Mathematics Subject Classification. 46F25

The real beta distribution with parameters p > 0 and q > 0 may be pre-sented as the distribution of the random variable U/U+V, where U and Vare two independent random variables with gamma distributions γp,σ andγq,σ respectively, it is given by

βp,q(dx) =xp−1(1− x)q−1

B(p, q)1(0,1)(x)dx

where B(p, q) is the beta Euler function.Similarly, as the multivariate version of the gamma distribution is the

Wishart distribution on the cone of positive definite symmetric (r, r)−matricesor on any symmetric cone Ω, the multivariate beta random variable is de-fined as the “quotient” of some Wishart random variables, using a divisionalgorithm (see Hassairi, Lajmi and Zine (2005)), it is given by

βp,q(dz) = (BΩ(p, q))−1(∆(z))p−r+12 (∆(e− z))q−

r+12 1Ω∩(e−Ω)(z)dz,

where e is identity matrix, ∆(z) is the determinant of z and BΩ(p, q) isthe beta function of the symmetric cone Ω.

Recently Seshadri and Wesolowski (2003) have given two characteri-zation results concerning the real beta distribution based on some con-stancy of regression conditions. For intance, let X and Y be two indepen-dent, non-degenerate random variables in (0, 1) and let V = 1 − XY andU = (1− Y )/V . The first result says that if

E(U | V ) = c and E(U2 | V ) = d, (1)

where c and d are real constants, then (X,Y ) ∼ βp,q ⊗ βp+q,s and conse-quently (U, V ) ∼ βr,q ⊗ βr+q,p. where q = (1−c)(d−c)c2−d > 0, s =

c(d−c)c2−d > 0

and p > 0.The second result says that if

E(U | V ) = c and E(U−1 | V ) = b, (2)

where c and b are real constants, then (X,Y ) ∼ βp,q ⊗ βp+q,s and conse-quently (U, V ) ∼ βr,q ⊗ βr+q,p where q = (b−1)(1−c)(bc−1) > 0, s =

c(b−1)bc−1 > 0 and

ICM 2006 – Madrid, 22-30 August 2006 1


p > 0. In the present work, we first establish some properties of the matrixBeta distribution, in particular, we give the distribution of some generalizedpowers and we show the independence of some related statistics. We thengive multivariate versions of the regression formulas (1) and (2) and weshow that each of the two extended versions characterizes the beta-Wishartdistribution on symmetric matrices.

References

[1] Hassairi, A., Lajmi, S. and Zine, R., Beta-Riesz distributions on symmetriccones, JSPI 133 (2005), 387-404.

[2] Seshadri,V. and Wesolowski, J., Constancy of regressions for beta distributions,Sankhya 65 (2003), 284-291.

2 ICM 2006 – Madrid, 22-30 August 2006


Combining Edgeworth’s and Berry–Esseen’s approaches to thecentral limit theorem

José A. Adell*, Alberto Lekuona

Departamento de Métodos Estad́ısticos, Universidad de Zaragoza, Pedro Cerbuna12, 50009 Zaragoza, [email protected]; [email protected]


We obtain near optimal Berry–Esseen bounds in the classical case. This isachieved by distinguishing the lattice and the nonlattice cases, as one–termEdgeworth expansions do. The main tool is an easy inequality involving theusual second modulus of continuity, in substitution of Esseen’s smoothinginequality.

To be more precise, let (Xk)k≥1 be a sequence of iid random variablessuch that EX1 = 0, EX21 = σ

2 > 0 and E|X1|3 = β < ∞. Denote by Fnthe distribution function of (X1 + · · ·+Xn)/(σ

√n) and by Φ the standard

normal distribution function. Depending on whether or not Fn is lattice,we prove that

12D(Fn) ≤ ‖Fn − Φ‖ ≤

12D(Fn) +

β

6σ3√

2π(1 + δ)n−1/2 +O(n−1), (1)

or

‖Fn − Φ‖ ≤β

6σ3√

2π(1 + δ)n−1/2 +O(n−1), (2)

respectively, where ‖ · ‖ stands for the usual supremum norm, and D(Fn)for the maximum jump of Fn. In (1)–(2), δ is a positive parameter as closeas we wish to zero and the “big o” terms are explicitly bounded, the upperbounds depending on some parameters arbitrarily chosen in certain ranges.No numerical computations are needed.Acknowledgement: This work has been supported by research grantsMTM2005–08376–C02–01 and DGA E–64, and by FEDER funds.

References

[1] Bentkus, V., On the asymptotical behavior of the constant in the Berry–Esseeninequality, J. Theoret. Probab. 7(2) (1994), 211–224 .

[2] Chen, L.H.Y. and Shao, Q.M., A non–uniform Berry–Esseen bound via Stein’smethod, Probab. Theory Related Fields 120 (2001), 236–254.



[3] Hall, P., Rates of Convergence in the Central Limit Theorem. Pitman, Boston,1982.

[4] Shiganov, I.S., Refinement of the upper bound of the constant in the centrallimit theorem, J. Sov. Math. 35 (1986), 2545–2550 .



Stochastic differential equations with fractional noise

Bijan Z. Zangeneh

Department of mathematics, Sharif University of Technology, Tehran, [email protected]

2000 Mathematics Subject Classification. 60H

Let Bt = (B1t , . . . , Bmt ) where the processes B

j ,j = 1, . . . ,m are inde-pendent fractional Brownian motion (fBm) with Hurst Parameter H > 12 .Consider the following d-dimensional stochastic integrals:

Xt = X0 +∫ t0g(Xs)dBs +

∫ t0f(Xs)ds (1)

where the integral with respect to B is a pathwise Riemann-Stieltjes integralof type Zahle[3]. This kind of stochastic integral equation has been studiedby several authors, see for example Lin[2], Nualart and Rascanu[1]. In [1]existence and uniqueness of the solution is proved where

(g1) g satisfies Lipschitz nonlinearity,

(g2) partial derivative of g satisfies local Holder continuity properties,

(g3) there exist γ in [0, 1] and K0 > 0 such that for all x in Rd

‖g(x)‖ ≤ K0(1 + ‖x‖γ),

(f1) f satisfies Lipschitz continuity,

(f2) f has sublinear growth.

In this paper we cover a more general situation, for the above integralequation assumig that f is semimonotone i.e.

< f(x)− f(y), x− y >≤ ‖x− y‖2.

An existence and uniqueness result is proved, which relies heavily on Picarditeration method of Zangeneh[4].



References

[1] Nualart, David, Rascanu, Aurel,Differential equations driven by fractionalBrownian motion.Collect. Math. 53,1(2002), 55-81, 2002 Universitat deBarceona

[2] Lin, S. J. Stochastic analysis of fractional Brownian motions. StochasticsStochastics Rep. 55 (1995), no. 1-2, 121–140..

[3] Zahle, M. Integration with respect to fractal functions and stochastic calculus.I. Probab. Theory Related Fields 111 (1998), no. 3, 333–374. .

[4] Zangeneh, Bijan Z. Semilinear stochastic evolution equations with monotonenonlinearities. Stochastics Stochastics Rep. 53 (1995), no. 1-2, 129–174.



Geometrical probabilities of Buffon type in bounded lattices

Roger Böttcher

Ludwigshafen am Rhein, [email protected]

2000 Mathematics Subject Classification. 60D05, 65C05, 68U20

The story began in 1733 when Comte de Buffon was asking for the prob-ability that a needle which is dropped at random on a set of equidistantparallel lines in the plane intersects one of them. For the case that the linesare a units apart and the needle is of length l < a Buffon gave the solutionp = 2lπa in 1777. In the following years and centuries based on results ofCrofton, Cartan and Poincaré and other mathematicians the theory of ge-ometric probability was born and developed, for an excellent overview see[4]. In the last decade a lot of general examinations where considered like[2] and [3] varying the lattice and the objects of test elements.

In this work now we will go back to Buffon and consider in the first part abounded lattice and unbounded test elements i.e. Buffon’s needle problemwill be inverted. As well we will show some results for bounded latticesand bounded test elements in form of line segments which generalizes thetheorem of Diaconis for long needles, [1]. For that we will look for needlesH which have to intersect the convex hull K of the finite lattice C, i.e.H ∩ K 6= ∅, and furthermore under the restriction that they have to fallcompletely into K, H ⊂ K. For both scenarios we will present the geometricprobabilities pk of exactly k = 0, 1, ..., m, m+1 intersections of the needlesH with the lattice C of equidistant parallel line segments of length d anda units apart consisting of m elementary cells. For the case H ⊂ K a nicerecursive formula for the pk is derived which gives us an explicit functionfor pk. In all our results we will find a structure of the form pk = f(k −1) + g(k) + h(k + 1) with functions f, g, h : IN → IR, some times with abig number of case distinctions. In spite of this the formula for the meanvalue N of intersections between the needles and the lattice based on thedistribution pk in the case of the scenario H∩K 6= ∅ is very comprehensible:N =

1m

+11m

+π2

al+a

d

. Finally we will look at some computer based experiments

finding the geometrical probabilities numerically.

