Does Averaging Always Increase Concentration?

Question

Let $X_1,X_2,\ldots$ be i.i.d zero-mean real random variables and $\epsilon>0$. Is there a simple argument that shows $$\mathbb{P}(|X_1 + X_2 + \dots + X_n| > n\epsilon) \geq \mathbb{P}(|X_1 + X_2 + \dots + X_m| > m\epsilon)$$ whenever $n<m$? The normal case says yes. Assuming $X_i$ has a finite first moment is fine with me.

Answer 1

Community wiki answer so the question doesn't remain unanswered:

As was shown in the comments, this inequality doesn't hold in general. If the variables are $X_i=\pm1$ with probability $\frac12$ each, for $\epsilon\lt\frac13$ we have $\mathsf P(|X_1+X_2|\gt2\epsilon)=\frac12\lt1=\mathsf P(|X_1+X_2+X_3|\gt3\epsilon)$.