State-Discretization of V-Geometrically Ergodic Markov Chains and Convergence to the Stationary Distribution
Let ( X n ) n ∈ ℕ be a V -geometrically ergodic Markov chain on a measurable space X with invariant probability distribution π . In this paper, we propose a discretization scheme providing a computable sequence ( π ̂ k ) k ≥ 1 of probability measures which approximates π as k growths to infinity. Th...
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| Published in | Methodology and computing in applied probability Vol. 22; no. 3; pp. 905 - 925 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.09.2020
Springer Nature B.V Springer Verlag |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1387-5841 1573-7713 |
| DOI | 10.1007/s11009-019-09746-0 |
Cover
| Abstract | Let
(
X
n
)
n
∈
ℕ
be a
V
-geometrically ergodic Markov chain on a measurable space
X
with invariant probability distribution
π
. In this paper, we propose a discretization scheme providing a computable sequence
(
π
̂
k
)
k
≥
1
of probability measures which approximates
π
as
k
growths to infinity. The probability measure
π
̂
k
is computed from the invariant probability distribution of a finite Markov chain. The convergence rate in total variation of
(
π
̂
k
)
k
≥
1
to
π
is given. As a result, the specific case of first order autoregressive processes with linear and non-linear errors is studied. Finally, illustrations of the procedure for such autoregressive processes are provided, in particular when no explicit formula for
π
is known. |
|---|---|
| AbstractList | Let (Xn)n∈ℕ be a V -geometrically ergodic Markov chain on a measurable space X with invariant probability distribution π. In this paper, we propose a discretization scheme providing a computable sequence (π̂k)k≥1 of probability measures which approximates π as k growths to infinity. The probability measure π̂k is computed from the invariant probability distribution of a finite Markov chain. The convergence rate in total variation of (π̂k)k≥1 to π is given. As a result, the specific case of first order autoregressive processes with linear and non-linear errors is studied. Finally, illustrations of the procedure for such autoregressive processes are provided, in particular when no explicit formula for π is known. Let $(X_n)_{n \in\mathbb{N}}$ be a $V$-geometrically ergodic Markov chain on a measurable space $\mathbb{X}$ with invariant probability distribution $\pi$. In this paper, we propose a discretization scheme providing a computable sequence $(\widehat\pi_k)_{k\ge 1}$ of probability measures which approximates $\pi$ as $k$ growths to infinity. The probability measure $\widehat\pi_k$ is computed from the invariant probability distribution of a finite Markov chain. The convergence rate in total variation of $(\widehat\pi_k)_{k\ge 1}$ to $\pi$ is given. As a result, the specific case of first order autoregressive processes with linear and non-linear errors is studied. Finally, illustrations of the procedure for such autoregressive processes are provided, in particular when no explicit formula for $\pi$ is known. Let ( X n ) n ∈ ℕ be a V -geometrically ergodic Markov chain on a measurable space X with invariant probability distribution π . In this paper, we propose a discretization scheme providing a computable sequence ( π ̂ k ) k ≥ 1 of probability measures which approximates π as k growths to infinity. The probability measure π ̂ k is computed from the invariant probability distribution of a finite Markov chain. The convergence rate in total variation of ( π ̂ k ) k ≥ 1 to π is given. As a result, the specific case of first order autoregressive processes with linear and non-linear errors is studied. Finally, illustrations of the procedure for such autoregressive processes are provided, in particular when no explicit formula for π is known. |
