Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

If $X_i$ are symmetric independent random variables, is $\vert \sum X_i I_{\vert X_i \vert < N_i}\vert $ stochastically smaller than $\vert \sum X_i \vert$ ? Is it comparable in any way which would guarantee the former was stochastically bounded if the latter was ? The motivating example is i.i.d. cauchy, when the convex hull of the r.v.s consists entirely of Cauchys, but can the convex hull of the truncated guys contain stochastically unbounded guys ? (The $X_i$ are stochastically bounded if there is $S(x), S(x) \rightarrow 0 $ as $x \rightarrow \infty$ with $\mathbb P(\vert X_i \vert > x ) < S(x) \; \forall i$)

share|improve this question
Did you mean to take the modulus of the last expression in the first sentence? –  cardinal May 24 '12 at 0:11
You can deduce a lot from Hoffman Jorgensen's inequality (or Rosenthal's inequality, which follows from it, by getting moment estimates for large $p$ and looking at the asymptotics as $p\to \infty$). Look at the paper projecteuclid.org/DPubS/Repository/1.0/… by Hitczenko and Montgomery-Smith and references therein, especially the paper by Klass and Nowicki. –  Bill Johnson May 24 '12 at 4:40
The paper is MEASURING THE MAGNITUDE OF SUMS OF INDEPENDENT RANDOM VARIABLES Annals Prob. 29 no.1, 447-466, by P. Hitczenko and S. Montgomery-Smith. –  Bill Johnson May 24 '12 at 4:42
yes, I'll make that edit if it lets me. –  mike May 24 '12 at 20:56
I think second order stochastic dominance might be what I'm looking for. –  mike May 25 '12 at 21:06

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.