References

[1] Diaconis, P., Buffon’s problem with a long needle, Journal of Applied Proba-bility, 13, 614-618, 1976



[2] Duma, A., Stoka, M. I., Geometrical Probabilities for Convex Test Bodies,Contributions to Algebra and Geometry, Volume 40, No. 1, pp. 15-25, 1999

[3] Duma, A., Stoka, M. I., Problems of ”’Buffon”’ Type for the Dirichlet-VoronoiLattice, Pub. Inst. Stat. Univ. Paris, XXXXVII, fasc. 1-2, pp. 91-108, 2003

[4] Mathai, A. M., An Introduction to Geometric Probability, Gordon and BreachScience Publishers, Australia, 1999

[5] Böttcher, R., Geometrische Wahrscheinlichkeiten vom Buffonsche Typ in be-grenzten Gittern, FernUniversität in Hagen, 2005, German version in worldwide web:http://deposit.fernuni-hagen.de/frontdoor.php?source opus=39



Design of continuous regression tests by transforming theaccumulated residuals process

Alejandra Cabaña and Enrique M. Cabaña∗

Universidad de Valladolid, Prado de la Magdalena s/n, 47005 Valladolid, Españaand Universidad de la República, Eduardo Acevedo 1139, 11200 Montevideo,[email protected]; [email protected]

2000 Mathematics Subject Classification. 62G10, 62J02

Transformed empirical processes have been used since [1] (see [2] and nestedreferences) in the construction of goodness-of-fit tests, consistent for allalternatives, focused on the sequences of contiguous alternatives chosen bythe user, and asymptotically distribution free under the null hypothesis offit as well as under the alternatives of focusing. After transforming, theselimiting laws are the same, regardless there are estimated parameters ornot.

In this work we consider sequences of linear models with continuousregressors xh, h = 0, 1, . . . , p − 1, that, eventually changing variables toobtain a normalized form, is written as

Yn,j =p−1∑h=0

βhxh(j/(n+1))+σZn,j , Zn,j i.i.d.(j = 1, . . . , n), EZn,1 = 0, EZ2n,1 = 1.

With suitable consistent estimators β̂n,h(h = 0, . . . , p − 1), σ̂ of the pa-rameters, the estimated residuals are ên,j = σ̂−1n (Yn,j −

∑p−1h=0 β̂hxh(j/(n +

1))). We introduce the process of accumulated estimated residuals (AER)r̂n(t) =

∑[nt]j=1 ên,j that under infill asymptotics (n → ∞), converges in law

to a standard Wiener process on [0, 1].The same kind of transformation applied to the empirical process in the

articles referred above is applied in the present work to the AER process,leading to a transformed accumulated residual process that is also asymp-totically Gaussian, from which Cramér-von Mises tests statistics of Watsontype ([3]) are constructed, thus providing tests which are also consistentunder all alternatives, can be focused to the alternatives of interest of theuser, and have distribution free asymptotic laws.

As a particular application, a simple test for polynomial regression ex-ploiting properties of the Legendre Polynomials is constructed and its be-haviour is described empirically.



References

[1] Cabaña, A., Transformations of the empirical measure and Kolmogorov-Smirnov tests, Ann. Statist. 24 (1996) 2020-2035.

[2] Cabaña, A., Cabaña, E.M., Goodness-of-fit to the exponential distribution,focused on Weibull alternatives, Communications in Statistics, Simulation andComputation 34 (2005), 711-724.

[3] Watson, G. S., Goodness - of - fit tests on a circle, Biometrika 48 (1961)109-114.



Free-differentiability conditions on the free-energy functionimplying large deviations

Henri Comman

Department of Mathematics, University of Santiago de Chile, BernardoO’Higgins 3363 Santiago, [email protected]


In this talk we will present our recent results concerning a problem of largedeviation in the real line ([1]). Let (µα) be a net of Radon sub-probabilitymeasures on R and (tα) be a net in ]0,+∞[ converging to 0. Let C(R) denotethe set of [−∞,+∞[-valued continuous functions on R. For each h ∈ C(R),we define Λ(h) = log lim inf µtαα (e

h/tα) and Λ(h) = log lim supµtαα (eh/tα)

where µtαα (eh/tα) stands for (

∫R e

h(x)/tαµα(dx))tα , and write Λ(h) when bothexpressions are equal. For each pair of reals (λ, ν), let hλ,ν be the functiondefined by hλ,ν(x) = λx if x ≤ 0 and hλ,ν(x) = νx if x ≥ 0 (we write simplyhλ in place of hλ,λ). For each real λ, we put L(λ) = Λ(hλ) when Λ(hλ)exists.

A well-known problem of large deviations in R (usually stated for se-quences of probability measures) is the following ([5], pp. 48): assumingthat L(λ) exists and is finite for all λ in an open interval G containing 0,and that the map L|G is not differentiable on G, what conditions on L|G doimply large deviations, and with which rate function?

In relation with this problem, R. S. Ellis posed the following question([4]): assuming that Λ(hλ,ν) exists and is finite for all (λ, ν) ∈ R2, whatconditions on the functional Λ|{hλ,ν :(λ,ν)∈R2} do imply large deviations withrate function J(x) = sup(λ,ν)∈R2{hλ,ν(x)− Λ(hλ,ν)} for all x ∈ R?

In this paper, we solve the above problem by giving conditions on L|Ginvolving only its left and right derivatives; the rate function is obtainedas an abstract Legendre-Fenchel transform Λ|S∗, where S can be any setin C(R) containing {hλ : λ ∈ G}. When S = {hλ : λ ∈ G}, we get astrengthening of the Gärtner-Ellis theorem by removing the usual differ-entiability assumption. The answer to the Ellis question is obtained withS = {hλ,ν : (λ, ν) ∈ R2}.

The techniques used are refinements of those developed in previous au-thor’s works ([2], [3]), where variational forms for Λ(h) and Λ(h) were ob-tained.



References

[1] H. Comman, Free-differentiability conditions on the free-energy function implying large deviations. Available onhttp://www.arxiv.org/abs/math/0506044.

[2] H. Comman, Functional approach of large deviations in general spaces, J.Theoretical Prob. 18 (2005), No. 1, 187-207.

[3] H. Comman, Criteria for large deviations, Trans. Amer. Math. Soc. 355(2003), 2905-2923.

[4] R. S. Ellis, An overview of the theory of large deviations and applications tostatistical mechanics, Scand. Actuarial J. 1 (1995), 97-142.

[5] R. S. Ellis, Entropy, large deviations, and statistical mechanics, Springer-Verlag, New-York, 1985.



Particle approximations for a class of stochastic partialdifferential equations

Dan Crisan

Department of Mathematics, Imperial College London, 180 Queen’s Gate, LondonSW7 2AZ, Great [email protected]

2000 Mathematics Subject Classification. 60H15, 60G35, 62M20, 93E11

Let (Ω,F , P ) be a probability space on which we have defined an m-dimensional standard Brownian motion W which drives the following semi-linear stochastic partial differential equation

dϑt(ϕ) = ϑt(Aϕ)dt+m∑

k=1

(ϑt(γkϕ)−ϑt(γk)ϑt(ϕ)+ϑt(Bkϕ))(dW kt −ϑt(γk)dt).

(1)Here ϑ is a measure valued process, A is a second order differential opera-tor and Bk, k = 1, ...,m are first order differential operators. Equation (1)is known as the Fujisaki-Kallianpur-Kunita or the Kushner-Stratonovitchequation. It plays a central role in nonlinear filtering: The solution of (1)gives the conditional distribution of a stochastic process X (the signal)given an associated observation process W (the process W becomes a Brow-nian motion after a suitable change of measure).

The aim of the paper is to show how one can construct a particle ap-proximation to the solution of (1). The main idea is to merge the weightedapproximation approach, as presented in Kurtz and Xiong [5] for a generalclass of nonlinear stochastic partial differential equation to which (1) be-longs, with the branching corrections approach introduced by Crisan andLyons in [2].