| Author | Hervé, Loic Ledoux, James |
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| References | Loges (CR12) 2004; 25 Ferré, Hervé, Ledoux (CR7) 2013; 50 De Doná, Goodwin, Middleton, Raeburn (CR6) 2000; 21 CR8 Anděl, Hrach (CR1) 2000; 36 Shardlow, Stuart (CR15) 2000; 37 CR16 Borkovec, Klüppelberg (CR4) 2001; 11 Keller, Liverani (CR11) 1999; 28 Keller (CR10) 1982; 94 Anděl, Ranocha (CR3) 2005; 41 Chan, Tong (CR5) 1986; 73 Meyn, Tweedie (CR13) 1993 Anděl, Netuka, Ranocha (CR2) 2007; 41 Rudolf, Schweizer (CR14) 2018; 24 Hervé, Ledoux (CR9) 2014; 124 SP Meyn (9746_CR13) 1993 D Ferré (9746_CR7) 2013; 50 D Rudolf (9746_CR14) 2018; 24 J Anděl (9746_CR3) 2005; 41 KS Chan (9746_CR5) 1986; 73 W Loges (9746_CR12) 2004; 25 G Keller (9746_CR10) 1982; 94 T Shardlow (9746_CR15) 2000; 37 J Anděl (9746_CR2) 2007; 41 9746_CR16 JA De Doná (9746_CR6) 2000; 21 M Borkovec (9746_CR4) 2001; 11 cr-split#-9746_CR8.2 L Hervé (9746_CR9) 2014; 124 G Keller (9746_CR11) 1999; 28 J Anděl (9746_CR1) 2000; 36 cr-split#-9746_CR8.1 |
| References_xml | – volume: 25 start-page: 103 issue: 1 year: 2004 end-page: 125 ident: CR12 article-title: The stationary marginal distribution of a threshold AR(1) process publication-title: J. Time Ser. Anal. doi: 10.1111/j.1467-9892.2004.00339.x – volume: 36 start-page: 311 issue: 3 year: 2000 end-page: 319 ident: CR1 article-title: On calculation of stationary density of autoregressive processes publication-title: Kybernetika (Prague) – volume: 11 start-page: 1220 issue: 4 year: 2001 end-page: 1241 ident: CR4 article-title: The tail of the stationary distribution of an autoregressive process with ARCH(1) errors publication-title: Ann. Appl. Probab. doi: 10.1214/aoap/1015345401 – volume: 37 start-page: 1120 year: 2000 end-page: 1137 ident: CR15 article-title: A perturbation theory for ergodic Markov chains and application to numerical approximations publication-title: SIAM J. Numer. Anal. doi: 10.1137/S0036142998337235 – ident: CR16 – volume: 21 start-page: 1102 issue: 4 year: 2000 end-page: 1111 ident: CR6 article-title: Convergence of eigenvalues in state-discretization of linear stochastic systems publication-title: SIAM J. Matrix Anal. Appl. doi: 10.1137/S0895479897329874 – volume: 50 start-page: 184 year: 2013 end-page: 194 ident: CR7 article-title: Regular perturbation of V -geometrically ergodic Markov chains publication-title: J. Appl. Probab. doi: 10.1239/jap/1363784432 – volume: 28 start-page: 141 year: 1999 end-page: 152 ident: CR11 article-title: Stability of the spectrum for transfer operators publication-title: Annali della Scuola Normale Superiore di Pisa - Classe di Scienze Sér. 4 – volume: 24 start-page: 2610 issue: 4A year: 2018 end-page: 2639 ident: CR14 article-title: Perturbation theory for Markov chains via Wasserstein distance publication-title: Bernoulli doi: 10.3150/17-BEJ938 – volume: 73 start-page: 153 issue: 1 year: 1986 end-page: 158 ident: CR5 article-title: A note on certain integral equations associated with nonlinear time series analysis publication-title: Probab. Theory Relat. Fields doi: 10.1007/BF01845999 – volume: 124 start-page: 613 year: 2014 end-page: 638 ident: CR9 article-title: Approximating Markov chains and V -geometric ergodicity via weak perturbation theory publication-title: Stoch. Process. Appl. doi: 10.1016/j.spa.2013.09.003 – volume: 94 start-page: 313 issue: 4 year: 1982 end-page: 333 ident: CR10 article-title: Stochastic