Weak Hölder convergence of processes with application to the perturbed empirical process

• Autorzy: Djamel Hamadouche , Charles Suquet
• Języki publikacji: EN

Abstrakty

EN

We consider stochastic processes as random elements in some spaces of Hölder functions vanishing at infinity. The corresponding scale of spaces $C^{α,0}_0$ is shown to be isomorphic to some scale of Banach sequence spaces. This enables us to obtain some tightness criterion in these spaces. As an application, we prove the weak Hölder convergence of the convolution-smoothed empirical process of an i.i.d. sample $(X_1,...,X_n)$ under a natural assumption about the regularity of the marginal distribution function F of the sample. In particular, when F is Lipschitz, the best possible bound α<1/2 for the weak α-Hölder convergence of such processes is achieved.

Słowa kluczowe

EN

triangular functions; Schauder decomposition; Hölder space; tightness; Brownian bridge; perturbed empirical process

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Applicationes Mathematicae
• Rocznik: 1999
• Tom: 26
• Numer: 1
• Strony: 63-83

Daty

wydano

1999

otrzymano

1998-07-20

Twórcy

autor

Djamel Hamadouche

• Laboratoire de Statistique et Probabilités, Bât. M2, Université des Sciences et Technologies de Lille, F-59655 Villeneuve d'Ascq Cedex, France

autor

Charles Suquet

• Laboratoire de Statistique et Probabilités, Bât. M2, Université des Sciences et Technologies de Lille, F-59655 Villeneuve d'Ascq Cedex, France

Bibliografia

• [1] P. Baldi and B. Roynette, Some exact equivalents for the Brownian motion in Hölder norm, Probab. Theory Related Fields 93 (1993), 457-484.
• [2] P. Billingsley, Convergence of Probability Measures, Wiley, 1968.
• [3] Z. Ciesielski, On the isomorphisms of the spaces $H_α$ and m, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 8 (1960), 217-222.
• [4] Z. Ciesielski, Hölder conditions for realizations of Gaussian processes, Trans. Amer. Math. Soc. 99 (1961), 403-413.
• [5] Z. Ciesielski, G. Kerkyacharian et B. Roynette, Quelques espaces fonctionnels associés à des processus gaussiens, Studia Math. 107 (1993), 171-204.
• [6] L. T. Fernholz, Almost sure convergence of smoothed empirical distribution functions, Scand. J. Statist. 18 (1991), 255-262.
• [7] D. Hamadouche, Invariance principles in Hölder spaces, Portugal. Math. (1998), to appear.
• [8] D. Hamadouche, Weak convergence of smoothed empirical process in Hölder spaces, Statist. Probab. Letters 36 (1998), 393-400.
• [9] T. Hida, Brownian Motion, Springer, 1980.
• [10] G. Kerkyacharian et B. Roynette, Une démonstration simple des théorèmes de Kolmogorov, Donsker et Ito-Nisio, C. R. Acad. Sci. Paris Sér. I 312 (1991), 877-882.
• [11] J. Lamperti, On convergence of stochastic processes, Trans. Amer. Math. Soc. 104 (1962), 430-435.
• [12] Yu. V. Prohorov [Yu. V. Prokhorov], Convergence of random processes and limit theorems in probability theory, Theory Probab. Appl. 1 (1956), 157-214.
• [13] G. R. Shorack and J. A. Wellner, Empirical Processes with Applications to Statistics, Wiley, 1986.
• [14] I. Singer, Bases in Banach Spaces II, Springer, 1981.
• [15] S. Sun, Perturbed empirical distribution functions and quantiles under dependence, J. Theoret. Probab. 8 (1995), 763-777.
• [16] C. Suquet, Tightness in Schauder decomposable Banach spaces, Translations of A.M.S., Proceedings of the St Petersburg Math. Soc., to appear.