On dependence structure of copula-based Markov chains

Longla, Martial

doi:10.1051/ps/2013052

Cited by 11 publications

(22 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Hence the label ‘

ℒ^{2}

spectral gap’ for condition (R1) in Theorem 3.3. (cf. also Longla (2014, proof of Lemma 2.1)) Since the operator T 0 is self‐adjoint, it satisfies

‖ T_{0}^{n} ‖ = ‖ T_{0} ‖^{n}

for every positive integer n by a theorem in functional analysis. Also, as a consequence of (again) (2.6), it satisfies

‖ T_{0}^{m} ‖ = ρ_{X} (m)

for every positive integer m .…”

Section: Strictly Stationary Reversible Markov Chainsmentioning

confidence: 98%

“…Proposition Under the hypothesis (the entire first paragraph, including reversibility ) of Theorem 3.3, the following three statements hold: In the notations of Theorem 3.3, if condition (A2) holds, then condition (R1) (the

ℒ^{2}

spectral gap condition) holds. (cf. Longla (2014, Lemma 2.1)). Regardless of the value (in [0, 1]) of

ρ_{X} (1)

, one has that for every

n \in ℕ

ρ_{X} (n) = {[ρ_{X} (1)]}^{n}

. In the notations of Theorem 3.3, if condition (A3) holds, then condition (A2) holds. …”

Section: Strictly Stationary Reversible Markov Chainsmentioning

confidence: 99%

“…(cf. Longla (2014, Lemma 2.1)). Regardless of the value (in [0, 1]) of

ρ_{X} (1)

, one has that for every

n \in ℕ

ρ_{X} (n) = {[ρ_{X} (1)]}^{n}

.…”

Section: Strictly Stationary Reversible Markov Chainsmentioning

confidence: 99%

“…Proposition 3.5(II) is taken essentially from Longla (2014, Lemma 2.1). Its formulation there was formally in a context involving certain types of copulas, but it easily extends beyond that context.…”

Section: Strictly Stationary Reversible Markov Chainsmentioning

confidence: 99%

“…It is a well known feature of Theorem 1.1 that the geometric ergodicity and

ℒ^{2}

spectral gap conditions therein can be formulated in terms of the dependence coefficients associated with the absolute regularity (

β

‐mixing) and

ρ

‐mixing conditions respectively. See for example the articles of Longla and Peligrad (2012), Longla (2013) and Longla (2014). It turns out that the dependence coefficients associated with the strong mixing (

α

‐mixing) condition of Rosenblatt (1956) apparently can bring some extra clarity to, and thereby perhaps facilitate a relatively gentle exposition of, Theorem 1.1.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

On some basic features of strictly stationary, reversible Markov chains

Bradley

2021

Journal Time Series Analysis

View full text Add to dashboard Cite

It has been well known for some time that for strictly stationary Markov chains that are 'reversible', the special symmetry (with the distribution of the Markov chain as a whole being invariant under a reversal of the 'direction of time') provides special extra features in the mathematical theory. This article here is in part an exposition of some of the basic aspects of that special theory. The mathematical techniques employed in this review are relatively gentle, involving only some basic measure-theoretic probability theory. To that special theory, a couple of new results are contributed here that are connected with the Rosenblatt strong mixing condition; and those new results in turn assist in bringing further clarity to the exposition of the theory.

show abstract

“…Hence the label ‘

ℒ^{2}

spectral gap’ for condition (R1) in Theorem 3.3. (cf. also Longla (2014, proof of Lemma 2.1)) Since the operator T 0 is self‐adjoint, it satisfies

‖ T_{0}^{n} ‖ = ‖ T_{0} ‖^{n}

for every positive integer n by a theorem in functional analysis. Also, as a consequence of (again) (2.6), it satisfies

‖ T_{0}^{m} ‖ = ρ_{X} (m)

for every positive integer m .…”

Section: Strictly Stationary Reversible Markov Chainsmentioning

confidence: 98%

ℒ^{2}

spectral gap condition) holds. (cf. Longla (2014, Lemma 2.1)). Regardless of the value (in [0, 1]) of

ρ_{X} (1)

, one has that for every

n \in ℕ

ρ_{X} (n) = {[ρ_{X} (1)]}^{n}

. In the notations of Theorem 3.3, if condition (A3) holds, then condition (A2) holds. …”

Section: Strictly Stationary Reversible Markov Chainsmentioning

confidence: 99%

“…(cf. Longla (2014, Lemma 2.1)). Regardless of the value (in [0, 1]) of

ρ_{X} (1)

, one has that for every

n \in ℕ

ρ_{X} (n) = {[ρ_{X} (1)]}^{n}

.…”

Section: Strictly Stationary Reversible Markov Chainsmentioning

confidence: 99%

Section: Strictly Stationary Reversible Markov Chainsmentioning

confidence: 99%

“…It is a well known feature of Theorem 1.1 that the geometric ergodicity and

ℒ^{2}

spectral gap conditions therein can be formulated in terms of the dependence coefficients associated with the absolute regularity (

β

‐mixing) and

ρ

α

‐mixing) condition of Rosenblatt (1956) apparently can bring some extra clarity to, and thereby perhaps facilitate a relatively gentle exposition of, Theorem 1.1.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On some basic features of strictly stationary, reversible Markov chains

Bradley

2021

Journal Time Series Analysis

View full text Add to dashboard Cite

show abstract

New copula families and mixing properties

Longla

2024

Stat Papers

View full text Add to dashboard Cite

We characterize absolutely continuous symmetric copulas with square integrable densities in this paper. This characterization is used to create new copula families, that are perturbations of the independence copula. The full study of mixing properties of Markov chains generated by these copula families is conducted. An extension that includes the Farlie–Gumbel–Morgenstern family of copulas is proposed. We propose some examples of copulas that generate non-mixing Markov chains, but whose convex combinations generate $$\psi $$ ψ -mixing Markov chains. Some general results on $$\psi $$ ψ -mixing are given. The Spearman’s correlation $$\rho _S$$ ρ S and Kendall’s $$\tau $$ τ are provided for the created copula families. Some general remarks are provided for $$\rho _S$$ ρ S and $$\tau $$ τ . A central limit theorem is provided for parameter estimators in one example. A simulation study is conducted to support derived asymptotic distributions for some examples.

show abstract