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}
}
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… 
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References

SHOWING 1-10 OF 35 REFERENCES
Persistence exponents in Markov chains
We prove the existence of the persistence exponent $$-\log\lambda:=\lim_{n\to\infty}\frac{1}{n}\log \mathbb{P}_\mu(X_0\in S,\ldots,X_n\in S)$$ for a class of time homogeneous Markov chains
Persistence of One-Dimensional AR(1)-Sequences
For a class of one-dimensional autoregressive sequences $$(X_n)$$ ( X n ) , we consider the tail behaviour of the stopping time $$T_0=\min \lbrace n\ge 1: X_n\le 0 \rbrace $$ T 0 = min { n ≥ 1 : X n
Persistence Probabilities and Exponents
This article deals with the asymptotic behavior as \(t \rightarrow +\infty \) of the survival function \(\mathbb{P}[T > t]\), where T is the first passage time above a non negative level of a random
Persistence versus stability for auto-regressive processes
The stability of an Auto-Regressive (AR) time sequence of finite order $L$, is determined by the maximal modulus $r^\star$ among all zeros of its generating polynomial. If $r^\star 1$ it is
Persistence of a continuous stochastic process with discrete-time sampling.
TLDR
For a>1, corresponding to motion in an unstable potential, there is a nonzero probability of having no zero-crossings in infinite time, and it is shown how to calculate it.
Survival probabilities of autoregressive processes
Given an autoregressive process X of order p (i.e. X n  = a 1 X n −1  + ··· + a p X n −p  + Y n where the random variables Y 1 , Y 2 ,... are i.i.d.), we study the asymptotic behaviour of the
Quasi-stationary distributions and population processes
This survey concerns the study of quasi-stationary distributions with a specific focus on models derived from ecology and population dynamics. We are concerned with the long time behavior of
First-passage phenomena and their applications
Arrival Statistics and Exploration Properties of Mortal Walkers (S B Yuste et al.) First-Passage of a Randomly Accelerated Particle (T Burkhardt) First-Passage Problems in Anomalous Diffusion (A
Banach Lattices and Positive Operators
I. Positive Matrices.- 1. Linear Operators on ?n.- 2. Positive Matrices.- 3. Mean Ergodicity.- 4. Stochastic Matrices.- 5. Doubly Stochastic Matrices.- 6. Irreducible Positive Matrices.- 7. Primitive
Survival probabilities of weighted random walks
We study the asymptotic behaviour of the probability that a weighted sum of centered i.i.d. random variables Xk does not exceed a constant barrier. For regular random walks, the results follow easily
...
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