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Yes, the condition $EY_i=0$ cannot be dropped. Indeed, if e.g. the $Y_i$'s are iid with $\mu=EY_i\ne0$, then, by the law of large numbers, the left-hand side of the inequality in question will go to $1$ for each $\epsilon\in(0,\|\mu\|)$, whereas the right-hand side of the inequality will go to $0$.