zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
U-processes: Rates of convergence. (English) Zbl 0624.60048
Let $\xi\sb 1,\xi\sb 2,..$. be independent, identically distributed random variables and denote by $$ S\sb n(f)=\sum\sb{1\le i\ne j\le n}f(\xi\sb i,\xi\sb j) $$ the U-statistic with respect to the kernel f. The authors obtain almost sure convergence results for $S\sb n(f)$ uniformly over $f\in F$ where F belongs to certain classes of kernels. Assumptions and proofs are motivated by the corresponding theory for empirical processes, though there are several significant differences in this case. Finally, an application to cross validation in density estimation is given.
Reviewer: M.Denker

60F15Strong limit theorems
62G05Nonparametric estimation
62E20Asymptotic distribution theory in statistics
Full Text: DOI