A Note on Sharp Inequality of Stopped Sumsof Non-Negative IID Random Variables

Hitczenko 2 studied the following inequality \mathrmE\left(\sum_i=1^\tau \xi_i\right)^r \leq C_r \mathrmE\left(\sum_i=1^\tau^\prime \xi_i\right)^r, \quad 1 \leq r<\infty where （ξ_i） is a sequence of independent non-negative random variables, r is a stopping time, r’ is a copy of r independent of the sequence （ξ_i）, and C_r is a constant which does not dependent on （ξ_i）. He obtained that the best possible constant for above inequality is C_r = 2^r-1. A question concerning the best possible constant for the above inequality when （ξ_i） is a sequence of non-negative i.i.d. random variables was raised in Hitczenko 2. We show here that the constant C_r = 2^r-1 is also sharp for sums of non-negative i.i.d. random variables.