For $n\ge 1$ independent random signs, $\mathbb{P}(\mathrm{\mid }S\mathrm{\mid }\ge t)\le 2\exp (-t^2\mathrm{/}(2n))$ for $t>0$

For every real $u$, [L2] groups the finite outcome sum by the sign tuple $a\in \{-1,1{\}}^n$. Mutual atom independence [L6] and exponential addition [L8] give $\mathbb{E}{e}^{uS}={\sum }_{a\in \{-1,1{\}}^n}{\prod }_{i<n}(e^{u{a}_i}\mathbb{P}({\epsilon }_i=a_i))={\prod }_{i<n}{\sum }_{a_i\in \{-1,1\}}{e}^{u{a}_i}\mathbb{P}({\epsilon }_i=a_i)={\prod }_{i<n}\mathbb{E}{e}^{u{\epsilon }_i},$ where finite Fubini [L7] justifies the middle equality. All sums and factors are finite.

For $u>0$, strict monotonicity gives $\{S\ge t\}=\{\exp (uS)\ge \exp (ut)\}$, so [L1] gives $\mathbb{P}(S\ge t)\le \exp (-ut)\mathbb{E}[\exp (uS)]$.

By step 1.1 and [L3], $\mathbb{E}[\exp (uS)]\le \exp (n{u}^2\mathrm{/}2)$.

Choose $u=t\mathrm{/}n>0$. Substitution in steps 1.2 and 2.1 gives $\mathbb{P}(S\ge t)\le \exp (-t^2\mathrm{/}(2n))$.

Negation merely relabels the two attained values of each sign, so the joint-value factorization in [L6] shows that the variables $-{\epsilon }_i$ are again mutually independent uniform signs. Step 3.1 applied to $-S$ gives the same bound for $\mathbb{P}(S\le -t)$.

Since $\{\mathrm{\mid }S\mathrm{\mid }\ge t\}=\{S\ge t\}\cup \{S\le -t\}$, [L5] and steps 3.1 and 4.1 give the result. The excluded boundary $t=0$ would only give the valid but uninformative bound $1\le 2$.