The result is a generalization of existing results (see for example [1, 3, 4])where the differential terms ϑt(Bkϕ), k = 1, ...,m are missing. The additionof these differential terms in (1) is important. They appear in the case wherethe signal noise and the observation noise are correlated. This feature isquite common, for example, in financial applications. To our knowledgethis is the first particle algorithm that treats the correlated case.

References

[1] D. Crisan, T. Lyons, A particle approximation of the solution of the Kushner-Stratonovitch equation. Probab. Theory Related Fields 115, no. 4, 549–578,1999.



[2] D. Crisan, T. Lyons, Minimal entropy approximations optimal algorithms forthe Filtering Problem, Monte Carlo Methods and Applications, Vol 8, No 4 pp343-356, 2002.

[3] D. Crisan, P. Del Moral, T. Lyons, Interacting particle systems approximationsof the Kushner-Stratonovitch equation, Adv. in Appl. Probab. 31, no. 3, 819–838, 1999.

[4] P. Del Moral Feynman-Kac formulae. Genealogical and interacting particle sys-tems with applications. Probability and its Applications, Springer-Verlag, NewYork, 2004.

[5] T. G Kurtz, J. Xiong, Particle representations for a class of nonlinear SPDEs.Stochastic Process. Appl. 83 (1999), no. 1, 103–126.



Hitting probabilities for systems of non-linear stochastic heatequations

Robert C. Dalang*, Davar Khoshnevisan, Eulalia Nualart

Institut de mathématiques, Ecole Polytechnique Fédérale de Lausanne, Station 8,CH-1015 Lausanne, Switzerland; Department of Mathematics, University ofUtah, 155 S 1400 E, Salt Lake City, UT 84112–0090, U.S.A.; Institut Galilée,Université Paris 13, 93430 Villetaneuse, [email protected]; [email protected];[email protected]

2000 Mathematics Subject Classification. 60H

We consider a system of d ≥ 1 coupled non linear stochastic heat equationsin spatial dimension 1, driven by d-dimensional space-time white noise. Thesolution of this system is a process indexed by space-time, with values inRd. The main objective is to determine, for a given subset of Rd, whetheror not this set is hit by the space-time process. Using Malliavin calculus, weobtain in [1, 2] upper and lower bounds on the univariate and bivariate jointdensities of the solution. This leads to upper and lower bounds on hittingprobabilities for the space-time process, in terms of capacity and Hausdorffmeasure of the sets. We also obtain related estimates when one of the space-time parameters is fixed. This makes it possible to determine the criticaldimension above which points are polar, as well as the Hausdorff dimensionsof the range of the process and of its level sets. Similar results were obtainedin [3] for systems of stochastic wave equations in spatial dimension 1.

References

[1] Dalang, R.C., Khoshnevisan, D., Nualart, E. Hitting probabilities for the non-linear stochastic heat equation with additive noise (preprint, 2006).

[2] Dalang, R.C., Khoshnevisan, D., Nualart, E. Hitting probabilities for the non-linear stochastic heat equation with multiplicative noise (preprint, 2006).

[3] Dalang, R.C., Nualart, E. Potential theory for hyperbolic spde’s, AnnalsProbab. 32 (2004), 2099–2148.



Sequential analysis under composite hypotheses

Boris Brodsky, Boris Darkhovsky*

Central Institute for Mathematics and Economics RAS, Institute for SystemsAnalysis RAS∗, Moscow, Russia

2000 Mathematics Subject Classification. 62C25, 62L15

Let Θi ∈ Rn, i = 0, 1,Θ0⋂

Θ1 = ∅,Θ0⋃

Θ1 = Θ are compact sets, f(x, θi), θi ∈Θi are density functions (d.f.) w.r.t. some σ-finite measure µ, for µ-a.s.x f(x, ·) is continuous and positive. The problem of sequential testing ofhypotheses {Hi : θi ∈ Θi}, i = 0, 1 based on i.i.d.r.v. {xk} with d.f. fis considered. Put η(θ) = ln f(x,θi)f(x,θj) , i 6= j, i, j = 0, 1. Suppose that ∀u =(θi, θj , θ∗i ), i 6= j, i, j = 0, 1∃ t(u) > 0 s.t. Eθ∗i exp (t(u)η(θ)) 0, i = 0, 1. Define the stopping times (st.t) Ti = inf

{n : minθ∈Θ

∑nk=1 ln

f(xk,θi)f(xk,θj)

> Ai}

and T = T0∧T1. The decision rule is: to accept Hi at instant T if T = Ti.

Theorem 1. If Ai, i, j = 0, 1, i 6= j is large enough then ∀ st.t.τ withgiven constraints on maximal (over θ) error probabilities the inequalitieshold: maxθi∈Θi E (T |Hi, θi ∈ Θi) ≤ maxθi∈Θi E (τ |Hi, θi ∈ Θi).

This result is a generalization for a composite hypotheses of classi-cal Wald’s result [1]. Denote by Pm,θ(Em,θ) the measure (mathematicalexpectation) corresponding to a sequence {xk}∞k=1 with the change-point(c-p) [2] at the instant m: the d. f. of xn is equal to f(x, θ0), n < mand f(x, θ1), n ≥ m. The problem consists in sequential detection of ac-p m. All known methods of c-p detection contain some ”large param-eter” N (f. e., the threshold). For any θ∗1 ∈ Θ1 define θ∗0(θ∗1) = θ∗0(·) :maxθ0∈Θ0

(∫ln f(x,θ

∗1)

f(x,θ0)f(x, θ∗1)dµ(x)

)−1= I−101 (θ

∗0(·), θ∗1)). Put θ∗ = (θ∗0(·), θ∗1)

and define the false alarm probability αaN(θ∗) = supn P∞,θ∗

{daN(n) = 1

}for

any method a with ”large parameter” N (here daN(n) = 1 (daN(n) = 0) cor-

responds to the decision about the presence (absence) of a change at theinstant n).

Theorem 2.Let τaN = min{n : daN(n) = 1}. If ∀θ∗ ∈ Θ,∀m∃ limN→∞Em,θ∗(τaN−m)

+

N

0 then limN→∞

Em,θ∗(τaN−m)+

| ln αaN (θ∗)|≥ I−101 (θ∗).

A method of sequential c-p detection is called asymptotically optimal(a.o) if the lastinequality turns into a strict equality (the left-hand side can be proposed asa naturalperformance index for c-p problems). Put L(n, θ) = max1≤k≤n

∑ni=k ln

f(xi,θ1)f(xi,θ0)



and T ∗N =inf {n : minθ0∈Θ0 maxθ1∈Θ1 L(n, θ) > N}.

Theorem 3. The detection method coresponding to the st.t. T ∗N is a.o.This result is a generalization of classical CUSUM [3] procedure for the

case of composite hypothesis not only after but also before the c-p.

References

[1] Wald, A., Sequential tests of statistical hypotheses, Ann. Math. Statist. 16(1945, 117-186.

[2] Shiryaev, A.N.,Optimal Stopping Rules. Nauka, Moscow, 1976.

[3] Page, E.S., Continuous inspection schemes, Biometrika, 41 (1954), 100-114.



Wiener chaos solutions of linear forward-backward stochasticdifferential equations

N. E. Frangos*, A. N. Yannacopoulos

Department of Statistics, Athens University of Economics and Business, 76Patission Str, Athens, Greece; Department of Statistics and Actuarial-FinancialMathematics, University of the Aegean, Karlovassi 83200, [email protected]; [email protected]

2000 Mathematics Subject Classification. 65C30

Forward-backward stochastic differential equations (FBSDEs) have attractedthe attention of the stochastic analysis community on account of their in-teresting mathematical theory and their widespread applicability in areassuch as control theory or mathematical finance.

Starting from a recent construction of solutions for linear stochastic dif-ferential equations, by Mikulevicius and Rozovskii (see e.g. [1]), which usesthe Wiener chaos expansion, we propose a new scheme for the solution oflinear forward-backward stochastic differential equations using the decom-position of the solution in a chaos expansion. Through duality arguments,we propose a variational formulation of FBSDEs which when coupled withthe chaos expansion provides a constructive way of solving the system.

References

[1] Mikulevicius R., Rozovskii, B. L. Linear parabolic stochastic PDEs and wienerchaos SIAM J. Math. Anal. 29 (1998), 452–480.



