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\(L_1\)-norm of combinations of products of independent random variables. (English) Zbl 1350.60012

Let \(X\), \(X_i\), \(1\leq i \leq n\) be independent and identically distributed (i.i.d.) non-negative random variables such that \(\operatorname{E}(X)=1\) and \(\operatorname{P}(X=1)<1\). Define \(R_0=1\) and \(R_i=\Pi_{j=1}^iX_j\), \(1\leq j \leq n\). Let \(a_i\), \(1\leq i \leq n\) be real numbers. It is obvious that \(\operatorname{E}(|\sum_{i=0}^na_iR_i|)\leq \sum_{i=0}^n|a_i|\). The author proves the converse of this observation. Suppose that \(X\), \(X_i\), \(1\leq i \leq n\) are i.i.d. non-negative non-degenerate random variables such that \(\operatorname{E}(X)=1\). Then there exists a constant \(c\) that depends only on the distribution of \(X\) such that for any \(v_0\), \(v_1\),…, \(v_n\) in a normed space \((F,\|\cdot\|)\), \(\operatorname{E}(\|\sum_{i=0}^nv_iR_i\|)\geq c\sum_{i=0}^n\|v_i\|\). This result has also been extended to independent non-negative random variables \(X_i\), \(1\leq i \leq n\) with mean one satisfying the conditions \(\operatorname{E}(\sqrt{X_i})\leq \lambda <1\), \(\operatorname{E}(|X_i-1|)>\mu >0\), \(1 \leq i \leq n\) for some \(\lambda <1\) and \(\mu >0\). Some additional related inequalities are proved.

MSC:

60E15 Inequalities; stochastic orderings
60B11 Probability theory on linear topological spaces
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References:

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