Largest-summand bound for independent symmetric variables

On ${\mathbb{R}}^n$ let $B_j$ be the Borel set where $j$ is the least index attaining the largest coordinate magnitude. The sets $B_j$ partition the space and are invariant under flipping the sign of coordinate $j$. The product joint law and symmetry of each marginal make that sign flip measure preserving. Ties and zero coordinates are included by the least-index rule.

Fix $j$, and write $R={\sum }_{k\mathrm{\ne }j}{y}_k$. Since $2\mathrm{\mid }{y}_j\mathrm{\mid }=\mathrm{\mid }(R+y_j)-(R-y_j)\mathrm{\mid }\le \mathrm{\mid }R+y_j\mathrm{\mid }+\mathrm{\mid }R-y_j\mathrm{\mid }$, at least one of $\mathrm{\mid }R+y_j\mathrm{\mid },\mathrm{\mid }R-y_j\mathrm{\mid }$ is at least $\mathrm{\mid }{y}_j\mathrm{\mid }$. On $B_j\cap \{\mathrm{\mid }{y}_j\mathrm{\mid }\ge t\}$ the two indicators of a final magnitude at least $t$, before and after the sign flip, therefore sum to at least one. Integrate using flip invariance to obtain $2\mathbb{P}(B_j\cap \{\mathrm{\mid }{S}_n\mathrm{\mid }\ge t\})\ge \mathbb{P}(B_j\cap \{\mathrm{\mid }{Y}_j\mathrm{\mid }\ge t\})$. The identical argument on $\mathrm{\mid }{y}_j\mathrm{\mid }>t$ uses strict final events and proves their version directly.

Summing over $j$ gives both bounds. Under IID, independence gives $\mathbb{P}({\max }_k\mathrm{\mid }{Y}_k\mathrm{\mid }>t)=1-(1-p{)}^n$. Finally $1-p\le e^{-p}$ for $0\le p\le 1$, so $(1-p{)}^n\le e^{-np}$. This includes $p=0$, $p=1$, and $n=1$.