Persistence of iterated partial sums

Let p_n denote the persistence probability that the first n iterated partial sums of integrable, zero-mean, i.i.d. random variables X_k, are negative. We show that p_n is bounded above up to universal constant by the square root of the expected absolute value of the empirical average of {X_k}. A converse bound holds whenever P(-X_1>t) is up to constant exp(-b t) for some b>0 or when P(-X_1>t) decays super-exponentially in t. Consequently, for such random variables we have that p_n decays as n^{-1/4} if X_1 has finite second moment. In contrast, we show that for any 0 < c < 1/4 there exist integrable, zero-mean random variables for which the rate of decay of p_n is n^{-c}.