Probability on independent probability spaces

Paul F. Gibson

Department of Mathematics, Delaware State University, Dover, DE 19901, [email protected]

In [1], an earlier paper by the author (publication pending), an algebrastructure called a reliability algebra is introduced. A reliability algebra is aset M together with two binary operations • and ⊕ and a unary operation¯̄satisfying:

R1 : (M, •) is a commutative monoid,

R2 : ¯̄a = a for a ∈M ,

R3 : a⊕ b = ā • b̄ for all a, b,∈M (De Morgan’s law).

The use of reliability algebra for finding the reliability of an organizationalsystem and the conditions necessary for a reliability algebra to be a BooleanAlgebra [1] is discussed in the paper. The functioning components placedin series or parallel can be expressed as a member of the reliability systemM = [0, 1] with a⊕ b = a+ b− a • b and unary − defined by ā = 1− a.

This paper generalized paper [1] in finding expressions for the prob-ability of more general events expressed in terms of a finite number ofindependent events {Ei}ni=1 of probabilities {pi}ni=1.

We define an independent probability space in terms of basic events{Ei}ni=1 and values {pi}ni=1 in [0, 1]. We also define a reliability algebra. Onecan view the probability measure as a function from the reliability algebraB[E1, ···, En] of Boolean polynomials into the reliability algebraM [p1, ···, pn]of polynomials in {pi}ni=1. Conditions under which the probability measureP is a homomorphism from B[E1, · · ·, En] into M [p1, · · ·, pn] is discussedand that P maps symmetric functions into symmetric function is proven.We develop formulas for the image of every symmetric function under P.

References

[1] Gibson P.F., A New Structure for Finding the Reliability of Organization Sys-tems,2005.

[2] Ghahramani, S., Fundamentals of Probability With Stochastic Processes, ThirdEdition, 2005.

[3] Jacobson, N., Basic Algebra 1, W.H. Freeman and Company , 1974.

[4] Folland, G., Real Analysis, John Wiley & Sons, 1984.



A tool for detecting customers’ leaving based on time seriesmodels

Ángeles Calduch, Miguel Ángel Edo, Vicent Giner*, Susana San Mat́ıas

Department of Applied Statistics and Operations Research, and Quality,Polytechnic University of Valencia, Camı́ de Vera, s/n 46022 Valencia, [email protected]

2000 Mathematics Subject Classification. 62M10

Nowadays, customer fidelity is a very important issue within the marketingstrategy of any company (i.e. banking or communication business). There-fore, research on the behavior of customers is a critical task to succeed.

In this work, our aim is to provide marketing researchers with a tool thatallows them to follow customers’ path along the time, in a single way. Wepropose a tool based on time series models which is intended to representthe evolution of the fidelity level of each customer. Our proposal defines anew combined variable that will be taken as the index to be considered andpredicted by the time series model.

The predictions given by the model for each customer at several mo-ments of time in the future are considered in order to determine if thatcustomer is at risk of leaving. In this case, suitable retention commercialactions over that customer are suggested to be performed by the company.

References

[1] Adomavicius, G., Tuzhilin, A. Using Data Mining Methods to Build CustomerProfiles. Computer. 34 (2001), 74–82.

[2] Box, G.E.P., Jenkins, G.M., Time Series Analysis : Forecasting and Control.Holden-Day Series in Time Series Analysis, Holden-Day, San Francisco, 1970

[3] Greene, W., Econometric Analysis. Prentice Hall, 2003



On a characterization of the generalized gamma distribution

Tea-Yuan Hwang*, Ping-Huang Huang

Institute of Statistics, National Tsing Hua University, Hsinchu, 30043, Taiwan,R. O. C.; Department of Statistics and Insurance, Aletheia University, Tamsui,Taipei, 25103, Taiwan, R. O. [email protected]

2000 Mathematics Subject Classification. 62E10

The generalized gamma distribution is a fairly flexible family of distributionwhich includes as special cases the exponential, Weibull, gamma, and log-normal distributions. They are widely used and play an important role in thereliablility field and the survival analysis, therefore a successful estimationof its parameters will be very important.

Ten years ago, we at first do research on gamma distribution, using anon-linear transformations published in Reference 1, and concentrated onits characterization. The results: The independence of sample coefficient ofvariation, sample Gini’s mean difference and sample range Rn and samplemean Xn respectively are equivalent to gamma distribution are obtained.More results are presented in the References 2, 3, 4, and 5.

Since same result holds for generalized gamma distribution can be con-cluded with same procedure using for gamma distribution, therefore moreresults are obtained such as the p.d.f., mean, variance, skewness and kurto-sis of the coefficient of sample range Rn/Xn for mall sample under gammaand its application to quality control. Further research will be concentratedon with the similar topics mentioned above under Weibull distribution inthe near future.

References

[1] Hwang, T.Y. and Hu, C.Y., (1994). On the joint distribution of studentizedorder statistics, Ann. Inst. Statist. Math.,46(1),165-177.

[2] Hwang, T.Y. and Hu, C.Y., (1999). On the characterization of the gammadistribution: The independence of the sample mean and the sample coefficientof variation, Ann. Inst. Statist. Math., 51(4), 749-753.

[3] Hwang, T.Y. and Hu, C.Y., (2000). On some characterizations of populationdistributions, Taiwanese J. Math. 4(3),427-437.

[4] Hwang, T.Y. and Hu, C.Y., (2002). On new moment estimation of parametersof gamma distribution using its characterization, Ann. Inst. Statist. Math.,54(4),840-847.



[5] Huang, P.H. and Hwang, T.Y., (2002). The inference of Gini’s mean difference,Internat. J. of Pure and Applied Math., 25(1), 39-49.



Robustness in statistical forecasting of time series

Yuriy Kharin

Department of Mathematical Modelling and Data Analysis, Belarusian StateUniversity, Independence avenue, 4, Minsk 220030, [email protected]

2000 Mathematics Subject Classification. 62M20, 62F35

Statistical forecasting is intensively used to solve many significant appliedproblems. Optimality of the majority of the known statistical forecastingprocedures is proved w.r.t. the mean square risk of prediction under theassumptions of the underlying hypothetical model. In practice, however, theobserved time series satisfy the hypothetical models with distortions of somelevel ε [1]. Unfortunately, the forecasting procedures, which are optimalunder hypothetical models, often lose their optimality and become unstableunder “a little distorted hypothetical models”. That is why the followingmain problems considered in this paper are very topical: A) robustnessevaluation of the traditionally used statistical forecasting procedures underdistortions; B) evaluation of δ-admissible distortion levels; C) constructionof new robust forecasting procedures by the minimax risk criterion.

By the method of asymptotic expansions of the risk functional w.r.t. ε[2, 3] we solve these problems for the following cases:

• trend models of time series under “outliers” and functional distortionsof trends;

• multivariate linear regression models under “interval” distortions, “rel-ative” distortions and distortions in lp-metrics;

• autoregressive time series under non-homogeneities in the innovationprocess and under bilinearities;

• vector autoregressive time series under “misspecifications”;

• vector autoregressive time series under missing values.

Theoretical results are illustrated by the results of computer experi-ments on simulated and real time series.

References

[1] Huber, P., Robust Statistics, Wiley, New York, 1981.



[2] Kharin, Yu., Robustness in Statistical Pattern Recognition, Kluwer, Dordrecht,1996.

[3] Kharin, Yu., “Plug-in” Statistical Forecasting of Vector Autoregressive TimeSeries with Missing Values, Austrian Journal of Statistics. 34 (2005), 163–175.



Large deviations of dependent discrete random variables

A. V.Kolodzey

Lebedev Institute of Precision Mechanics and Computer Engineering, RussianAcamemy of Sciences, 119991, Moscow, Leninsky pr-t, 51, [email protected]


Let ξ1, ξ2, . . . are independent Poisson variables of means λ1, λ2, . . . . Fol-lowing [1], we consider for n > 1 random variables µ̄n = (µ1n ,

µ2n , . . . ,

µnn )

such that

P{µν = kν , ν = 1, . . . , n} = P{ξν = kν , ν = 1, . . . , n|n∑

ν=1

νξν = n}.

Consider the function ρ(x̄, ȳ) = supν |lnxνyν |, where 0 ln 0, ln00 are taken to

be 0 and | ln 10 |, | ln 0| are taken to be +∞. It is easy to prove that functionρ(x̄, ȳ) is a metric which may reach infinite value on

Ω = {x̄ = (x1, . . . ), xν ≥ 0, ν = 1, 2, . . . ,∞∑

ν=1

νxν = 1}.

Let function φ(x̄) be continuous on space Ω with metric ρ(x̄, ȳ) and forany x̄ ∈ Ω

limn→∞

φ(x1, ..., xn, 0, 0, . . . ) = φ(x̄).

