Weighted exponential inequality for differentially subordinate martingales

The paper contains a study of weighted exponential inequalities for differentially subordinate martingales, under the assumption that the underlying weight satisfies Muckenhoupt’s condition $$A_{\infty }$$ A ∞ . The proof exploits certain functions enjoying appropriate size conditions and concavity. The martingales are adapted, uniformly integrable, and càdlàg - we do not assume any path-continuity restrictions. Because of this generality, we need to handle jump parts of processes which forces us to construct a Bellman function satisfying a stronger condition than local concavity. As a corollary, we will establish some new weighted $$L^p$$ L p estimates for differential subordinates of bounded martingales.

An exponential inequality for $U$-statistics of i.i.d. data

Устанавливается экспоненциальное неравенство для вырожденных $U$-статистик порядка $r$, основанных на н.о.р. данных. Это неравенство позволяет оценить хвост максимума абсолютной величины $U$-статистики суммой двух членов: экспоненциального и второго члена, содержащего хвост $h(X_1,…,X_n)$. Также предлагается вариант неравенства для необязательно вырожденной $U$-статистики с симметричным ядром. Рассмотрены приложения к оценке скорости сходимости в законе больших чисел Марцинкевича и принципу инвариантности в гeльдеровом пространстве.

On the strong convergence forweighted sums of negatively superadditive dependent random variables

In this paper, we present some results on the complete convergence for arrays of rowwise negatively superadditive dependent (NSD, in short) random variables by using the Rosenthal-type maximal inequality, Kolmogorov exponential inequality and the truncation method. The results obtained in the paper extend the corresponding ones for weighted sums of negatively associated random variables with identical distribution to the case of arrays of rowwise NSD random variables without identical distribution.