On the moments of \(\sup_n\|S_n/(n\varphi(n))^{1/p}\|\) in \(p\)-type space (Q2716403)

In 1967, Teicher proved that: NEWLINE\[NEWLINEE\left(\underset {n}{\text{Sup}} \frac{|S_n|}{\sqrt{nL_2n}}\right)^p<\infty\quad \text{iff}\quad \begin{cases} E\;\frac{X^2\log^+|X|}{L_2|X|}<\infty, & p=2\\ E|X|^p<\infty, & p>2,\end{cases}NEWLINE\]NEWLINE where \(\{X,X_n,n\geq 1\}\) is a sequence of i.i.d. real random variables with means zero. In 1955 similar results have been set up for i.i.d. random variables \(\{X,X_n,n\geq 1\}\) with means zero and valued in a 2-type Banach space.NEWLINENEWLINENEWLINEThe authors study the same question in \(p\)-type Banach spaces. Considering \(B\) a \(p\)-type Banach space \((1<p<2)\) and \(\varphi(x)\) some slowly increasing function on \([0,\infty)\), if \(\{X,X_n,n\geq 1\}\) is a sequence of i.i.d \(B\)-valued random variables, then a necessary and sufficient condition for the moments of \(\text{Sup}_n \|S_n/(n\varphi(n))^{1/p}\|\) being bounded is given.