Prediction for discrete time series

Let {X_n} be a stationary and ergodic time series taking values from a finite or countably infinite set Assume that the distribution of the process is otherwise unknown. We propose a sequence of stopping times _n along which we will be able to estimate the conditional probability P(=x|X₀,...,) from data segment (X₀,...,) in a pointwise consistent way for a restricted class of stationary and ergodic finite or countably infinite alphabet time series which includes among others all stationary and ergodic finitarily Markovian processes. If the stationary and ergodic process turns out to be finitarily Markovian (among others, all stationary and ergodic Markov chains are included in this class) then almost surely. If the stationary and ergodic process turns out to possess finite entropy rate then _n is upperbounded by a polynomial, eventually almost surely.Mathematics Subject Classification (2000): 62G05, 60G25, 60G10     

Path regularity for Feller semigroups via Gaussian kernel estimates and generalizations to arbitrary semigroups on <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>

Given γ ∈ (−1,1), we present a dyadic growth condition on the finite dimensional distributions of operator semigroups on C₀(E which - for γ>0 and Feller semigroups - assures that the corresponding Feller process has paths in local Hölder spaces and in weighted Besov spaces of order γ. We show that, for operator semigroups satisfying Gaussian kernel estimates of order m>1, condition holds for all and even for all in the case of Feller semigroups. Such Gaussian kernel estimates are typical for Feller semigroups on fractals of walk dimension m and for semigroups generated by elliptic operators on ℝ^D of order m≥D.     

The divergence of fluctuations for shape in first passage percolation

We consider the first passage percolation model on Z ^d for d ≥ 2. In this model, we assign independently to each edge the value zero with probability p and the value one with probability 1−p. We denote by T(0, ν) the passage time from the origin to ν for ν ∈ R ^d and It is well known that if p < p _c , there exists a compact shape B _d ⊂ R ^d such that for all > 0, t B _d (1 − ) ⊂ B(t) ⊂ tB _d (1 + ) and G(t)(1 − ) ⊂ B(t) ⊂ G(t)(1 + ) eventually w.p.1. We denote the fluctuations of B(t) from tB _d and G(t) by In this paper, we show that for all d ≥ 2 with a high probability, the fluctuations F(B(t), G(t)) and F(B(t), tB _d ) diverge with a rate of at least C log t for some constant C. The proof of this argument depends on the linearity between the number of pivotal edges of all minimizing paths and the paths themselves. This linearity is also independently interesting. Research supported by NSF grant DMS-0405150     

Associated primes and cofiniteness of local cohomology modules

Let be an ideal of Noetherian ring R and let s be a non-negative integer. Let M be an R-module such that is finite R-module. If s is the first integer such that the local cohomology module is non -cofinite, then we show that is finite. In particular, the set of associated primes of is finite. Let be a local Noetherian ring and let M be a finite R-module. We study the last integer n such that the local cohomology module is not -cofinite and show that n just depends on the support of M.The research of the first author was supported in part by a grant from IPM (No. 83130114).The second author was supported by a grant from University of Tehran (No. 6103023/1/01).     

Cumulants are universal homomorphisms into Hausdorff groups

This is a contribution to the theory of sums of independent random variables at an algebraico-analytical level: Let Prob denote the convolution semigroup of all probability measures on with all moments finite, topologized by polynomially weighted total variation. We prove that the cumulant sequence regarded as a function from Prob into the additive topological group ofall real sequences, is universal among continuous homomorphisms from Prob into Hausdorff topological groups, in the usual sense that every other such homomorphism factorizes uniquely through . An analogous result, referring to just the first cumulants,holds for the semigroup of all probability measures with existing rth moments. In particular, there is no nontrivial continuous homomorphism from the convolution semigroup of all probability measures, topologized by the total variation metric, into any Hausdorff topological group.Mathematics Subject Classification (2000): 60B15, 60E10, 60G50     

Non-commutative polynomials of independent Gaussian random matrices. The real and symplectic cases.

