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Niveau: Supérieur, Master, Bac+5

1Central limit the- o- rems in Holder spaces Random fields and central limit theorem in some generalized Holder spaces A. RACˇKAUSKAS?, CH. SUQUET? Department of Mathematics, Vilnius University, Naugarduko 24, Lt-2006 Vilnius, Lithuania Laboratoire de Statistique et Probabilites, E.P. CNRS 1765 Bat. M2, Cite Scientifique, F-59655 Villeneuve d'Ascq Cedex, France Abstract. For rather general moduli of smoothness ? (like e.g. ?(h) = h? ln?(c/h) ) the Holder spaces H?([0, 1]d), are characterized by the rate of coefficients in the skew pyramidal basis. With this analytical tool, we study in terms of second differences the existence of a version in H? for a given random field. In the same spirit, central limit theorems are obtained both for i.i.d. and martingale differences sequences of random elements in H?. 1. INTRODUCTION In many situations, stochastic processes and random fields have a smoothness intermediate between the continuity and differentiability. The scale of Holder spaces is then a natural functional framework to investigate the regularity of such processes and fields. And weak convergence in this setting is a stronger result than in the space of continuous functions. In this paper we consider the scale of generalized Holder spaces H?([0, 1]d), where ? is a modulus of smoothness (precise definition is given in Section 1 below) and discuss two questions: (I) For a given random field indexed by [

1Central limit the- o- rems in Holder spaces Random fields and central limit theorem in some generalized Holder spaces A. RACˇKAUSKAS?, CH. SUQUET? Department of Mathematics, Vilnius University, Naugarduko 24, Lt-2006 Vilnius, Lithuania Laboratoire de Statistique et Probabilites, E.P. CNRS 1765 Bat. M2, Cite Scientifique, F-59655 Villeneuve d'Ascq Cedex, France Abstract. For rather general moduli of smoothness ? (like e.g. ?(h) = h? ln?(c/h) ) the Holder spaces H?([0, 1]d), are characterized by the rate of coefficients in the skew pyramidal basis. With this analytical tool, we study in terms of second differences the existence of a version in H? for a given random field. In the same spirit, central limit theorems are obtained both for i.i.d. and martingale differences sequences of random elements in H?. 1. INTRODUCTION In many situations, stochastic processes and random fields have a smoothness intermediate between the continuity and differentiability. The scale of Holder spaces is then a natural functional framework to investigate the regularity of such processes and fields. And weak convergence in this setting is a stronger result than in the space of continuous functions. In this paper we consider the scale of generalized Holder spaces H?([0, 1]d), where ? is a modulus of smoothness (precise definition is given in Section 1 below) and discuss two questions: (I) For a given random field indexed by [

- gaussian process
- ho? always
- banach isomorphism between
- analytical tool
- smoothness ?
- central limit theorems
- differences sequences
- odic function
- functions

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