Moderate deviation probabilities for open convex sets: nonlogarithmic behavior

Precise asymptotics for moderate deviation probabilities are established for open convex sets in both the finite- and infinite-dimensional settings. Our results are based on the existence of dominating points for these sets, a related representation formula, and asymptotics for the integral term in this formula.Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000021

Uniform in bandwidth consistency of kernel-type function estimators

We introduce a general method to prove uniform in bandwidth consistency of kernel-type function estimators. Examples include the kernel density estimator, the Nadaraya-Watson regression estimator and the conditional empirical process. Our results may be useful to establish uniform consistency of data-driven bandwidth kernel-type function estimators.Comment: Published at http://dx.doi.org/10.1214/009053605000000129 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Some results on two-sided LIL behavior

Let {X,X_n;n\geq 1} be a sequence of i.i.d. mean-zero random variables, and let S_n=\sum_{i=1}^nX_i,n\geq 1. We establish necessary and sufficient conditions for having with probability 1, 0<lim sup_{n\to \infty}|S_n|/\sqrtnh(n)<\infty, where h is from a suitable subclass of the positive, nondecreasing slowly varying functions. Specializing our result to h(n)=(\log \log n)^p, where p>1 and to h(n)=(\log n)^r, r>0, we obtain analogues of the Hartman-Wintner LIL in the infinite variance case. Our proof is based on a general result dealing with LIL behavior of the normalized sums {S_n/c_n;n\ge 1}, where c_n is a sufficiently regular normalizing sequence.Comment: Published at http://dx.doi.org/10.1214/009117905000000198 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Law of the iterated logarithm type results for random vectors with infinite second moments

Ten tekst jest rozszerzona wersja prezentacji autora na konferencji A path through probability in honour of F.Thomas BRUSS kt&oacute;re odbyła sie na Unversite Libre de Bruxelles w Brukseli w dniach 9-11 wrzesnia 2015 roku. W pierwszej czesci przedstawiam niekt&oacute;re wyniki uog&oacute;lniajac klasyczne prawo Hartmana-Wintnera iterowanego logarytmu dla zmiennych 1-wymiarowych z nieskonczonym drugim momentem, a nastepnie pokazuje, jak te wyniki moga byc rozszerzone do zagadnien d-wymiarowych. Wywody koncze na og&oacute;lnym funkcjonalnym prawie tego typu.This survey paper is an extended version of the author&rsquo;s presentation at the conference in honor of Professor F. Thomas Bruss at the occasion of his retirement as Chair of Math&acute;ematiques G&acute;en&acute;erales from the Unversit&acute;e Libre de Bruxelles which was held September 9-11, 2015 in Brussels. I first present some results generalizing the classical Hartman-Wintner law of the iterated logarithm to 1-dimensional variables with infinite second moments and then I show how these results can be further extended to the d-dimensional setting. Finally, I look at general functional law of the iterated logarithm type results