In [HT2] Haagerup and Thorbjo rnsen prove the following extension of Voiculescus random matrix model (cf. [V2, Theorem 2.2]): For each n , let X₁⁽ⁿ⁾,..., X_r⁽ⁿ⁾ be r independent complex self-adjoint random matrices from the class and let x₁,...,x_r be a semicircular system in a C*-probability space. Then for any polynomial p in r non-commuting variables the convergenceholds almost surely. We generalize this result to sets of independent Gaussian random matrices with real or symplectic entries (the GOE- and the GSE-ensembles) and random matrix ensembles related to these.This work was partially supported by MaPhySto – A Network in Mathematical Physics and Stochastics, funded by The Danish National Research Foundation.As a student of the PhD-school OP-ALG-TOP-GEO the author is partially supported by the Danish Research Training Council.Acknowledgement I would like to thank my advisor, Uffe Haagerup, with whom I had many enlightening discussions, and who made some important contributions to this paper. Also, thanks to Steen Thorbjørnsen who took time to answer several questions.     

On the disconnection of a discrete cylinder by a random walk

We investigate the large N behavior of the time the simple random walk on the discrete cylinder needs to disconnect the discrete cylinder. We show that when d≥2, this time is roughly of order N ^{2 d} and comparable to the cover time of the slice , but substantially larger than the cover timer of the base by the projection of the walk. Further we show that by the time disconnection occurs, a massive ``clogging' typically takes place in the truncated cylinders of height . These mechanisms are in contrast with what happens when d=1.     

Asymptotic laws for regenerative compositions: gamma subordinators and the like

For a random closed set obtained by exponential transformation of the closed range of a subordinator, a regenerative composition of generic positive integer n is defined by recording the sizes of clusters of n uniform random points as they are separated by the points of . We focus on the number of parts K_n of the composition when is derived from a gamma subordinator. We prove logarithmic asymptotics of the moments and central limit theorems for K_n and other functionals of the composition such as the number of singletons, doubletons, etc. This study complements our previous work on asymptotics of these functionals when the tail of the Lévy measure is regularly varying at 0+. Research supported in part by N.S.F. Grant DMS-0071448     

Modified Ariki-Koike algebras and cyclotomic <Emphasis Type="Italic">q</Emphasis>-Schur algebras

The modified Ariki-Koike algebra is a variation of the original Ariki-Koike algebra over an integral domain R. When R is a rational function field over the independent parameters, But for general R, is not isomorphic to , and has a simpler structure than . In this paper, we construct a cellular basis of which has a similar property as the cellular basis of introduced by Dipper-James-Mathas. By comparing these two cellular bases, we obtain some estimate on the decomposition numbers of in terms of the decomposition numbers of . We also prove the integral form of the Schur-Weyl reciprocity between a certain quantum algebra U_q and on the tensor space      

Distributions diophantiennes et théorème limite local sur

We study the speed of convergence of n^d/2fd^*n in the local limit theorem on under very general conditions upon the function f and the distribution . We show that this speed is at least of order and we give a simple characterization (in diophantine terms) of those measures for which this speed (and the full local Edgeworth expansion) holds for smooth enough f. We then derive a uniform local limit theorem for moderate deviations under a mild moment assumption. This in turn yields other limit theorems when f is no longer assumed integrable but only bounded and Lipschitz or Hölder. We finally give an application to equidistribution of random walks.     

Translation invariant Gibbs states for the Ising model

We prove that all the translation invariant Gibbs states of the Ising model are a linear combination of the pure phases for any . This implies that the average magnetization is continuous for . Furthermore, combined with previous results on the slab percolation threshold [B2] this shows the validity of Pisztora's coarse graining [Pi] up to the critical temperature. We would like to thank C. Pfister and Y. Velenik for very useful comments.     

New dependence coefficients. Examples and applications to statistics

To measure the dependence between a real-valued random variable X and a -algebra , we consider four distances between the conditional distribution function of X given and the distribution function of X. The coefficients obtained are weaker than the corresponding mixing coefficients and may be computed in many situations. In particular, we show that they are well adapted to functions of mixing sequences, iterated random functions and dynamical systems. Starting from a new covariance inequality, we study the mean integrated square error for estimating the unknown marginal density of a stationary sequence. We obtain optimal rates for kernel estimators as well as projection estimators on a well localized basis, under a minimal condition on the coefficients. Using recent results, we show that our coefficients may be also used to obtain various exponential inequalities, a concentration inequality for Lipschitz functions, and a Berry-Esseen type inequality.Mathematics Subject Classification (2000): 62G07, 60J10, 60E15, 37C30     

Inversion de l'espace et du temps des processus de Lévy stables

We proove a new duality relation between stable Lévy processes with index and those with index . This duality appears to be the trajectorial version of the duality of Zolotarev which concerns one dimensional stable laws. We give an application of this result to the behaviour of the paths at small and large times of the process ``conditioned to stay positive'. Mots-clefs: Lévy process Stable process Fluctuation identities Ladder process     

On Cross-intersecting Families of Sets

A family of -element subsets and a family of k-element subsets of an n-element set are cross-intersecting if every set from has a nonempty intersection with every set from . We compare two previously established inequalities each related to the maximization of the product , and give a new and short proof for one of them. We also determine the maximum of for arbitrary positive weights ,_k.     

