Persistence of iterated partial sums

Dembo, Amir; Ding, Jian; Gao, Fuchang

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 3, pp. 873-884.

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

$$p_n^{\left(2\right)}:=\mathbb{P}\left(\underset{1\le i\le n}{max}{S}_i^{\left(2\right)}<0\right)\le c\sqrt{\frac{𝔼|S_{n+1}|}{(n+1)𝔼|X_1|}},$$

avec $c\le 6\sqrt{30}$ (et $c=2$ dès que $X_1$ est symétrique). En outre, l’inégalité inverse est vraie quand $\mathbb{P}(-X_1>t)\asymp e^{-\alpha t}$ pour un $\alpha >0$ ou si $\mathbb{P}{(-X_1>t)}^{1/t}\to 0$ 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 $n^{-\gamma }$.

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

$$p_n^{\left(2\right)}:=\mathbb{P}\left(\underset{1\le i\le n}{max}{S}_i^{\left(2\right)}<0\right)\le c\sqrt{\frac{𝔼|S_{n+1}|}{(n+1)𝔼|X_1|}},$$

with $c\le 6\sqrt{30}$ (and $c=2$ whenever $X_1$ is symmetric). The converse inequality holds whenever the non-zero $min(-X_1,0)$ 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 $n^{-\gamma }$.

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