Let $(\xi\_n)\_{n\ge 1}$, $(\eta\_n)\_{n\ge 1}$ be independent mean-zero random variables with values in a Banach space $X$ such that 
$$\sum\_n\mathbb P(\xi\_n\in A)\le\sum\_n\mathbb P(\eta\_n\in A)$$for any Borel set $A\subset X\setminus\{0\}$. 


Let $1\le p<\infty$. Is there a constant $C$ (perhaps depending on $p$ and $X$) such that


$$\textstyle\mathbb E\|\sum\_n\xi\_n\|^p\le C\,\mathbb E\|\sum\_n\eta\_n\|^p?$$




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This problem was posed on 28.06.2019 on page [133](http://www.math.lviv.ua/szkocka/viewpage.php?vol=2&page=133) of [Volume 2](http://www.math.lviv.ua/szkocka/viewbook.php?vol=2) of the [Lviv Scottish Book](http://www.math.lviv.ua/szkocka). 



On sums of independent random variables in Banach spaces
Certainly not always. The most trivial example seems to be $X=\ell^\infty$, $\eta\_n=\pm e\_n$ (with probability $1/2$ for each sign), and $\xi\_n$ being uniformly distributed on $\pm e\_1,\dots,\pm e\_N$ with large $N$ for $n=1,\dots,N$ (as usual, $e\_n$ is the vector with the $n$-th coordinate $1$). The rest of $\xi\_n$ and $\eta\_n$ can be put to $0$, say.


Then the sums of probabilities in question are equal (and equal to one half times the number of vectors $\pm e\_n, n=1,\dots,N$ lying in $A$). However, $\|\sum\eta\_n\|$ is always $1$ while $\|\sum\xi\_n\|$ is typically (i.e., with probability $\ge \frac12$, say) at least of order $\sqrt{\frac{\log N}{\log\log N}}$ (that is just the classical balls into bins problem combined with the random choice of signs for the balls in the maximal bin, and in this computation I ignore the fact that the bin with nearly maximal number of balls is typically not unique, which may drive the estimate up even more). 


Next, the "perhaps, depending on $X$" construct looks very fishy in the question as it is posed now (nobody prevents us from taking the appropriate sum of spaces with large constants to get a space without any constant, say; also, since the question is essentially finite-dimensional, it is clear that $\ell^\infty$ must be a counterexample if anything at all is). 


Are you sure that the problem is not "Describe (in some familiar terms) the Banach spaces $X$ such that..."?