stability in some chaotic dynamical systems publication-title: Monatsh. Math. doi: 10.1007/BF01667385 – ident: CR8 – year: 1993 ident: CR13 publication-title: Markov chains and stochastic stability doi: 10.1007/978-1-4471-3267-7 – volume: 41 start-page: 735 issue: 6 year: 2005 end-page: 742 ident: CR3 article-title: Stationary distribution of absolute autoregression publication-title: Kybernetika (Prague) – volume: 41 start-page: 279 issue: 4 year: 2007 end-page: 287 ident: CR2 article-title: Methods for calculating stationary distribution in linear models of time series publication-title: Statistics doi: 10.1080/02331880701270663 – volume: 124 start-page: 613 year: 2014 ident: 9746_CR9 publication-title: Stoch. Process. Appl. doi: 10.1016/j.spa.2013.09.003 – volume: 37 start-page: 1120 year: 2000 ident: 9746_CR15 publication-title: SIAM J. Numer. Anal. doi: 10.1137/S0036142998337235 – volume: 41 start-page: 735 issue: 6 year: 2005 ident: 9746_CR3 publication-title: Kybernetika (Prague) – volume-title: Markov chains and stochastic stability year: 1993 ident: 9746_CR13 doi: 10.1007/978-1-4471-3267-7 – volume: 36 start-page: 311 issue: 3 year: 2000 ident: 9746_CR1 publication-title: Kybernetika (Prague) – volume: 41 start-page: 279 issue: 4 year: 2007 ident: 9746_CR2 publication-title: Statistics doi: 10.1080/02331880701270663 – volume: 24 start-page: 2610 issue: 4A year: 2018 ident: 9746_CR14 publication-title: Bernoulli doi: 10.3150/17-BEJ938 – volume: 21 start-page: 1102 issue: 4 year: 2000 ident: 9746_CR6 publication-title: SIAM J. Matrix Anal. Appl. doi: 10.1137/S0895479897329874 – volume: 25 start-page: 103 issue: 1 year: 2004 ident: 9746_CR12 publication-title: J. Time Ser. Anal. doi: 10.1111/j.1467-9892.2004.00339.x – volume: 11 start-page: 1220 issue: 4 year: 2001 ident: 9746_CR4 publication-title: Ann. Appl. Probab. doi: 10.1214/aoap/1015345401 – ident: #cr-split#-9746_CR8.1 doi: 10.1016/B978-044450083-0/50047-2 – ident: #cr-split#-9746_CR8.2 – volume: 94 start-page: 313 issue: 4 year: 1982 ident: 9746_CR10 publication-title: Monatsh. Math. doi: 10.1007/BF01667385 – volume: 50 start-page: 184 year: 2013 ident: 9746_CR7 publication-title: J. Appl. Probab. doi: 10.1239/jap/1363784432 – volume: 28 start-page: 141 year: 1999 ident: 9746_CR11 publication-title: Annali della Scuola Normale Superiore di Pisa - Classe di Scienze Sér. 4 – ident: 9746_CR16 – volume: 73 start-page: 153 issue: 1 year: 1986 ident: 9746_CR5 publication-title: Probab. Theory Relat. Fields doi: 10.1007/BF01845999 |
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| Snippet | Let
(
X
n
)
n
∈
ℕ
be a
V
-geometrically ergodic Markov chain on a measurable space
X
with invariant probability distribution
π
. In this paper, we propose a... Let (Xn)n∈ℕ be a V -geometrically ergodic Markov chain on a measurable space X with invariant probability distribution π. In this paper, we propose a... Let $(X_n)_{n \in\mathbb{N}}$ be a $V$-geometrically ergodic Markov chain on a measurable space $\mathbb{X}$ with invariant probability distribution $\pi$. In... |
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| SubjectTerms | Autoregressive processes Business and Management Convergence Discretization Economics Electrical Engineering Ergodic processes Invariants Life Sciences Markov analysis Markov chains Mathematics Mathematics and Statistics Probability Probability distribution Statistics |
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| Title | State-Discretization of V-Geometrically Ergodic Markov Chains and Convergence to the Stationary Distribution |
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