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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