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Consider $N$ pairs of random variables $(X_i, Y_i)$. $X_i$ are iid, with $EX_i=0$ and $EX_i^2=1$. The same conditions hold for $Y_i$. Moreover all $X_i$ are independent of all $Y_j$. It seems very reasonable that $\sup_i|X_i+Y_i| > \sup_i|X_i|$ with high probability:

$$P\left(\sup_i |X_i| \le \sup_i |X_i+Y_i| +\epsilon \right) < \exp(-c\epsilon^2)$$ But is it true? I can't find counterexamples and yet don't know how to attack the problem.

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  • $\begingroup$ maybe, another sign of the main inequality? $\endgroup$
    – Fedor Petrov
    Commented Oct 13, 2015 at 15:16
  • $\begingroup$ in any case, $N$ must appear somehow in the inequality $\endgroup$
    – Fedor Petrov
    Commented Oct 13, 2015 at 15:20
  • $\begingroup$ Yes, I agree. I believe the RHS could be an even stronger $\exp(-cN\epsilon^2)$, but would be happy with the original conjecture. $\endgroup$
    – gappy3000
    Commented Oct 13, 2015 at 15:42
  • $\begingroup$ I guess $c$ here is allowed to depend on the distribution of the $X_i$ somehow? (If not, what's preventing the $X_i$ from, say, each being $0$ with probability $1-\frac{1}{N^2}$). $\endgroup$
    – Kevin P. Costello
    Commented Oct 13, 2015 at 19:43
  • $\begingroup$ how can you exclude with exponentially small probability that $Y_i\approx -X_i$? I would expect a power law in $\epsilon^2$, not an exponential. $\endgroup$
    – Carlo Beenakker
    Commented Oct 13, 2015 at 20:00

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