Then for any sequence an that tends to a as n→∞

limn→∞

− 1n

lnP{φ(µ̄n) > an} = J(λ̄, {x̄ ∈ Ω, φ(x̄ > a})− J(λ̄,Ω),

where λ̄ = (λ1, λ2, . . . ), J(x̄, ȳ) is Kullback-Leibler divergence [2],

J(x̄, ȳ) =∞∑

ν=1

xν lnxνyν, J(x̄, A) = inf

ȳ∈AJ(x̄, ȳ).

References

[1] Ivchenko G.I.,Medvedev Yu.I., Random Combinatorial Objects. Doklady Math-ematics. Interperiodica Translat. 69 (2004), 2, 344–347.

[2] Kullback S., Information Theory and Statistics. New York, Dover, 1968.



The configurational measure on mutually avoiding SLE paths

Michael J. Kozdron

Department of Mathematics and Statistics, University of Regina, College West307, Regina, Saskatchewan, Canada S4S [email protected]

2000 Mathematics Subject Classification. 60G50, 60J45, 60J65

We discuss a new approach to the study of multiple chordal SLEs in a simplyconnected domain by considering configurational measures on paths insteadof infinitesimal descriptions. We construct these measures so that they areconformally covariant, and satisfy boundary perturbation and Markov prop-erties, as well as a cascade relation. As an application of our construction,we derive the scaling limit of Fomin’s identity [1] in the case of two pathsdirectly.

Theorem (Kozdron-Lawler): If γ : [0,∞) → H is an SLE2 in the upperhalf plane H from 0 to ∞, and β : [0, 1] → H is a Brownian excursion fromx to y in H where 0 < x < y


[2] M. J. Kozdron. On the scaling limit of simple random walk excursion measurein the plane. To appear in Latin American J. Probab. Math. Stat. (ALEA),2006.

[3] M. J. Kozdron and G. F. Lawler. Estimates of random walk exit probabilitiesand application to loop-erased random walk. Electron. J. Probab., 10:1442–1467, 2005.

[4] M. J. Kozdron and G. F. Lawler. The configurational measure on mutuallyavoiding SLE paths. Preprint, 2006.

[5] G. F. Lawler, O. Schramm, and W. Werner. Conformal invariance of planarloop-erased random walks and uniform spanning trees. Ann. Probab., 32:939–995, 2004.



Lévy white noise measures on infinite dimensional spaces

Yuh-Jia Lee

Department of Applied Mathematics, National University of Kaohsiung, No. 700,Kaohsiung University Rd., Kaohsiung, Taiwan [email protected]

2000 Mathematics Subject Classification. Primary: 28C20, 46T12; Sec-ondary: 60H40, 46G12

It is shown that a Lévy white noise measure Λ[1] always exist as a Borelmeasure on the dual K′ of the space K of C∞ functions on R with compactsupport. Then a characterization theorem that ensures that the measurablesupport of Λ is contained in S ′ is proved. In the course of the proofs, arepresentation of the Lévy process as a function on K′ is obtained andstochastic Lévy integrals are studied. The results is a key to extend [2] toa more general class of Lévy white noise functionals, such as functionalsassociated with stable processes.

References

[1] Y.-J. Lee, H.-H. Shih, The Segal-Bargmann transform for Lévy functionals, J.Funct. Anal. 168(1999), 46-83

[2] Y.-J. Lee, H.-H. Shih, Analysis of generalized Lévy white noise functionals, J.Funct. Anal. 211(2004), 1-70



A mechanical model of Markov processes

Shigeo Kusuoka, Song Liang*

Graduate School of Mathematical Sciences, the University of Tokyo, Tokyo,Japan; Graduate School of Information Sciences, Tohoku University, Sendai,[email protected]; [email protected]

2000 Mathematics Subject Classification. 60J25, 70F10

We consider the motion of several massive particles (molecules) in an idealgas of identical point particles (atoms) in a d-dimensional Euclidean space,with certain interactions. It is well-believed that the motion of the moleculesconverges to a Markov process when the mass m of atoms converges to 0,heuristically because the central limit theorem for “independent identicallydistributed” atoms. However, the atoms are actually not independent, sincethey could be the ones interacted with some molecule(s) already.

This problem was first considered by Holley, in the case that the particlesare moving in a one dimensional space. The corresponding question forgeneral dimensional case has been considered by, e.g., [1], [2], but for thesystem of collision with only one molecule.

In this study, we consider the problem of multi-molecules without theindependent assumption (which, as explained, actually does not hold). Weprove the existence of the solution of the corresponding equation, and studythe limit when m converges to 0.

References

[1] Claderoni, P., Durr. D., Kusuoka, S., A mechanical Model of Brownian Motionin Half-Space, J. Statist. Phys. 55 (1989), 649–693.

[2] Durr, D., Goldstein, S., Lebowitz, J. L. A Mechanical Model of Brownian Mo-tion, Comm. Math. Phys. 78 (1980/81), 507–530.



An analysis of the last round matching problem

Wenbo Lia, Fengshan Liu* b, Xiquan Shib

bDepartment of Mathematical Sciences, University of Delaware, Newark, DE19716, USA; bDepartment of Applied Mathematics and Theoretical Physics,Delaware State University, Dover, DE 19901, [email protected]

2000 Mathematics Subject Classification. 91A06

The matching problem or the “Hats Problem” goes back to at least 1713when it was proposed by the French mathematician Pierre de Montmortin his book [1] on games of gambling and chance. Detailed progress madeon this problem can be found in [2] and [3]. In this paper, we considerthe multi-round matching problem: Suppose that each of n men at a partythrows his hat into the center of the room. The hats are first mixed up, andthen each man randomly selects a hat. Suppose that those choosing theirown hats depart, while the others put their selected hats in the center of theroom, mixed the hats up, and then reselect. Also, suppose that this processcontinues until each individual has his own hat. Assume we start with nmen. We study the probability distribution Ln of the number of playersin the last round of the matching problem and obtain the existence of thelimiting distribution by using convolution method. We give the recursiverelations for the expectation E(Ln) and estimate the distribution P (Ln =m). We show that the limiting distribution of Ln exists and the expectationE(Ln) converges to a constant l ≈ 2.26264703816.

References

[1] P. R. de Montmort. Essay d’Analyse sur les Jeux de Hazard. Chelsea PublishingCo., New York, second edition, 1980.

[2] Ross, S., Introduction to Probability Models, eighth edition, Academic Press,2003.

[3] Hathout, H., The old hats problem revisited, College Math. J., 35, 97-102,2004.



Optimum designs for the description of sorption data usingisotherm equations

Licesio J. Rodŕıguez-Aragón, Jesús López-Fidalgo*

Department of Mathematics, University of Castilla-La Mancha, E.T.S.Ingenieros Industriales, Av./ Camilo José Cela s/n, E-13071 Ciudad Real, [email protected]; [email protected]

2000 Mathematics Subject Classification. 62K05

Gas-solid adsorption can be described phenomenologically in terms of anadsorption function n = f(P, T ) where n is the amount adsorbed, P thepressure and T the temperature at which the process occurs. As a mat-ter of experimental convenience, the adsorption isotherm is determined byn = fT (P ) and in a detailed study this is done for several temperatures.Adsorption isotherms are widely used to determine the surface area andpore size distribution of a variety of different solid materials, to character-ize retention of chemicals in soils, to model the water transport occurringin foods during processing, and many other industrial processes. Thereforea correct characterization is crucial for the best quality of many goods.

Simple equations, such as Langmuir, BET [1] or GAB[2] are commonlyused to describe adsorption data. Certain parameters in these equations areconsidered independent from P and their estimation has a great importancefor the characterization of the mentioned phenomena.

In order to improve the accuracy of the parameters estimations, op-timum designs have been discussed [3]. In this work T−optimum designsmaximizing the differences between competing equations have been cal-culated to discriminate. Besides, for each equation, D−optimum designsminimizing the determinant of the covariance matrix of the estimates andc−optimum designs to individually estimate a parameter have been found.All the optimum designs have been checked using the Equivalence theorem.

Traditional used common experimental designs have been compared tothe optimum designs obtained, and as result, a valuable tool has been pro-vided for researches to choose the most appropriate design compared to anoptimum with the help of the efficiency concept.

References

[1] Adamson, A. W., Phisical Chemistry of Surfaces. Wiley, 1990.

[2] Guggenhenim, E. A., Application of Stratistical Mechanics. Clarendon Press,Oxford, 1966.



[3] Atkinson, A. C., Donev, A. N., Optimum Experimental Designs. ClarendonPress, Oxford, 1992.



