Affine stochastic equation with triangular matrices

Damek, Ewa; Zienkiewicz, Jacek

doi:10.1080/10236198.2017.1422249

Cited by 9 publications

(10 citation statements)

References 34 publications

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“…Regarding the case κ + = κ − , we further remark that the methods used in the present paper are not strong enough and thus need to be refined. It seems reasonable to believe that in this case a first order expansion of P (θ) in κ is required with a possible extra term in the power tail of ν, see again [16] for similar considerations in the case of the afore-mentioned two-dimensional recursions.…”

Section: Basic Assumptions and Main Resultsmentioning

confidence: 98%

Asymptotically linear iterated function systems on the real line

Alsmeyer¹,

Brofferio²,

Buraczewski³

2021

Preprint

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Given a sequence of i.i.d. random functions Ψ n : R → R, n ∈ N, we consider the iterated function system and Markov chain which is recursively defined by X x 0 := x and X x n := Ψ n−1 (X x n−1 ) for x ∈ R and n ∈ N. Under the two basic assumptions that the Ψ n are a.s. continuous at any point in R and asymptotically linear at the "endpoints" ±∞, we study the tail behavior of the stationary laws of such Markov chains by means of Markov renewal theory. Our approach provides an extension of Goldie's implicit renewal theory [20] and can also be viewed as an adaptation of Kesten's work on products of random matrices [24] to one-dimensional function systems as described. Our results have applications in quite different areas of applied probability like queuing theory, econometrics, mathematical finance and population dynamics, e.g. ARCH models and random logistic transforms.

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Section: Basic Assumptions and Main Resultsmentioning

confidence: 98%

Asymptotically linear iterated function systems on the real line

Alsmeyer¹,

Brofferio²,

Buraczewski³

2021

Preprint

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“…This is not clear even in the case of 2 × 2 matrices. A particular case A 11 = A 22 > 0 and A 12 ∈ R was studied in [11] where the result is…”

Section: Kesten's Condition and Previous Resultsmentioning

confidence: 99%

“…The example is given in [11], which we briefly see. Let A be an upper triangular matrix with A 11 = A 22 having the index α > 0.…”

Section: Open Questionsmentioning

confidence: 99%

Tail indices for $$\mathbf{A}\mathbf{X}+\mathbf{B}$$ Recursion with Triangular Matrices

Matsui

Świątkowski

2020

J Theor Probab

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We study multivariate stochastic recurrence equations (SREs) with triangular matrices. If coefficient matrices of SREs have strictly positive entries, the classical Kesten result says that the stationary solution is regularly varying and the tail indices are the same in all directions. This framework, however, is too restrictive for applications. In order to widen applicability of SREs, we consider SREs with triangular matrices and we prove that their stationary solutions are regularly varying with component-wise different tail exponents. Several applications to GARCH models are suggested.

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“…This was probably the main motivation of Mentemeier, Wintenberger to study diagonal SRE models and to apply their result to particular cases of BEKK-ARCH and CCC-GARCH [29]. Finally, also BEKK-ARCH with triangular matrices has been of interest, [27] but then only regular behavior of components X j follows from [11], [12], [28].…”

Section: Applicationsmentioning

confidence: 99%