Contrat doctoral Probabilités et statistiques Orléans Tours M/F

Job Description

Thesis Subject

PhD Topic: Persistence Properties of Random Walks and Autoregressive Processes on ℝ
Proposed by
L. Delsol & M. Zani (University of Orléans)
M. Peigné & K. Raschel (University of Tours)

Probabilistic and Statistical Perspectives

Let θ be a fixed parameter greater than 1. Consider a sequence of centered, independent, and identically distributed random variables (ξₙ)ₙ≥₀ with distribution μ, defined on a probability space (Ω, ℱ, ℙ). We define:
• Y₀(θ) = 0
• Yₙ(θ) = ξ₁ + θξ₂ + ... + θⁿ⁻¹ξₙ
We also define:
• p(θ) = limₙ→∞ pₙ(θ), where pₙ(θ) = ℙ(S₁(θ) ..., Sₙ(θ) positive).
Here, θ is called the coupling parameter, and pₙ(θ) represents the persistence probability up to time n for the sequence (Yₖ(θ))ₖ≥₀.

The inequality p(θ) positive holds for θ greater than 1. We are particularly interested in the behavior of the function θ ↦ p(θ) near 1. It is known that p(1) = 0, as the autoregressive process reduces to a random walk when the coupling parameter equals 1.

It has been conjectured that this function behaves universally like square root of (θ - 1) in a neighborhood of 1. In recent work currently in preparation, L. H. Ngo (Hanoi National University of Education), M. Peigné, and K. Raschel established this property under restrictive conditions (notably, the measure μ must admit a density with compact support). Their approach builds on an idea by Z. Kabluchko (University of Münster) and relies on the convergence of the process (Yₖ(θ))ₖ≥₀, suitably normalized, to an explosive Ornstein-Uhlenbeck process. This work heavily depends on papers by D. Denisov, A. Sakhanenko, and V. Wachtel concerning random walks with non-identically distributed increments.

Research Questions:
1. Extending Universality Results:
The first question is to extend this universality result to distributions with unbounded support that do not necessarily admit a density with respect to the Lebesgue measure. This requires extending the results from previous work on the fluctuations of random walks constructed from triangular arrays, adapting the approach to the specific features of the autoregressive model.

2. Persistence Probability for Oscillating Random Walks:
The question of the behavior of the persistence probability as n → ∞ also arises naturally for models other than "classical" random walks. In particular, it remains completely open for "oscillating" random walks, where increments are governed by distinct probability measures on ℝ⁺ and ℝ⁻. The approach introduced in recent work, which relies on the renewal theory of aperiodic sequences of operators, appears promising for addressing this challenging problem. Applications are numerous, especially in population dynamics in non-homogeneous random environments.

3. Statistical Tests for Oscillating Random Walks:
Under sufficiently strong moment assumptions, oscillating random walks satisfy a central limit theorem-type result, with the limiting distribution being the marginal of a skew Brownian motion whose parameters can be explicitly determined. The development of statistical tests for these models remains to be explored: estimation of the means and variances of each increment distribution, estimation of the parameters of the limiting law, etc. This work aligns with the theory of statistical testing for autoregressive systems exhibiting transition phenomena and will build on recent results on statistical tests for skew Brownian motion.

4. Persistence Probabilities for Higher-Dependence Autoregressive Processes:
One may also study persistence probabilities for autoregressive processes with a higher dependence index. Very little is known about the persistence of such processes, apart from the existence of the persistence exponent. Recent work suggests connections with random walks in cones, although these links have not yet been formalized. This represents a new and broad research direction, adaptable depending on the progress of the doctoral student.


Supervision:
• L. Delsol: 25%
• M. Zani: 30%
• M. Peigné: 25%
• K. Raschel: 20%

References:
• Aurzada, F., Mukherjee, S., & Zeitouni, O. (2021): Persistence exponents in Markov chains. Ann. Inst. Henri Poincaré, Probab. Stat.
• Dembo, A., Ding, J., & Yan, J. (2019): Persistence versus stability for auto-regressive processes. arXiv:1906.00473
• Denisov, D., Sakhanenko, A., & Wachtel, V. (2018): First-passage times for random walks with nonidentically distributed increments. Ann. Probab.
• Denisov, D., Sakhanenko, A., & Wachtel, V. (2021): First-passage times for random walks in the triangular array setting. Prog. Probab.
• Lejay, A., & Pigato, P. (2018): Statistical estimation of the Oscillating Brownian Motion. Bernoulli.
• Lejay, A., & Mazzonetto, S. (2024): Maximum likelihood estimator for skew Brownian motion: The convergence rate. Scand. J. Stat.
• Peigné, M., Pham, D.C., & Vo, T.D. (2025): A local limit theorem for oscillating random walks. arXiv:2509.15647
• Ngo, L.H., Peigné, M., & Raschel, K. (2025+): Universal behavior of persistence probabilities for some autoregressive sequences. Work in progress.
• Teräsvirta, T. (1994): Specification, Estimation, and Evaluation of Smooth Transition Autoregressive Models.Journal of the American Statistical Association.

Your Work Environment

Research work in mathematics.

Compensation and benefits

Compensation

2300 € gross monthly

Annual leave and RTT

44 jours

Remote Working practice and compensation

Pratique et indemnisation du TT

Transport

Prise en charge à 75% du coût et forfait mobilité durable jusqu’à 300€

About the offer

About the CNRS

The CNRS is a major player in fundamental research on a global scale. The CNRS is the only French organization active in all scientific fields. Its unique position as a multi-specialist allows it to bring together different disciplines to address the most important challenges of the contemporary world, in connection with the actors of change.

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