Given a normalized Orlicz function M we provide an easy formula for a distribution such that, if X is a random variable distributed accordingly and X₁,...,Xₙ are independent copies of X, then $1/C_p{\left|\right|x\left|\right|}_M\le \left|\right|\left(x_i{X}_i\right){ⁿ}_{i=1}{\left|\right|}_p\le C_p{\left|\right|x\left|\right|}_M$, where $C_p$ is a positive constant depending only on p. In case p = 2 we need the function t ↦ tM’(t) - M(t) to be 2-concave and as an application immediately obtain an embedding of the corresponding Orlicz spaces into L₁[0,1]. We also provide a general result replacing the ${\ell }_p$-norm by an arbitrary N-norm. This...

We generalize some results concerning the classical notion of a spreading model to spreading models of order ξ. Among other results, we prove that the set $S{M}_{\xi }^w\left(X\right)$ of ξ-order spreading models of a Banach space X generated by subordinated weakly null ℱ-sequences endowed with the pre-partial order of domination is a semilattice. Moreover, if $S{M}_{\xi }^w\left(X\right)$ contains an increasing sequence of length ω then it contains an increasing sequence of length ω₁. Finally, if $S{M}_{\xi }^w\left(X\right)$ is uncountable, then it contains an antichain of size...