Uniformly Accurate Quantile Bounds Via The Truncated Moment Generating Function: The Symmetric Case

Let X1,X2, ... be independent and symmetric random variables such that Sn = X1+ ...+Xn converges to a finite valued random variable S a.s. and let S = sup1≤n<∞ Sn (which is finite a.s.). We construct upper and lower bounds for sy and s ∗ y, the upper 1 y th quantile of Sy and S , respectively. Our approximations rely on an explicitly computable quantity q y for which we prove that 1 2 q y/2 < s∗y < 2q2y and 1 2 q y 4 (1+ √ 1−8/y) < sy < 2q2y. The RHS’s hold for y ≥ 2 and the LHS’s for y ≥ 94 and y ≥ 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.