Incomplete samples from stationary sequences and extremevalues

Pavle Mladenović*, Vladimir Piterbarg

University of Belgrade, Faculty of Mathematics, Studentski trg 16, 11000Belgrade, Serbia and Montenegro; Lomonosov Moscow State University, [email protected]

2000 Mathematics Subject Classification. 60G70

We consider a strictly stationary random sequence (Xn) with the marginaldistribution function F (x) = P{X1 ≤ x}. Suppose that some of randomvariables X1, X2, X3, . . . can be observed, and let εk be the indicator of theevent that random variable Xk is observed and Sn = ε1 + · · ·+ εn. Supposethat εk is independent of the sequence (Xn). Let us denote M̃n = max{Xj :1 ≤ j ≤ n, εj = 1}, and Mn = max{X1, . . . , Xn}. Suppose that d.f. Fbelongs to the maximum domain of attraction of some of extreme valuedistributions. The limiting distribution of the random vector (M̃n,Mn) and”asymptotic independency” of M̃n and Mn are obtained under a conditionimposed on asymptotic behavior of the sum Sn and a condition of weakdependency of random variables from the sequence (Xn).

References

[1] Hüsler, J. and Piterbarg, V., Limit theorem for maximum of the storage processwith fractional Brownian motion as input. Stoch. Proc. Appl. 114 (2004), 231-250.

[2] Mladenović, P. and Piterbarg, V., On asymptotic distribution of maxima ofcomplete and incomplete samples from stationary sequences. Preprint.



Stochastic comparisons of Jones-Larsen’s model

Félix Belzunce, Julio Mulero*, José M. Ruiz

Dpto. Estad́ıstica e I.O., Universidad de Murcia, Campus de Espinardo 30100Espinardo (Murcia), [email protected]; [email protected]; [email protected]


Recently, in [1], a general family of distributions for the empirical modellingof ordered multivariate data is proposed. This model extends the joint distri-bution of order statistics from an independent and identically distributedunivariate sample. This model was considered first by [5] and [2] from atheoretical point of view. Let us consider n + 1 real positive parametersa = (a1, . . . , an+1), and consider the joint density

fa(x1, . . . , xn) =Γ(a1 + · · ·+ an+1)∏n

j=1 Γ(aj)

n∏

j=1

f(xj)

×n∏

j=2

{F (xj)− F (xj−1)}F (x1)(1− F (xn)),

on x1 < x2 < · · · < xn, where F is a continuous distribution function andf the corresponding density function.

The purpose of this work is to provide conditions for the stochasticcomparison of two of these models. In particular, given two joint densitiesas above, based on distributions functions F and G, we provide conditionson F and G, to compare in stochastic, likelihood ratio and hazard rateorders (see [3] and [4]), the corresponding random vectors or the marginaldistributions.

References

[1] Jones, M.C., Larsen, P.V., Multivariate distributions with support above thediagonal, Biometrika 91 (2004), 975–986.

[2] Papadatos, N., Intermediate order statistics with applications to nonparametricestimation, Statistics and Probabilty Letters 22 (1995), 231–238.

[3] Müller, A., Stoyan, D., Comparison Methods for Stochastic Models and Risks.Wiley Series in Probability and Statistics. John Wiley & Sons, Ltd., Chichester,2002.

[4] Shaked, M., Shanthikumar, J.G., Stochastic Orders and Their Applications.Academic Press, INC, Boston, MA, 1994.

[5] Stigler, S.M., Fractional order statistics, with applications, Journal of theAmerican Statistical Association 72 (1977), 544–550.



Limit theorems for sequences of blockwise negatively associatedrandom variables

A. Nezakati

Faculty of Mathematics, Shahrood University of Technology, Shahrood, P.O.BOX36155-316, [email protected]


In this paper, we defined blockwise negatively associated random variablesand prove Marcinkiewicz strong law and classical strong law of large num-bers for a sequence of blockwise negatively associated random variables.

References

[1] Joag-Dev, K., Proschan, F. Negative association random variable with appli-cation, Ann.Statist. 11 (1983), 286–295.

[2] Liu, J., Gan, S., Chen, P.The Hajek-Renyi inequality for the NA random vari-ables and its application, Statist. Probab. Letters, 43 (1999), 99–105.

[3] Loeve, M., Probability Theory. Foundations. Random Sequences. Van Nostrand,New York, 1955.

[4] Matula, P., A note on the almost sure convergence of negatively dependentvariables, Statist. Probab. Letters, 15 (1992) 209–213.

[5] Newman, C.M., Asymption independence and limit theorems for positivelyand negatively dependent random variables, In: Y.L.Tong(Ed.), Inequalitiesin Statistics Probability, Institute of Mathematics Statistics. Hayward, CA,(1984) 127–140.



A problem of stochastic approximation in genetics

Gabriel V. Orman*, Irinel Radomir

Department of Mathematical Analysis and Probability, ”Transilvania” University,50 Iuliu Maniu Street, 500091 Brasov, [email protected]

2000 Mathematics Subject Classification. 60J65, 60J70

We refer to an application to genetics discussed especially by W. Feller, D.Ludwig, M. Mangel, Z. Schuss, A.T. Bharucha-Reid; also some problemsof convergence are discussed by G.V. Orman. Now we emphasize some as-pects concerning the approximation of Markov chains by a solution of astochastic differential equation to determine the probability of extinctionof a genotype. Thus, the Markovian nature of the problem will be pointedout again, and we think that this is a very important aspect.

Let us consider a population of N individuals consisting of XN (n) = iindividuals of type A in the nth generation. Then the next generation con-sists of N individuals randomly selected from a practically infinite offspringof the previous generation. Obviously, the selection process is binomial andthe probability of extinction of a genotype, or the time until extinction be-come very hard to calculate.

For this reason the Markov chain {XN (n)} can be approximated by adiffusion process, or more exactly, by a solution of a stochastic differentialequation. We come to a problem of convergence and it is shown that theconvergence is sufficiently rapid as to satisfy some imposed conditions pro-vided that a specific stochastic differential equation has a unique solution.Thus, the existence and uniqueness of the solution are shown.

A natural conclusion is found again: if x(t) is the solution of a specificstochastic differential equation with absorbing boundaries at x=0 and x=1,then the probability of extinction is the probability of exit of x(t) from theinterval (0,1).

Remark. Obviously, various situation may exist when the survival of aparti-cular genotype can be very dinamic. In general, the interaction of apopulation can have a great complexity, which lead to the enhancement ofthe interdisciplinary coordination in these studies.

References

[1] Bharucha-Reid, A. T., Elements Of The Theory Of Markov Processes AndTheir Applications, Dover Publications, Inc., Mineola, New York, 1997.



[2] Feller, W., Introduction to Probability Theory and Its Applications, vol.I, JohnWiley & Sons, Inc. New York, London, 1960.

[3] Mangel, M., Ludwig, D., Probability of extinction in a stochastic competition,SIAM J. Appl. Math. 33 (1977), 92, 256-266.

[4] Orman, G. V., Capitole de matematici aplicate, MicroInformatica, Ed. Albas-tra, Cluj-Napoca, 1999.

[5] Schuss, Z., Theory and Applications of Stochastic Differential Equations, JohnWiley & Sons, New York, 1980.



Bayesian sample size determination for a 2 × 2 contingency tablewith dependent proportions

T. Pham-Gia

Université de Moncton, [email protected]

2000 Mathematics Subject Classification. K62c10

Several measures used in the analysis of a 2×2 contingency table are basedon the two conditional probabilities, π1|1 and π1|2 , with π1|1 and π1|2 de-pendent,or their ratio θ = π1+/π+1. They can also use the relative riskassociated with rows ρ, defined by ρ = π1|1/π1|2 =

π11π21.π2+π1+ , or the odds

ratio ψ = π1|1/π2|1π1|2/π2|2 =π11π21.π22π12 . In a Bayesian context, we consider the vector

π = (π11, π12, π21, π22), with∑2

i=1

∑2j=1 πij = 1. Let the prior distribution

of π be a Dirichlet distribution, denoted π ∼ Dir (α1, α2, α3, α4) with αi >0, 0 ≤ πi ≤ 1, i = 1, ..., 4, and density f (π1, π2, π3) =

∏3i=1 π

αi−1i .

