Persistence of One-Dimensional AR(1)-Sequences

Hinrichs, Günter; Kolb, Martin; Wachtel, Vitali

doi:10.1007/s10959-018-0850-0

Cited by 10 publications

(7 citation statements)

References 22 publications

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“…The idea of relating the persistence exponent to an eigenvalue of an integral operator has already received great attention (see [2,3,12,17], also see [8,9,20,25,26] for the quasi-stationary approach). Among these, [12] is a very recent work, which uses functional analytical methods, such as the Fredholm alternative to obtain a precise asymptotic behaviour of the persistence probabilities. However, the persistence exponent is given there only implicitly.…”

Section: Related Work and Discussionmentioning

confidence: 99%

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

Аурзада

Kettner

2019

J Stat Phys

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For AR(1)-processes X n = ρX n−1 + ξ n , n ∈ N, where ρ ∈ R and (ξ i ) i∈N is an i.i.d. sequence of random variables, we study the persistence probabilities P(X 0 ≥ 0, . . . , X N ≥ 0) for N → ∞. For a wide class of Markov processes a recent result [3] 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 ρ for normally distributed AR(1)-processes.

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Section: Related Work and Discussionmentioning

confidence: 99%

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

Аурзада

Kettner

2019

J Stat Phys

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“…, k} such that θ 0,i = θ, the process Y (i ) satisfies Assumption (A) with the objects θ 0,i , j i , α i ,n , W i , ν i and η i such that j i ≡ 0, I i = {1} and η i > 0 on E i (note that we omit the second index for η i ,1 and ν i ,1 ). Many references provide practical criteria to check Assumption (A) with I S = {1}, j S ≡ 0, η S > 0 and with α S,n converging exponentially fast to 0, which corresponds to the classical irreducible situation, see for example [16,9,10,11,15,20,1,18,19,21,4]. The process Y (i ) is called the process X restricted to E i .…”

Section: Quasi-stationary Distributions In Reducible State Spacesmentioning

confidence: 99%

Quasi-stationary distributions in reducible state spaces

Champagnat¹,

Villemonais²

2022

Preprint

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We study quasi-stationary distributions and quasi-limiting behavior of Markov chains in general reducible state spaces with absorption. We propose a set of assumptions dealing with particular situations where the state space can be decomposed into three subsets between which communication is only possible in a single direction. These assumptions allow us to characterize the exponential order of magnitude and the exact polynomial correction, called polynomial convergence parameter, for the leading order term of the semigroup for large time. They also provide explicit convergence speeds to this leading order term. We apply these results to general Markov chains with finitely or denumerably many communication classes using a specific induction over the communication classes of the chain. We are able to explicitely characterize the polynomial convergence parameter, to determine the complete set of quasistationary distributions and to provide explicit estimates for the speed of convergence to quasi-limiting distributions in the case of finitely many communication classes. We conclude with an application of these results to the case of denumerable state spaces, where we are able to prove that, in general, there is existence of a quasi-stationary distribution without assuming irreducibility before absorption. This actually holds true assuming only aperiodicity, the existence of a Lyapunov function and the existence of a point in the state space from which the return time is finite with positive probability.

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“…> n) converges towards the corresponding quasi-stationary distribution, see [13]. It is worth mentioning that one can compute the persistence exponent λ in some special cases only.…”

Section: Introductionmentioning

confidence: 99%

Persistence of autoregressive sequences with logarithmic tails

Denisov¹,

Hinrich²,

Kolb³

et al. 2022

Preprint

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We consider autoregressive sequences Xn = aX n−1 + ξn and Mn = max{aM n−1 , ξn} with a constant a ∈ (0, 1) and with positive, independent and identically distributed innovations {ξ k }. It is known that if P(ξ 1 > x) ∼ d log x with some d ∈ (0, − log a) then the chains {Xn} and {Mn} are null recurrent. We investigate the tail behaviour of recurrence times in this case of logarithmically decaying tails. More precisely, we show that the tails of recurrence times are regularly varying of index −1 − d/ log a. We also prove limit theorems for {Xn} and {Mn} conditioned to stay over a fixed level x 0 . Furthermore, we study tail asymptotics for recurrence times of {Xn} and {Mn} in the case when these chains are positive recurrent and the tail of log ξ 1 is subexponential.

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Persistence of One-Dimensional AR(1)-Sequences

Cited by 10 publications

References 22 publications

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

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

Quasi-stationary distributions in reducible state spaces

Persistence of autoregressive sequences with logarithmic tails

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