[Paper Review] Persistence probabilities for stationary increment processes

Frank Aurzada, Nadine Guillotin‐Plantard+1 more|arXiv (Cornell University)|Jun 1, 2016

Stochastic processes and statistical mechanics29 references3 citations

TL;DR

This paper introduces a novel approach to estimating persistence probabilities for discrete-time processes with stationary increments, using the expected range of the process as a key tool. It establishes tight upper and lower bounds for persistence exponents without requiring exponential moments, achieving optimal order estimates—particularly for fractional Brownian motion, random walks in random scenery, and the Matheron-de Marsily model in Z² with random orientations.

ABSTRACT

We study the persistence probability for processes with stationary increments. Our results apply to a number of examples: sums of stationary correlated random variables whose scaling limit is fractional Brownian motion, random walks in random sceneries, random processes in Brownian scenery, and the Matheron-de Marsily model in Z^2 with random orientations of the horizontal layers. Using a new approach, strongly related to the study of the range, we obtain an upper bound of optimal order in the general case and improved lower bounds (compared to previous literature) for many processes.

Motivation & Objective

• To develop a general, non-Markovian method for estimating persistence probabilities in processes with stationary increments.
• To provide sharp upper and lower bounds for persistence exponents across diverse stochastic processes.
• To eliminate the need for exponential moments, which are often infeasible in non-Markovian settings.
• To apply the method to key examples: fractional Brownian motion, random walks in random scenery, and the Matheron-de Marsily model.
• To establish the exact persistence exponent for the Matheron-de Marsily model in Z² with random layer orientations, confirming a long-standing conjecture.

Proposed method

• Use the expectation of the maximum process, E[Zₙ*], as a proxy for persistence probability, showing P(Zₙ* ≤ a) ≈ E[Zₙ*]/n up to n^o(1) terms.
• Leverage the range Rₙ = #{Z₁,…,Zₙ} as a measurable proxy for the maximum, especially in lattice-valued processes.
• Apply uniform integrability and weak convergence in Skorokhod space to derive limit laws for the range and maximum in scaling limits.
• Use conditional positive association and moment bounds (via Theorem 2.1 of [14]) to control the growth of the maximum under random walk environments.
• Establish convergence in distribution of Rₙ^(1)/n^{3/4} to a functional of Brownian local time, enabling exact asymptotics.
• Combine limit theorems with the first-passage time relation P(T₀ > n) ≈ c n^{-1/4} to derive persistence exponents.

Experimental results

Research questions

• RQ1 What is the persistence exponent for the Matheron-de Marsily model in Z² with random layer orientations?
• RQ2 Can persistence probabilities for non-Markovian processes with stationary increments be bounded without requiring exponential moments?
• RQ3 How does the expected range of a process relate to its persistence probability?
• RQ4 What is the exact asymptotic behavior of the first return time to zero in the Matheron-de Marsily model?
• RQ5 Can the persistence exponent be derived for processes like fractional Brownian motion and random walks in random scenery using a unified framework?

Key findings

• For the Matheron-de Marsily model in Z² with random layer orientations, the persistence probability decays as P(max_{k=1,…,n} Mₖ^(1) ≤ -1) ∼ (3/4) × [p / (1−p)^{1/4}] × E[sup_{t∈[0,1]} Δₜ^(0)] × n^{-1/4}, confirming a conjecture by Redner and Majumdar.
• The persistence exponent for the Matheron-de Marsily model is exactly 1/4, derived from the scaling limit of the range and maximum of the process.
• The range Rₙ^(1) of the Matheron-de Marsily process satisfies Rₙ^(1)/n^{3/4} → Kₚ × (sup Δₜ^(0) − inf Δₜ^(0)) in distribution, with E[Rₙ^(1)]/n^{3/4} → 2Kₚ × E[sup Δₜ^(0)].
• The first return time to zero satisfies lim_{n→∞} n^{1/4} P(T₀^(1) > n) = (3/2)Kₚ × E[sup_{t∈[0,1]} Δₜ^(0)], providing a precise asymptotic for recurrence behavior.
• The method achieves optimal-order upper bounds for persistence probabilities without requiring exponential moments, applicable to fractional Brownian motion and other non-Markovian processes.
• For bounded increments, the method yields the exact persistence exponent, as E[Zₙ*]/n captures the correct decay rate up to n^o(1) factors.