# Persistence exponents in Markov chains

@article{Aurzada2021PersistenceEI, title={Persistence exponents in Markov chains}, author={Frank Aurzada and Sumit Mukherjee and Ofer Zeitouni}, journal={Annales de l'Institut Henri Poincar{\'e}, Probabilit{\'e}s et Statistiques}, year={2021} }

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 $\{X_i\}_{i\geq 0}$ in a Polish space, where $S$ is a Borel measurable set and $\mu$ is the initial distribution. Focusing on the case of AR($p$) and MA($q$) processes with $p,q\in N$ and continuous innovation distribution, we study the existence of $\lambda$ and its continuity in the parameters, for $S…

## 9 Citations

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

- MathematicsJournal of Statistical Physics
- 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…

Persistence probabilities of weighted sums of stationary Gaussian sequences

- Mathematics
- 2020

With $\{\xi_i\}_{i\ge 0}$ being a centered stationary Gaussian sequence with non negative correlation function $\rho(i):=\mathbb{E}[ \xi_0\xi_i]$ and $\{\sigma(i)\}_{i\ge 1}$ a sequence of positive…

Persistence versus stability for auto-regressive processes

- Mathematics
- 2019

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 autoregressive sequences with logarithmic tails

- Mathematics
- 2022

We consider autoregressive sequences Xn = aXn−1 + ξn and Mn = max{aMn−1, ξn} with a constant a ∈ (0, 1) and with positive, independent and identically distributed innovations {ξk}. It is known that…

Universality for Persistence Exponents of Local Times of Self-Similar Processes with Stationary Increments

- ArtJournal of Theoretical Probability
- 2021

<jats:p>We show that <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathbb {P}( \ell _X(0,T] \le 1)=(c_X+o(1))T^{-(1-H)}$$</jats:tex-math><mml:math…

Persistence of heavy-tailed sample averages occurs by infinitely many jumps

- Mathematics
- 2019

We consider the sample average of a centered random walk in $\mathbb{R}^d$ with regularly varying step size distribution. For the first exit time from a compact convex set $A$ not containing the…

Persistence for a class of order-one autoregressive processes and Mallows-Riordan polynomials

- Mathematics
- 2021

We establish exact formulae for the persistence probabilities of an AR(1) sequence with symmetric uniform innovations in terms of certain families of polynomials, most notably a family introduced by…

Persistence of heavy-tailed sample averages: principle of infinitely many big jumps

- Mathematics
- 2019

: We consider the sample average of a centered random walk in R d with regularly varying step size distribution. For the ﬁrst exit time from a compact convex set A not containing the origin, we show…

Persistence and Ball Exponents for Gaussian Stationary Processes

- Mathematics
- 2021

Consider a real Gaussian stationary process fρ, indexed on either R or Z and admitting a spectral measure ρ. We study θ ρ = − lim T→∞ 1 T logP ( inft∈[0,T ] fρ(t) > l ) , the persistence exponent of…

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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…

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