Let $X_1, X_2, dots$ be independent and symmetric random variables such that $S_n = X_1 + cdots + X_n$ converges to a finite valued random variable $S$ a.s. and let $S^* = sup_{1 leq n leq infty} S_n$ (which is finite a.s.). We construct upper and lower bounds for $s_y$ and $s_y^*$, the upper $1/y$-th quantile of $S_y$ and $S^*$, respectively. Our approximations rely on an explicitly computable quantity $underline q_y$ for which we prove that $$rac 1 2 underline q_{y/2} < s_y^* < 2 underline q_{2y} quad ext{ and } quad rac 1 2 underline q_{ (y/4) ( 1 + sqrt{ 1 - 8/y})} < s_y < 2 underline q_{2y}. $$ The RHS's hold for $y geq 2$ and the LHS's for $y geq 94$ and $y geq 97$, respectively. Although our results are derived primarily for symmetric random variables, they apply to non-negative variates and extend to an absolute value of a sum of independent but otherwise arbitrary random variables.
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