Sudakov Minoration Principle and Supremum of Some Processes

Let Xi be a sequence of independent symmetric real random variables with logarithmically concave tails. We find a new version of Sudakov minoration principle for estimating from below \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ E \sup_{t \in T} \sum t_{i}X_{i} $\end{document}. If we moreover assume that tails of Xi are of moderate growth this enables us to give geometric conditions on set T equivalent to boundedness of the process \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ (\sum t_{i}X_{i})_{t \in T} $\end{document}. This generalize previous results of Talagrand [T4].