# On sums of independent random variables in Banach spaces

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

$$\textstyle\mathbb E\|\sum_n\xi_n\|^p\le C\,\mathbb E\|\sum_n\eta_n\|^p?$$

This problem was posed on 28.06.2019 on page 133 of Volume 2 of the Lviv Scottish Book.

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..."?

• Thank you very much for the answer. Concerning the proper form of this question, I do not know what to answer as I even cannot decrypt the name of the person that posed this problem, see math.lviv.ua/szkocka/viewpage.php?vol=2&page=133 – Lviv Scottish Book Nov 7 '19 at 11:24
• @LvivScottishBook As I understand the name is Ivan Yaroslavtsev (written as Iv. Yar.) – Mikhail Ostrovskii Nov 10 '19 at 5:53
• @MikhailOstrovskii Thank you very much for the suggestion. I will try to contact with Iv.Yar. Probably this will be a kind of surprise for him. – Lviv Scottish Book Nov 11 '19 at 11:48

First of all, many thanks to @LvivScottishBook for putting the problem online (that was indeed very surprising for me) and to @fedja for finding a counterexample.

While writing the problem I was sure that it has a positive answer for any Banach space (for some reason this was kind of intuitive for me) with $$C$$ independent of $$X$$, so @fedja answered the original question. Nonetheless, as @fedja noticed, it is not clear for which Banach spaces such an inequality holds true. The only thing that I can guarantee so far is that the necessary assumption on the Banach space is having a finite cotype thanks to @fedja counterexample and the Maurey–Pisier theorem (see Corollary 7.3.14 in Analysis in Banach spaces, Volume II by Hytönen, van Neerven, Veraar, and Weis).

It remains open whether finite cotype is a sufficient condition as well.

• P.S. I guess I better make a comment out of this text but I could not as I do not have enough reputation. – Iv Yar Nov 11 '19 at 14:26
• You should ask a separate question out of this and make a link to the old one. This way no one finds your comment/question. – András Bátkai Nov 11 '19 at 14:46
• I think it merits a new question rather. – Amir Sagiv Nov 11 '19 at 16:17
• Thanks! Here is the new question. – Iv Yar Nov 11 '19 at 17:04