论文信息 - Some limit theorems for the empirical process indexed by functions

Some limit theorems for the empirical process indexed by functions

SummaryLet,n≧1, be a sequence of classes of real-valued measurable functions defined on a probability space (S,,P). Under weak metric entropy conditions on,n≧1, and under growth conditions on we show that there are non-zero numerical constantsC1 andC2 such that where α(n) is a non-decreasing function ofn related to the metric entropy of. A few applications of this general result are considered: we obtain a.s. rates of uniform convergence for the empirical process indexed by intervals as well as a.s. rates of uniform convergence for the empirical characteristic function over expanding intervals.

J. Yukich

[1] R. Dudley. Central Limit Theorems for Empirical Measures , 1978 .

[2] V. Vapnik,et al. Necessary and Sufficient Conditions for the Uniform Convergence of Means to their Expectations , 1982 .

[3] M. Ledoux. Loi du logarithme itéré dans CS) et fonction caractéristique empirique , 1982 .

[4] J. Wellner,et al. Limit Theorems and Inequalities for the Uniform Empirical Process Indexed by Intervals , 1982 .

[5] Richard M. Dudley,et al. Invariance principles for sums of Banach space valued random elements and empirical processes , 1983 .

[6] R. Dudley. A course on empirical processes , 1984 .

[7] E. Giné,et al. Some Limit Theorems for Empirical Processes , 1984 .

[8] École d'été de probabilités de Saint-Flour,et al. École d'Été de Probabilités de Saint-Flour XII - 1982 , 1984 .

[9] J. Yukich. Laws of large numbers for classes of functions , 1985 .

[10] J. E. Yukich,et al. Weak convergence of the empirical characteristic function , 1985 .

[11] J. Yukich. Rates of convergence for classes of functions: the non-i.i.d. case , 1986 .

[12] K. Alexander,et al. Rates of growth and sample moduli for weighted empirical processes indexed by sets , 1987 .

[13] M. Talagrand. The Glivenko-Cantelli Problem , 1987 .