Persistence of iterated partial sums

Abstract

Let $S_n^{(2)}$ denote the iterated partial sums. That is, $S_n^{(2)}=S_1+S_2+ ... +S_n$, where $S_i=X_1+X_2+ ... s+X_i$. Assuming $X_1, X_2,....,X_n$ are integrable, zero-mean, i.i.d. random variables, we show that the persistence probabilities $$p_n^{(2)}:=\PP(\max_{1\le i \le n}S_i^{(2)}< 0) \le c\sqrt{\frac{\EE|S_{n+1}|}{(n+1)\EE|X_1|}},$$ 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, $p_n^{(2)}\asymp n^{-1/4}$ 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 $p_n^{(2)}$ is $n^{-\gamma}$.Comment: overlaps and improves upon an earlier version by Dembo and Gao at arXiv:1101.574

Persistence of integrated stable processes

“…It is worth mentioning that the same persistence exponent (α − 1)/2α appears for the integrals of random walks which are attracted towards this spectrally positive Lévy process-see Remark 1.2 in [5] and the main result of [26]. Our result leads therefore to a natural conjecture on the persistence exponent of general integrated random walks in a stable domain of attraction.…”

“…The universality of the persistence exponent 1/4 for integrals of real-valued Lévy processes having exponential moments on both sides has been shown in [1], with the help of strong approximation arguments. Recently, it was proved in [5] that all integrated real random walks with finite variance also have 1/4 as persistence exponent, extending [25] in which the particular case of the integrated simple random walk was studied. Let us also mention that the survival function of the nth hitting time of zero for the integrated Brownian motion exhibits the same power decay up to a logarithmic term in ct −1/4 (ln(t)) n−1 with an explicit constant c, as shown by the first author in [18].…”

Persistence and first-passage properties in nonequilibrium systems

“…We end up this section by mentioning that the study of integrated processes, including integrated Lévy processes, has recently attracted much attention in the mathematics literature [92] and we refer the reader to Ref. [7] for a review on them.…”

Probability to be positive for the membrane model in dimensions 2 and 3

“…In dimension n = 1 the model corresponds to an integrated Gaussian random walk, see [CD08]. Dembo, Ding and Gao [DDG13] proved that for such processes with zero mean and finite variance the probability to be positive on an interval of lenght N is of order N −1/4 , extending a result by Sinai [Sin92] for integrated simple random walk. We consider here the membrane model defined on a box of side-length 2N + 1, N ∈ N, and focus on dimensions n = 2, 3.…”