Recurrence and transience of contractive autoregressive processes and related Markov chains

Zerner, Martin

doi:10.1214/18-ejp152

Cited by 15 publications

(16 citation statements)

References 43 publications

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“…This proves (24) because s − s * can be chosen arbitrarily small. Assertion (25) for s > 0 is proved in a similar manner. Indeed, pick any s ∈ (0, s * ) (or even s * itself unless s * = ∞) and define…”

Section: Tail Lemmatamentioning

confidence: 82%

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Null recurrence and transience of random difference equations in the contractive case

2017

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Given a sequence (M k , Q k ) k≥1 of independent, identically distributed random vectors with nonnegative components, we consider the recursive Markov chain (X n ) n≥0 , defined by the random difference equation X n = M n X n−1 + Q n for n ≥ 1, where X 0 is independent of (M k , Q k ) k≥1 . Criteria for the null recurrence/transience are provided in the situation where (X n ) n≥0 is contractive in the sense that M 1 · . . . · M n → 0 a.s., yet occasional large values of the Q n overcompensate the contractive behavior so that positive recurrence fails to hold. We also investigate the attractor set of (X n ) n≥0 under the sole assumption that this chain is locally contractive and recurrent.

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Section: Tail Lemmatamentioning

confidence: 82%

“…Remark 3.1. Zerner [25] studied the recurrence/transience of (X n ) n≥0 defined by (1.1) in the more general setting when M is a and that either lim t→∞ t β P(log Q > t) = 0 for some β ∈…”

Section: Resultsmentioning

confidence: 99%

Null recurrence and transience of random difference equations in the contractive case

2017

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“…where the sequence (α n , β n ) n is i.i.d. with positive coefficients, α n < 1 and β n with logarithmic tails, P(β 1 > t) ∼ c/ ln t for large t. Although autoregressive processes AR(1) of the type (1.4) are usually addressed with exponential or power-law tail for β n , see [5], the case of logarithmic tail has been also considered, see [15], [31], [3], and also both papers [1] and [32] for a recent account. Interestingly, our model is critical in the perspective of the Markov chain S n , in the sense that the actual value of the constant c is precisely the transition from recurrence to transience for the chain.…”

Section: Rate Of Escape Of Conditioned Brownian Motionmentioning

confidence: 99%

“…and satisfy a < 0 (contractive case), 0 < b < ∞ (very heavy tail). Following [1] and [32], this prevents the Markov chain S n to be positive recurrent: though the contraction brings stability to the process, yet occasional large values of β n overcompensate this behavior so that positive recurrence fails to hold. In our case, we easily check from (3.…”

Section: Tail Estimate For Umentioning

confidence: 99%

Rate of escape of conditioned Brownian motion

Comets

2022

Electron. J. Probab.

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We study the norm of the two-dimensional Brownian motion conditioned to stay outside the unit disk at all times. By conditioning the process is changed from barely recurrent to slightly transient. We obtain sharp results on the rate of escape to infinity of the process of future minima:(i) we find an integral test on the function g so that the future minima process drops below the barrier exp{ln t × g(ln ln t)} at arbitrary large times; (ii) we show that the future minima process exceeds K √ t × ln ln ln t at arbitrary large times with probability 0 [resp., 1] if K is larger [resp., smaller] than some positive constant.For this, we introduce a renewal structure attached to record times and values. Additional results are given for the long time behavior of the norm.

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“…as θ ↓ 0, all assertions are easily verified. ⊓ ⊔ Remark 5.2 As for the autoregressive Markov chain (M n ) n≥0 , it must be acknowledged that, unlike positive recurrence, there seems to be no complete classification of null recurrence and transience of that chain in terms of the random parameter (A, B); for the contractive case when Π n → 0 a.s. we mention the work by the author with Buraczewski and Iksanov [3] and by Zerner [27], and for the critical case considered in this section the classical…”

mentioning

confidence: 97%

Linear fractional Galton-Watson processes in random environment and perpetuities

Alsmeyer¹

2021

Preprint

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Linear fractional Galton-Watson branching processes in i.i.d. random environment are, on the quenched level, intimately connected to random difference equations by the evolution of the random parameters of their linear fractional marginals. On the other hand, any random difference equation defines an autoregressive Markov chain (a random affine recursion) which can be positive recurrent, null recurrent and transient and which, as the forward iterations of an iterated function system, has an a.s. convergent counterpart in the positive recurrent case given by the corresponding backward iterations. The present expository article aims to provide an explicit view at how these aspects of random difference equations and their stationary limits, called perpetuities, enter into the results and the analysis, especially in quenched regime. Although most of the results presented here are known, we hope that the offered perspective will be welcomed by some readers.

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Recurrence and transience of contractive autoregressive processes and related Markov chains

Cited by 15 publications

References 43 publications

Null recurrence and transience of random difference equations in the contractive case

Null recurrence and transience of random difference equations in the contractive case

Rate of escape of conditioned Brownian motion

Linear fractional Galton-Watson processes in random environment and perpetuities

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