• DOI:10.1007/s10955-019-02384-3
• Corpus ID: 119590874

Persistence Exponents via Perturbation Theory: AR(1)-Processes

@article{Aurzada2019PersistenceEV,
  title={Persistence Exponents via Perturbation Theory: AR(1)-Processes},
  author={Frank Aurzada and Marvin Kettner},
  journal={Journal of Statistical Physics},
  year={2019}
}

• F. Aurzada, Marvin Kettner
• Published 23 October 2018
• Mathematics
• Journal of Statistical Physics

For AR(1)-processes $X_n=\rho X_{n-1}+\xi_n$, $n\in\mathbb{N}$, where $\rho\in\mathbb{R}$ and $(\xi_i)_{i\in\mathbb{N}}$ is an i.i.d. sequence of random variables, we study the persistence probabilities $\mathbb{P}(X_0\ge 0,\dots, X_N\ge 0)$ for $N\to\infty$. For a wide class of Markov processes a recent result [Aurzada, Mukherjee, Zeitouni; arXiv:1703.06447; 2017] shows that these probabilities decrease exponentially fast and that the rate of decay can be identified as an eigenvalue of some integral operator. We discuss a perturbation technique to determine a series expansion of the eigenvalue in the parameter $\rho$ for normally distributed AR(1)-processes.