LIL and the Approximation of Rectangular Sums of B-valued Random Variables when Extreme Terms are Excluded

Let {X,Xn̄; n̄ ∈ ℕd} be a field of i.i.d. random variables indexed by d-tuples of positive integers and taking values in a Banach space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\user1{\mathcal{B}}} $$\end{document} and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ X^{{{\left( r \right)}}}_{{\ifmmode\expandafter\bar\else\expandafter\=\fi{n}}} = X_{{\ifmmode\expandafter\bar\else\expandafter\=\fi{m}}} \;{\text{if}}\;{\left\| {X_{{\ifmmode\expandafter\bar\else\expandafter\=\fi{m}}} } \right\|} $$\end{document} is the r-th maximum of {∥Xk̄∥ ; k̄ ≤ n̄}. Let Sn̄ =∑k̄ ≤ nXk̄ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {}^{{{\left( r \right)}}}S_{{\ifmmode\expandafter\bar\else\expandafter\=\fi{n}}} = S_{{\ifmmode\expandafter\bar\else\expandafter\=\fi{n}}} - {\left( {X^{{{\left( 1 \right)}}}_{{\ifmmode\expandafter\bar\else\expandafter\=\fi{n}}} + \cdots + X^{{{\left( r \right)}}}_{{\ifmmode\expandafter\bar\else\expandafter\=\fi{n}}} } \right)} $$\end{document}. We approximate the trimmed sums (r)Sn̄ by a Brownian sheet and obtain sufficient and necessary conditions for (r)Sn to satisfy the compact and functional laws of the iterated logarithm. These results improve the previous works by Morrow (1981), Li and Wu (1989) and Ledoux and Talagrand (1990).