Not looking for a proof

I have the following inequality given to me: If $X_i$ are i.i.d symmetric random variables and $S_n$ their partial sum, then

$$P(|S_n| > x) \geq \frac{1}{2} P(\max |X_i| > x)$$

Now, I have successfully proven this inequality for random variables that are symmetric around $0$. But, the inequality merely states "symmetric" not "symmetric around 0".

I don't think this inequality is true for any symmetric random variables in general, unless they're symmetric around $0$, but I am having trouble showing otherwise.

Is this indeed a typo, or is it actually true for any symmetric random variables?


Not true.Take $X_i=1$ for all $i$ and $0<x<1$. When you say that a distribution is symmetric you usually mean that it is symmetric about $0$. With that understanding the inequality is correct.


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