Persistence of iterated partial sums

Abstract

Let Sn(2) denote the iterated partial sums. That is, Sn(2) = S1 +S2+· · ·+Sn, where Si = X1 +X2+· · ·+Xi. Assuming X1,X2, . . . , X n are integrable, zero-mean, i.i.d. random variables, we show that the persistence probabilities pn(2) := ℙ(max 1≤i≤n Si(2) √ E|Sn+1|/(n+ 1)E|X1| with c ≤ 6 √ 30 (and c = 2 whenever X1 is symmetric). The converse inequality holds whenever the non-zero min(-X1, 0) is bounded or when it has only finite third moment and in addition X1 is squared integrable. Furthermore, p n(2)= n -1/4 for any nondegenerate squared integrable, i.i.d., zero-mean Xi . In contrast, we show that for any 0 <1/4 there exist integrable, zero-mean random variables for which the rate of decay of pn(2) is n -γ . © Association des Publications de l'Institut Henri Poincaré, 2013.