Persistence of iterated partial sums
Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 3, pp. 873-884.

Let ${S}_{n}^{\left(2\right)}$ denote the iterated partial sums. That is, ${S}_{n}^{\left(2\right)}={S}_{1}+{S}_{2}+\cdots +{S}_{n}$, where ${S}_{i}={X}_{1}+{X}_{2}+\cdots +{X}_{i}$. Assuming ${X}_{1},{X}_{2},...,{X}_{n}$ are integrable, zero-mean, i.i.d. random variables, we show that the persistence probabilities

 ${p}_{n}^{\left(2\right)}:=ℙ\left(\underset{1\le i\le n}{max}{S}_{i}^{\left(2\right)}<0\right)\le c\sqrt{\frac{𝔼|{S}_{n+1}|}{\left(n+1\right)𝔼|{X}_{1}|}},$
with $c\le 6\sqrt{30}$ (and $c=2$ whenever ${X}_{1}$ is symmetric). The converse inequality holds whenever the non-zero $min\left(-{X}_{1},0\right)$ is bounded or when it has only finite third moment and in addition ${X}_{1}$ is squared integrable. Furthermore, ${p}_{n}^{\left(2\right)}\asymp {n}^{-1/4}$ for any non-degenerate squared integrable, i.i.d., zero-mean ${X}_{i}$. In contrast, we show that for any $0<\gamma <1/4$ there exist integrable, zero-mean random variables for which the rate of decay of ${p}_{n}^{\left(2\right)}$ is ${n}^{-\gamma }$.

Soit ${S}_{n}^{\left(2\right)}$ la somme partielle itérée, c’est à dire ${S}_{n}^{\left(2\right)}={S}_{1}+{S}_{2}+\cdots +{S}_{n}$, où ${S}_{i}={X}_{1}+{X}_{2}+\cdots +{X}_{i}$. Pour des variables aléatoires ${X}_{1},{X}_{2},...,{X}_{n}$ i.i.d. intégrables et de moyenne nulle, nous montrons que les probabilités de persistance satisfont

 ${p}_{n}^{\left(2\right)}:=ℙ\left(\underset{1\le i\le n}{max}{S}_{i}^{\left(2\right)}<0\right)\le c\sqrt{\frac{𝔼|{S}_{n+1}|}{\left(n+1\right)𝔼|{X}_{1}|}},$
avec $c\le 6\sqrt{30}$ (et $c=2$ dès que ${X}_{1}$ est symétrique). En outre, l’inégalité inverse est vraie quand $ℙ\left(-{X}_{1}>t\right)\asymp {e}^{-\alpha t}$ pour un $\alpha >0$ ou si $ℙ{\left(-{X}_{1}>t\right)}^{1/t}\to 0$ quand $t\to \infty$. Pour ces variables, on a donc ${p}_{n}^{\left(2\right)}\asymp {n}^{-1/4}$ si ${X}_{1}$ admet un moment d’ordre 2. Par contre nous montrons que pour tout $0<\gamma <1/4$, il existe des variables intégrables de moyenne nulle pour lesquelles ${p}_{n}^{\left(2\right)}$ décroît comme ${n}^{-\gamma }$.

DOI: 10.1214/11-AIHP452
Classification: 60G50, 60F10
Keywords: first passage time, iterated partial sums, persistence, lower tail probability, one-sided probability, random walk
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Dembo, Amir; Ding, Jian; Gao, Fuchang. Persistence of iterated partial sums. Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 3, pp. 873-884. doi : 10.1214/11-AIHP452. http://archive.numdam.org/articles/10.1214/11-AIHP452/

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