On the distribution of random variables corresponding to Musielak-Orlicz norms (Q2866771)

Given \(p>1\) and an Orlicz function \(M\) satisfying certain natural conditions, the authors provide an easy formula for a probability distribution \(\mu\) such that for i.i.d. random variables \(X_1\),\dots, \(X_n\), drawn from this distribution, for every \(x\in \mathbb{R}^n\) one has NEWLINE\[NEWLINEC_p^{-1} \, \|x\| _M \leq \mathbb{E} \|(x_i X_i)_{i=1}^n\| _p \leq C_p \, \|x\| _M ,NEWLINE\]NEWLINE where \(C_p>0\) depends only on \(p\), \(\|\cdot\|_p\) and \(\|\cdot\|_M\) denote the \(\ell _p\)-norm and the Orlicz norm defined by \(M\), respectively. Then the authors consider the case when the norm \(\|\cdot\|_p\) is substituted by a given Orlicz norm \(N\), providing an implicit formula for \(\mu\). This result is, in a sense, an inverse theorem to the theorem proved by \textit{Y. Gordon} et al. in [Ann. Probab. 30, No. 4, 1833--1853 (2002; Zbl 1016.60008)] and in [Positivity 16, No. 1, 1--28 (2012; Zbl 1263.62090)]. The authors also discuss an embedding of the corresponding Orlicz spaces into \(L_1([0, 1])\), simplifying a representation by \textit{C. Schütt} [Stud. Math. 113, No. 1, 73--80 (1995; Zbl 0835.46023)].