Positivity of integrated random walks

Vysotsky, Vladislav

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 1, pp. 195-213

Take a centered random walk $S_n$ and consider the sequence of its partial sums $A_n:={\sum }_{i=1}^n{S}_i$. Suppose $S_1$ is in the domain of normal attraction of an $\alpha$-stable law with $1<\alpha \le 2$. Assuming that $S_1$ is either right-exponential (i.e. $\mathbb{P}(S_1>x|S_1>0)={\mathrm{e}}^{-ax}$ for some $a>0$ and all $x>0$) or right-continuous (skip free), we prove that

$$\mathbb{P}\{A_1>0,\cdots ,A_N>0\}\sim C_{\alpha }{N}^{1/\left(2\alpha \right)-1/2}$$

as $N\to \infty$, where $C_{\alpha }>0$ depends on the distribution of the walk. We also consider a conditional version of this problem and study positivity of integrated discrete bridges.

Soit $S_n$ une marche aléatoire centrée, nous considérons la suite de ses sommes partielles $A_n:={\sum }_{i=1}^n{S}_i$. Nous supposons que $S_1$ est dans le domaine d’attraction normale d’une loi $\alpha$-stable avec $1<\alpha \le 2$. En supposant que $S_1$ est soit exponentielle à droite (i.e. $\mathbb{P}(S_1>x|S_1>0)={\mathrm{e}}^{-ax}$), soit continue à droite (i.e. $\mathbb{P}(S_1=1|S_1>0)=1$), nous prouvons que

$$\mathbb{P}\{A_1>0,\cdots ,A_N>0\}\sim C_{\alpha }{N}^{1/\left(2\alpha \right)-1/2}$$

quand $N\to \infty$, où $C_{\alpha }>0$ dépend de la distribution de la marche. Nous considérons aussi une version conditionnelle de ce problème et nous étudions la positivité de ponts discrets intégrés.

Classification : 60G50, 60F99
Keywords: integrated random walk, persistence, one-sided exit probability, unilateral small deviations, area of random walk, Sparre-Andersen theorem, stable excursion, area of excursion

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