(1−

∑3i=1 πi

)α4−1/B (α1, α2, α3, α4) ,

defined in the simplex∑3

i=1 πi ≤ 1 .Sample size determination: We now proceed in two steps:

A) ” Mapping out ” the positive axis:a) First, for each measure,we compute the (1− ε)100% hpd interval for ζ,using the expression of its posterior density fζ(α1∗, α2∗, α3∗, α4∗; t).Let `ζ (α1∗, α2∗ , α3∗, α4∗ ; n, {x}) be the total length of that interval.b) Secondly, for each of the above measures, we compute the expectation ofthis length: Kζ (n) = EX [`ζ (α, β;n, x)] , using the marginal Multinomial-Dirichlet distribution of (X1, X2, X3, ) .c) Thirdly, we let n vary from 1 to 30. We now have a curve Cζ1 , withvertical coordinate Kζ (n) , function of n , with 0 ≤ n ≤ N.B) Determination of the size n: Since each Kζ (n) is an decreasing functionof n , considered here as a continuous variable, if η1 is the desired valueof the average precision, we only need to numerically solve : Kζ (n) ≤ η1,and, in most cases, we have an interval open to the right as solution. Ournumerical application concerns well-known Salk’s vaccine results.

References

[1] Agresti, A. and Min, Y., Frequentist Performance of Bayesian Confidence In-tervals for Comparing Proportions in 2 x 2 Contingency Tables, Biometrics,61, 2005, 515-523.



[2] Chin, T., Marine, W., Hall, E., Gravelle, C. and Speers, J., The Influence ofSalk Vaccination on the Epidemic Pattern and the Spread of the Virus in theCommunity, the American Journ. of Hygiene, 1961, 73, 67-94

[3] Pham-Gia, T. , Bayes Inference, in Handbook of the Beta Distribution, A.K.Gupta and S. Nadarajah, Ed., Marcel Dekker, New York, 2004, 361-422.

[4] Pham-Gia, T. and Turkkan, N., Sample Size Determination in Bayesian Statis-tics, The Statistician, 1992, 41, 389-397.



Generalized Beta distributions, random continued fractions andThomae invariance

Claudio Asci, Gerard Letac, Mauro Piccioni*

Università di Roma La Sapienza and Université Paul Sabatier, [email protected]


The aim of this paper is to generalize the following well known result aboutGamma and Beta distributions. If the independent random variables Xand G1 are distributed as Beta βa,a′ and Gamma γa+a′,1, respectively, thenXG1 ∼ γa,1. Furthermore, if the random variable G2 is distributed as γa′,1and it is independent of (X,G1), then

G2 +XG1 ∼ γa+a′,1, (1)G2

G2 +XG1=

11 +X G1G2

∼ βa′,a. (2)

and these two random variables are independent. Defining the random vari-able

W ′ ≡ G1G2

∼ β(2)a+a′,a′ , (3)

the beta law of the second kind β(2)b,a being the law of the ratio of twogamma variables with common scale parameter and shape parameters band a, respectively, we can reexpress (2) in the following way. If W ′ has thelaw (3) and it is independent of X ∼ βa,a′ , then

11 +W ′X

∼ βa′,a. (4)

Moreover, if W ∼ β(2)a+a′,a is independent of (X,W ′), then

11 + W1+W ′X

∼ X. (5)

This relation (5) characterizes the law βa,a′ [1]. A similar characterizationholds for the generalized inverse Gaussian distribution [2]. In this paperwe generalize this result to the general case W ∼ β(2)b,a , W ′ ∼ β

(2)b,a′ , with

b > 0 not necessarily equal to a+ a′. We prove that there is again a uniquelaw (parametrized by (a, a′, b)) for X, supported by the positive real line,which ensures the equality in law (5). This result allows a probabilisticinterpretation for a result due to Thomae (contained e.g. in [3]) concerningthe symmetry of certain hypergeometric functions.



References

[1] Chamayou, J.F. and Letac, G., Explicit stationary distributions for compositionof random functions and products of random matrices, J. Theor. Probab. 4(1991), 3–36.

[2] Letac, G. and Seshadri, V., Characterization of the Generalized Inverse Gaus-sian distribution and continued fractions, Z. Wahrsch. verw. Geb. 62 (1983),485–489.

[3] Maier, R. S., A generalization of Euler’s transformation, Trans. Amer. Math.Soc.. 358 (2005), 39-57.



A comparative analysis of different allotment methods forpreferential vote

Victoriano Ramı́rez González

Dept. Matemática Aplicada, University of Granada, Edif. Politécnico - Campusde Fuentenueva, [email protected]

We show different proportional methods for preferential vote and we give ananalysis of them. These methods can be used for unipersonal or pluriper-sonal elections. We study their behaviour and some properties like Con-dorcet, proportionality, manipulation, and so on. Then, we propose appli-cations to political elections and we obtain the winners.

References

[1] Balinski, M. L., Young H. P., Fair Representation: Meeting the Ideal of OneMan One Vote. New Haven, CT, 1982.

[2] Brams, S., Fishburn, P., Approval Voting. Birkhäuser, Boston, 1983.

[3] Ramı́rez V., Palomares, A., The Senatorial Election in Spain. ProportionalBorda methods for selecting several candidates, to appear in Annals of Opera-tions Research, 2006.

[4] Saari, D., Geometry of Voting. Springer-Verlag, 1994.

[5] Taylor A., Mathematics and Politics. Springer-Verlag, 1995.



On maxima and sum of random number of random variables

Sreenivasan Ravi

Department of Studies in Statistics, University of Mysore, Manasagangotri,Mysore 570006, [email protected]

2000 Mathematics Subject Classification. 60F

In this article, limit distributions of sum and maxima of random numberof i.i.d. random variables are studied. We consider several cases when therandom summand is (i) Geometric, (ii) Negative Binomial, (iii) Poisson,(iv) Discrete Uniform. Under the Poisson case, the random summand doesnot play any role in the limit distribution. Interesting limit laws arise andwe study these limit laws and their properties. An iterative result which isalso an invariance principle is obtained in the sequel. Domains of attractionof the limit laws are also characterized.

Keywords and phrases : Random Maxima, Limit Distributions, Geometric,Negative Binomial, Poisson, Discrete Uniform, random summands, Itera-tive property, Invariance Principle, Domains of Attraction, Extreme ValueTheory, Stable Laws.

References

[1] Christoph, G.; and Falk, M. (1996). A note on domains of attraction ofp-max stable laws. Stat. Probab. Lett. Vol. 28 279-284.

[2] Mohan, N.R.; and Ravi, S. (1993). Max domains of attraction of univari-ate and multivariate p-max stable laws. Theory Probab. Appl. Vol. 37, No.4,Translated from Russian Journal 632-643.

[3] Pancheva, E.I. (1985). Limit theorems for extreme order statistics undernonlinear normalization. Lecture Notes in Mathematics Vol. 1155 284-309.



On rough paths and stochastic PDEs

Massimiliano Gubinelli, Samy Tindel*

IECN, University of Nancy 1, BP 239, 54506 Vandœuvre-lès-Nancy, [email protected]

2000 Mathematics Subject Classification.

This talk is based on the preprint [2], which is a continuation of [1]. We aimat a pathwise definition of stochastic PDEs, based on rough path type tech-niques inspired by [3]. We consider then an equation on a given separableHilbert space B, of the form

dyt = Ayt + f(yt) dxt, y0 = ψ (1)

for t ∈ [0, T ], where A is the generator of an analytical semi-group on B,f : B → B is a regular coefficient, ψ a smooth enough initial condition,and x is a general irregular control, which is supposed to be γ-Hölder con-tinuous in time, with γ > 1/3. We show that, up to some algebraic andanalytic manipulations, one can get some existence and uniqueness resultsfor equation (1), provided some operator-valued increments of the type

nts =∫ t

sS(t− u)dxuS(u− s)

can be defined, with a given space-time regularity. These results are appliedto the basic example of the stochastic heat equation in dimension 1 drivenby an infinite dimensional fractional Brownian motion.

References

[1] Gubinelli, M., Lejay, A. and Tindel, S., Young integrals and SPDEs, to appearat Potential Analysis.

[2] Gubinelli, M. and Tindel, S., Rough evolution equations. Preprint.

[3] Lyons, T. and Qian, Z., System control and rough paths, Oxford UniversityPress, 2002.



Some results of generalized mixtures of Weibull distributions

Manuel Franco Nicolás and Juana Maŕıa Vivo Molina*

Departamento de Estad́ıstica e Investigación Operativa, Universidad de Murcia,Campus de Espinardo, 30.100 Murcia, Spain; Departamento de MétodosCuantitativos para la Economı́a, Universidad de Murcia, Campus de Espinardo,30100 Murcia, Spain[[email protected]

2000 Mathematics Subject Classification. 62E10, 60E05

Weibull distributions and their mixtures play a great role in the reliabilitytheory to model lifetime and failure time data, since they can incorporatewide varieties of failure rate functions.

