On maxima of independent random elements in Banach functional lattice (Q2740453)

The authors consider Banach ideal space (BIS) as an important example of functional Banach lattice. BIS is a Banach space \(E\) of measurable functions on measurable space \((T,F,P)\), \(P\) is a \(\sigma\)-finite, \(\sigma\)-additive and nonnegative measure, for which from \(|x(t)|\leq|y(t)|\) a.s. and \(y\in E\) it follows that \(x\in E\) and \(\|x\|\leq\|y\|,\) where \(\|\cdot\|\) and \(|\cdot|\) are the norm and the modul in the space \(E\), respectively. Let \(X\) be a random element (r.e.) in BIS \(E,\) let \(X_{n}\) be independent copies of \(X,\) and let \(Z_{n}:=\max_{1\leq k\leq n}X_{k}.\) The main problem of this paper is to obtain for some classes of r.e. and some classes of BIS the following asymptotic relations: \(\|Z_{n}/a_{n}-\Delta X\|@> \text{P} >>0\) and \(\|Z_{n}/a_{n}-\Delta X\|@>\text{a.s.}>>0\) as \(n\to\infty,\) where \(\Delta X \in E\) and \(a_{n}\in R.\) Sufficient conditions are found to obtain the above-mentioned relations. Problems related to the measurability of supremum of r.e. are also considered.