On the logarithmic plurigenera of complements of plane curves

Let B be a (not necessarily irreducible) plane curve in ². In the present article, we prove that if and only if Moreover, we determine the curve B when and Mathematics Subject Classification (2000): 14R05, 14H50, 14J26     

Interior regularity criteria in weak spaces for the Navier-Stokes equations

We study the interior regularity of weak solutions of the incompressible Navier-Stokes equations in ×(0,T), where and 0<T<. The local boundedness of a weak solution u is proved under the assumption that is sufficiently small for some (r,s) with and 3r<. Our result extends the well-known criteria of Serrin (1962), Struwe (1988) and Takahashi (1990) to the weak space-time spaces.Mathematics Subject Classification (2000): 35Q30, 76N10     

Diophantine inequalities involving several power sums

Let denote the ring of power sums, i.e. complex functions of the form for some and _iA, where is a multiplicative semigroup. Moreover, let We consider Diophantine inequalities of the form where >1 is a quantity depending on the dominant roots of the power sums appearing as coefficients in F(n,y), and show that all its solutions have y parametrized by some power sums from a finite set. This is a continuation of the work of Corvaja and Zannier [4–6] and of the authors [10, 18] on such problems.Mathematics Subject Classification (2000):11D45,11D61Revised version: 6 May 2004     

Universality of critical behaviour in a class of recurrent random walks

Let X₀=0, X₁, X₂,.. be an aperiodic random walk generated by a sequence ₁, ₂,... of i.i.d. integer-valued random variables with common distribution p(·) having zero mean and finite variance. For anN-step trajectory and a monotone convex functionV: withV(0)=0, define Further, let be the set of all non-negative paths compatible with the boundary conditionsX₀=a, X_N=b. We discuss asymptotic properties of under the probability distribution N and 0, Z^a,b_N,+, being the corresponding normalization. If V(·) grows not faster than polynomially at infinity, define H() to be the unique solution to the equation Our main result reads that as 0, the typical height of X_[, ^N] scales as H() and the correlations along decay exponentially on the scale H()². Using a suitable blocking argument, we show that the distribution tails of the rescaled height decay exponentially with critical exponent 3/2. In the particular case of linear potential V(·), the characteristic length H() is proportional to ^-1/3 as 0.Mathematics Subject Classification (2000):60G50, 60K35; 82B27, 82B41     

An estimate on the supremum of a nice class of stochastic integrals and U-statistics

Let a sequence of iid. random variables ξ ₁, . . . ,ξ _n be given on a space with distribution μ together with a nice class of functions f(x ₁, . . . ,x _k ) of k variables on the product space For all f ∈ we consider the random integral J _n,k (f) of the function f with respect to the k-fold product of the normalized signed measure where μ _n denotes the empirical measure defined by the random variables ξ ₁, . . . ,ξ _n and investigate the probabilities for all x>0. We show that for nice classes of functions, for instance if is a Vapnik–Červonenkis class, an almost as good bound can be given for these probabilities as in the case when only the random integral of one function is considered. A similar result holds for degenerate U-statistics, too. Supported by the OTKA foundation Nr. 037886     

On certain categories of modules for affine Lie algebras

In this paper, we exploit basic formal variable techniques to study certain categories of modules for an (untwisted) affine Lie algebra , motivated by Chari-Pressleys work on certain integrable modules. We define and study two categories and of -modules using generating functions, where is proved to contain the well known evaluation modules and to unify highest weight modules, evaluation modules and their tensor product modules. We classify integrable irreducible -modules in categories and and we determine the isomorphism classes of those irreducible modules. Finally we prove a result that relates fusion rules in the context of vertex operator algebras with integrable irreducible modules of Chari-Pressley.in final form: 12 November 2003Partially supported by a NSA grant and a grant from Rutgers Research Council.