Some extensions of these distributions arise under the formation of se-ries, parallel and other structures of systems, see e.g. [4], which are mixturesallowing negative weights (generalized mixtures). Generalized mixtures ex-tend the used models to describe the probabilistic behavior, providing moreflexibility in their estimations.

Moreover, the generalized mixtures of Weibull distributions with com-mon shape parameter can be considered as extensions of the generalizedmixtures of exponentials. In the literature, generalized mixtures of expo-nential distributions have been characterized and related results have beenobtained (among others, see [1], [2], [3] and [5]).

Nevertheless, the generalized mixtures of Weibull distributions have notbeen studied before. Therefore, in this work, we prove some results for thatan arbitrary finite mixture of Weibull distributions with common shapeparameter to be a valid probability model.

References

[1] Baggs, G. E., Nagaraja, H. N., Reliability properties of order statistics from bi-variate exponential distributions, Comm. Statist. Stochastic Models 12 (1996),611–631.

[2] Bartholomew, D. J., Sufficient conditions for a mixture of exponentials to be aprobability density function, Ann. Math. Statist. 40 (1969), 2183–2188.

[3] Franco, M., Vivo, J. M., Reliability properties of series and parallel systemsfrom bivariate exponential models, Comm. Statist. Theory Methods 31 (2002),2349–2360.

[4] Jiang, R., Zuo, M. J., Li, H. X., Weibull and inverse Weibull mixture modelsallowing negative weights, Reliab. Engrg. Syst. Safety 66 (1999), 227–234.



[5] Steutel, F. W., Note on the infinity divisibility of exponential mixtures, Ann.Math. Statist. 38 (1967), 1303–1305.



Growth of Lévy trees

Thomas Duquesne, Matthias Winkel*

Department of Statistics, University of Oxford, 1 South Parks Road, Oxford OX13TG, [email protected]

2000 Mathematics Subject Classification. 60J80

Evans, Pitman and Winter [4] developed a framework of compact real treesand the Gromov-Hausdorff metric for probabilistic applications. In [3] weconstruct random locally compact real trees called Lévy trees that are thegenealogical trees associated with continuous-state branching processes.

More precisely, we define a growing family of discrete Galton-Watsontrees with independent identically exponentially distributed branch lengthsthat is consistent under Bernoulli percolation on leaves; we define the Lévytree as the limit of this growing family with respect to the Gromov-Hausdorfftopology on metric spaces. This elementary approach notably includes su-percritical trees and does not make use of the height process introducedby Le Gall and Le Jan to code the genealogy of (sub)critical continuous-state branching processes and used in subsequent publications, notably [2]to study Lévy trees. The special case of Brownian trees was studied in [5],where the growing family of binary Galton-Watson trees was shown to havea property of independent growth increments expressed in form of a compo-sition rule. In the general case of [3] we get a Markovian growth structure.

Using Aldous’s [1] `1-representatives of continuum random trees, weconstruct the mass measure of Lévy trees and we give a decompositionalong the ancestral subtree of a Poisson sampling directed by the massmeasure.

References

[1] Aldous, D. J., The continuum random tree I, Ann. Probab. 19 (1991), 1-28.

[2] Duquesne, T., Le Gall, J-F., Probabilistic and fractal aspects of Lévy trees,Probab. Th. Rel. Fields 131-4 (2005), 553-603.

[3] Duquesne, T., Winkel, M., Growth of Lévy trees, Preprint available on ArXivat http://www.arxiv. org/abs/math/0509518

[4] Evans, S.N., Pitman, J., Winter, A., Rayleigh processes, real trees, and rootgrowth with re-grafting, Probab. Th. Rel. Fields 134-1 (2005), 81-126.

[5] Pitman, J., Winkel, M., Growth of the Brownian forest, Ann. Probab. 33(2005), 2188-2211.



On a successive approximation of the solution for a stochasticdifferential equation

Mounir Zili

University Studies Department, Preparatory Institute for Military Academies,Avenue Maréchal Tito, 4029 Sousse, [email protected]

2000 Mathematics Subject Classification. 60H20, 60G20

In this paper, in a first theorem we give a successive approximation solutionof the d− dimensional stochastic differential equation:

dX(t) = σ(t,X(t))dB(t) + α(t,X(t))dt, on t ≥ 0, X(0) = ξ ∈ Rd,

where, d ∈ N \ {0}, σ : R+ ×Rd −→ Rd ⊗Rd and α : R+ ×Rd −→ Rdare bounded measurable functions satisfying: ∃λ > 0, such that

∀δ ∈ Rd,∀t ∈ [0,+∞[,∀y ∈ Rd, δ?σ(t, y)δ ≥ λ | δ |2 .

We shall define the approximate solutions Xn for n ∈ N \ {0} by:

Xn(t) = ξ on − 1 ≤ t ≤ 0 and

Xn(t) = ξ+∫ t0σ(u,Xn(u− 1

n))dB(u)+

∫ t0α

(u,Xn(u− 1

n))du on t ≥ 0.

It shall be shown that

limn→+∞

E(| Xn(t)−X(t) |2

)= 0 for any t ≥ 0

whenever the pathwise uniqueness of the solution holds.Compared with the standard Itô’s successive approximation technique,

the advantage of this method is: given any n ∈ N \ {0}, one can calculateXn by step-wise iterated Itô integrals over the intervals

[0,

1n

[,[ 1n,2n

[, ...,

etc. We see there is no need to calculate Xi(t), i < n at all.In a second theorem, we give a successive approximation solution of

a 1− dimensional stochastic differential equation involving the local timein 0 of the unknown process. Our method consists in using a Zvonkin-transformation in order to come back to a stochastic differential equationsatisfying the conditions of the first theorem.

We consider that this work pave the way for the resolution of the succes-sive approximations convergence problem of an important class of stochasticdifferential equations involving the local time of the unknown process.



References

[1] Zili, M., Approximation of solutions of stochastic differential equations. Inter-national Journal of Differential Equations and Applications, V. 8, No. 3, 2003,271-287.

[2] Zili, M., Sur l’unicité forte des solutions d’une équation différentielle stochas-tique avec drift singulier dépendant du temps en dimension un. Stochastics andStocastics Reports, V. 1-2, 1997, 21-29.



White-noise functionals as infinite dimensional randomColombeau distributions

Uluğ Çapar

FENS, Sabancı University, Tuzla, Istanbul, [email protected]

2000 Mathematics Subject Classification. 60B, 60B40

It is well-known that the white-noise is not a genuine process, but can berepresented as random linear distributions of the form Ω → D′ or as mem-bers of the white-noise space (S′,B, µ) , where S′ is the space of tempereddistributions, B a related σ-field and µ the Bochner measure. Howeverthese representations are rather smoothed versions of the white-noise ofthe form ξ : S × S′ → Rd. In order to define

.B (t) and its non-linear

functions like [.B (t)]2, e

.B(t) one has to resort to Hida, Kubo-Takenaka,

Kondratiev or some other distributions. These spaces are usually the dualsof the test function spaces constructed on (S′,B, µ). However in all casesthe reprentation of the non-linear functions and functionals of the white-noise is a painstaking process, e.g. in the case of Hida distributions onehas to utilize the second quantization operator and renormalization in eachseparate case.In this communiqué, after a critical evaluation of the afore-mentioned dis-tributional methods, an attempt is made of the construction of an infinitedimensional Colombeau distribution G(S′,B, µ) . In this space , like inG(Rn), the algebra operations would be possible; so that the products ofthe white-noise functions and functionals can be defined in a natural way.This is the extension of the work on random Colombeau distributions asintroduced and developed in [1] and [2]. It turns out that the counterpartsof Sobolev derivatives, Cameron-Martin formula can be defined, moderategerms A and null germs N can be constructed on S′, so that an infinitedimensional Colombeau algebra G/N can be formulated.

References

[1] Çapar, U., Aktuğlu, H., An Overall View of Stochastics in Colombeau RelatedAlgebras. In Progress in Probability, Birkhauser (ed. by Çapar,U., Üstünel),A.S. 53 (2003), 67-90.

[2] Çapar, U., Aktuğlu, H., A New Construction of Random Colombeau Algebras,Stat. and Prob. Letters 54 (2001), 291-299.

[3] Kuo, H. H., White Noise Distribution Theory. C.R.C. Press Inc. 1996.


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