span of symmetrically truncated symmetric random variables

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 $\mathbb P(\vert X_i \vert > x ) < S(x) \; \forall i$)

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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