# 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}
}
• Published 19 March 2017
• Mathematics
• Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
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 $S… 9 Citations Persistence Exponents via Perturbation Theory: AR(1)-Processes • Mathematics Journal of Statistical Physics • 2019 For AR(1)-processes$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 <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 ## References SHOWING 1-10 OF 41 REFERENCES Persistence of Gaussian processes: non-summable correlations • Computer Science • 2015 It is shown that the persistence probabilities decay rate of -logP(supt∈[0,T]{Z(t)}<0) is precisely of order, thereby closing the gap between the lower and upper bounds of Newell and Rosenblatt (Ann. Math. Stat. 47:146–163, 2015). Persistence of One-Dimensional AR(1)-Sequences • Mathematics Journal of Theoretical Probability • 2018 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 • Mathematics • 2015 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 • 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 On the Probability That a Stationary Gaussian Process With Spectral Gap Remains Non-negative on a Long Interval • Mathematics International Mathematics Research Notices • 2018 Let$f$be a zero mean continuous stationary Gaussian process on$\mathbb{R}$whose spectral measure vanishes in a$\delta $-neighborhood of the origin. Then, the probability that$f$stays On the maximum of a critical branching process in a random environment Let {ξ,ι} be a critical branching process in a random environment with linear-fractional generating functions. We demonstrate that, under some conditions, as χ —>· <», where CQ is a positive Persistence of Gaussian stationary processes: A spectral perspective • Mathematics • 2017 We study the persistence probability of a centered stationary Gaussian process on$\mathbb{Z}$or$\mathbb{R}\$, that is, its probability to remain positive for a long time. We describe the delicate
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